Open YEC with time asymptotes.sagews

%typeset_mode true

version()

SageMath version 7.5.1, Release Date: 2017-01-15

Man = Manifold(4, 'M',latex_name=r'\mathcal{M}')
print(Man)
Man

4-dimensional differentiable manifold M

$\displaystyle \mathcal{M}$

var('t,r,th,phi', domain='real')

($\displaystyle t$, $\displaystyle r$, $\displaystyle \mathit{th}$, $\displaystyle \phi$)

fr.<t,r,th,ph> = Man.chart(r't r:[0,+oo) th:[0,pi):\theta ph:[0,2*pi):\phi')
fr
fr.frame()
fr.frame()[0]
u = Man.vector_field('u')
 = fr.frame()[0]
u

$\displaystyle \left(\mathcal{M},(t, r, {\theta}, {\phi})\right)$

$\displaystyle \left(\mathcal{M}, \left(\frac{\partial}{\partial t },\frac{\partial}{\partial r },\frac{\partial}{\partial {\theta} },\frac{\partial}{\partial {\phi} }\right)\right)$

$\displaystyle \frac{\partial}{\partial t }$

$\displaystyle u$

#var('tau,rho,th,phi', domain='real')
#Hyper.<tau,rh,th,ph> = Man.chart(r'tau rh:[0,+oo):\rho th:[0,pi):\theta ph:[0,2*pi):\phi')
#Hyper

#%var t , x, theta ,A , B
var('G, Lambda,A,Adot,B, alpha , beta , eta ,rho ,a ,  b, k, chi, M, N, t_0', domain='real')
assume(b>0)
assume(M>0)
b = function('b')(t) #function('b')(t)
#A = Man.scalar_field(function('A')(t),name='A')
A = function('A')(t)
#eta = Man.scalar_field(function('eta')(t,r), name =('\eta'))
eta.category()
eta  = t - A*cosh(r)
chi = B*sinh(r)
F = Man.scalar_field(function('F')(r), name='F')
a = Man.scalar_field(function('a')(t,r), name='a')
Adot = Man.scalar_field(function('Adot')(t),name='Adot')
Adot.category() #dA = A.exterior_derivative()
#dA.display()
#Adot = dA[0]
Adot = A.derivative(t) #A.expr().derivative(t)
Adot.show()
Bdot = B.derivative(t)
Bdot
gfactor = Man.scalar_field({fr:1 - Adot*cosh(r)},name = r'g_f')
N = Man.scalar_field(function('N')(t,r),name='N')
N = a*(1 - Adot*cosh(r))*B*cosh(r)/sqrt(B^2*cosh(r)^2 - A^2*sinh(r)^2)
N.display()
#N  = a*gfactor*B*cosh(r)/sqrt(B^2*cosh(r)^2 - A^2*sinh(r)^2)
rho = Man.scalar_field(function('rho')(t,r), name='rho')
#rho = M*F/(2*pi^2*a^3)
pressure = Man.scalar_field(function('p')(t), name='p')
B #.display()
#eta.display()

($\displaystyle G$, $\displaystyle \Lambda$, $\displaystyle A$, $\displaystyle \mathit{Adot}$, $\displaystyle B$, $\displaystyle \alpha$, $\displaystyle \beta$, $\displaystyle \eta$, $\displaystyle \rho$, $\displaystyle a$, $\displaystyle b$, $\displaystyle k$, $\displaystyle \chi$, $\displaystyle M$, $\displaystyle N$, $\displaystyle t_{0}$)

$\displaystyle \mathbf{Elt}_{\text{SR}}$

$\displaystyle \mathbf{Elt}_{C^{\infty}\left(\mathcal{M}\right)}$

$\displaystyle \frac{\partial}{\partial t}A\left(t\right)$

$\displaystyle 0$

$\displaystyle \begin{array}{llcl} & \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & -\frac{{\left(\cosh\left(r\right) \frac{\partial\,A}{\partial t} - 1\right)} B a\left(t, r\right) \cosh\left(r\right)}{\sqrt{B^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} \sinh\left(r\right)^{2}}} \end{array}$

$\displaystyle B$

#FLRW_to_Hyper = fr.transition_map(Hyper, (  t , B*sinh(r) , th , ph) )
#FLRW_to_Hyper.display()

g = Man.lorentzian_metric('g')
g[0,0] = -a*a*(1 - A.derivative(t)*cosh(r))*(1 - Adot*cosh(r))
g[0,1] = a*a*A*(1 - A.derivative(t)*cosh(r))*sinh(r)
g[1,0] = a*a*A*(1 - A.derivative(t)*cosh(r))*sinh(r)
g[1,1] = a*a*(B^2*cosh(r)^2 - A^2*sinh(r)^2)
g[2,2] = a*a*(chi)^2
g[3,3] = a*a*((chi)*sin(th))^2
#g[2,2] = a*a*chi^2
#g[3,3] = a*a*(chi*sin(th))^2
g.display()
ginv = g.inverse()
ginv.display()

$\displaystyle g = -{\left(\cosh\left(r\right) \frac{\partial\,A}{\partial t} - 1\right)}^{2} a\left(t, r\right)^{2} \mathrm{d} t\otimes \mathrm{d} t -{\left(\cosh\left(r\right) \frac{\partial\,A}{\partial t} - 1\right)} A\left(t\right) a\left(t, r\right)^{2} \sinh\left(r\right) \mathrm{d} t\otimes \mathrm{d} r -{\left(\cosh\left(r\right) \frac{\partial\,A}{\partial t} - 1\right)} A\left(t\right) a\left(t, r\right)^{2} \sinh\left(r\right) \mathrm{d} r\otimes \mathrm{d} t + {\left(B^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} \sinh\left(r\right)^{2}\right)} a\left(t, r\right)^{2} \mathrm{d} r\otimes \mathrm{d} r + B^{2} a\left(t, r\right)^{2} \sinh\left(r\right)^{2} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + B^{2} a\left(t, r\right)^{2} \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

$\displaystyle g^{-1} = \left( -\frac{B^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} \sinh\left(r\right)^{2}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \left(\frac{\partial\,A}{\partial t}\right)^{2} - 2 \, a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} + a\left(t, r\right)^{2} \cosh\left(r\right)^{2}\right)} B^{2}} \right) \frac{\partial}{\partial t }\otimes \frac{\partial}{\partial t } -\frac{A\left(t\right) \sinh\left(r\right)}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{2}\right)} B^{2}} \frac{\partial}{\partial t }\otimes \frac{\partial}{\partial r } -\frac{A\left(t\right) \sinh\left(r\right)}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{2}\right)} B^{2}} \frac{\partial}{\partial r }\otimes \frac{\partial}{\partial t } + \frac{1}{B^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{2}} \frac{\partial}{\partial r }\otimes \frac{\partial}{\partial r } + \frac{1}{B^{2} a\left(t, r\right)^{2} \sinh\left(r\right)^{2}} \frac{\partial}{\partial {\theta} }\otimes \frac{\partial}{\partial {\theta} } + \frac{1}{B^{2} a\left(t, r\right)^{2} \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{2}} \frac{\partial}{\partial {\phi} }\otimes \frac{\partial}{\partial {\phi} }$

#g[:].substitute(-b*cosh(r)+t==eta).substitute(b*sinh(r)==chi).simplify_full()

nablaM = g.connection()
g.christoffel_symbols_display()

$\displaystyle \begin{array}{lcl} \Gamma_{ \phantom{\, t} \, t \, t }^{ \, t \phantom{\, t} \phantom{\, t} } & = & \frac{{\left(a\left(t, r\right) \cosh\left(r\right)^{3} \frac{\partial^2\,A}{\partial t ^ 2} + {\left(\cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial t}\right)} B^{2} + {\left(A\left(t\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial t} - {\left(A\left(t\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial\,A}{\partial t}^{2} - 2 \, A\left(t\right) \cosh\left(r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right) \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right) \cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - a\left(t, r\right) \cosh\left(r\right)^{2}\right)} B^{2}} \\ \Gamma_{ \phantom{\, t} \, t \, r }^{ \, t \phantom{\, t} \phantom{\, r} } & = & \frac{A\left(t\right)^{3} \sinh\left(r\right)^{3} \frac{\partial\,a}{\partial t} + {\left(a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - {\left(\sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} A\left(t\right) \frac{\partial\,a}{\partial t} + {\left(\cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - {\left(A\left(t\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right) \cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - a\left(t, r\right) \cosh\left(r\right)^{2}\right)} B^{2}} \\ \Gamma_{ \phantom{\, t} \, r \, r }^{ \, t \phantom{\, r} \phantom{\, r} } & = & \frac{B^{4} \cosh\left(r\right)^{4} \frac{\partial\,a}{\partial t} + {\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t\right)^{4} \frac{\partial\,a}{\partial t} + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial\,a}{\partial t} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) + {\left(A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - A\left(t\right) \cosh\left(r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - {\left({\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right) \cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t}^{2} - 2 \, a\left(t, r\right) \cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} + a\left(t, r\right) \cosh\left(r\right)^{2}\right)} B^{2}} \\ \Gamma_{ \phantom{\, t} \, {\theta} \, {\theta} }^{ \, t \phantom{\, {\theta}} \phantom{\, {\theta}} } & = & \frac{A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} + B^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial t} - A\left(t\right)^{2} \sinh\left(r\right)^{4} \frac{\partial\,a}{\partial t} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} + {\left(A\left(t\right) \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - A\left(t\right) \sinh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}}{a\left(t, r\right) \cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t}^{2} - 2 \, a\left(t, r\right) \cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} + a\left(t, r\right) \cosh\left(r\right)^{2}} \\ \Gamma_{ \phantom{\, t} \, {\phi} \, {\phi} }^{ \, t \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & \frac{B^{2} \cosh\left(r\right)^{2} \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial t} + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \sinh\left(r\right)^{4} \frac{\partial\,a}{\partial t} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} + {\left(A\left(t\right) \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - A\left(t\right) \sinh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} \sin\left({\theta}\right)^{2}}{a\left(t, r\right) \cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t}^{2} - 2 \, a\left(t, r\right) \cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} + a\left(t, r\right) \cosh\left(r\right)^{2}} \\ \Gamma_{ \phantom{\, r} \, t \, t }^{ \, r \phantom{\, t} \phantom{\, t} } & = & -\frac{{\left(A\left(t\right) \cosh\left(r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} - A\left(t\right) \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial t} - {\left(\cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t}^{2} - 2 \, \cosh\left(r\right) \frac{\partial\,A}{\partial t} + 1\right)} \frac{\partial\,a}{\partial r}}{B^{2} a\left(t, r\right) \cosh\left(r\right)^{2}} \\ \Gamma_{ \phantom{\, r} \, t \, r }^{ \, r \phantom{\, t} \phantom{\, r} } & = & -\frac{A\left(t\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial t} - {\left(\sinh\left(r\right)^{2} + 1\right)} B^{2} \frac{\partial\,a}{\partial t} - {\left(A\left(t\right) \cosh\left(r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} - A\left(t\right) \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}}{B^{2} a\left(t, r\right) \cosh\left(r\right)^{2}} \\ \Gamma_{ \phantom{\, r} \, r \, r }^{ \, r \phantom{\, r} \phantom{\, r} } & = & -\frac{{\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial\,a}{\partial t} - {\left(a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial\,a}{\partial t} - a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right) + {\left(\cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - {\left(A\left(t\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right) \cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - a\left(t, r\right) \cosh\left(r\right)^{2}\right)} B^{2}} \\ \Gamma_{ \phantom{\, r} \, {\theta} \, {\theta} }^{ \, r \phantom{\, {\theta}} \phantom{\, {\theta}} } & = & -\frac{a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - A\left(t\right) \sinh\left(r\right)^{3} \frac{\partial\,a}{\partial t} - a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right) + {\left(\cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - \sinh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial r}}{a\left(t, r\right) \cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - a\left(t, r\right) \cosh\left(r\right)^{2}} \\ \Gamma_{ \phantom{\, r} \, {\phi} \, {\phi} }^{ \, r \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & -\frac{{\left(a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - A\left(t\right) \sinh\left(r\right)^{3} \frac{\partial\,a}{\partial t} - a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right) + {\left(\cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - \sinh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} \sin\left({\theta}\right)^{2}}{a\left(t, r\right) \cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - a\left(t, r\right) \cosh\left(r\right)^{2}} \\ \Gamma_{ \phantom{\, {\theta}} \, t \, {\theta} }^{ \, {\theta} \phantom{\, t} \phantom{\, {\theta}} } & = & \frac{\frac{\partial\,a}{\partial t}}{a\left(t, r\right)} \\ \Gamma_{ \phantom{\, {\theta}} \, r \, {\theta} }^{ \, {\theta} \phantom{\, r} \phantom{\, {\theta}} } & = & \frac{a\left(t, r\right) \cosh\left(r\right) + \sinh\left(r\right) \frac{\partial\,a}{\partial r}}{a\left(t, r\right) \sinh\left(r\right)} \\ \Gamma_{ \phantom{\, {\theta}} \, {\phi} \, {\phi} }^{ \, {\theta} \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & -\cos\left({\theta}\right) \sin\left({\theta}\right) \\ \Gamma_{ \phantom{\, {\phi}} \, t \, {\phi} }^{ \, {\phi} \phantom{\, t} \phantom{\, {\phi}} } & = & \frac{\frac{\partial\,a}{\partial t}}{a\left(t, r\right)} \\ \Gamma_{ \phantom{\, {\phi}} \, r \, {\phi} }^{ \, {\phi} \phantom{\, r} \phantom{\, {\phi}} } & = & \frac{a\left(t, r\right) \cosh\left(r\right) + \sinh\left(r\right) \frac{\partial\,a}{\partial r}}{a\left(t, r\right) \sinh\left(r\right)} \\ \Gamma_{ \phantom{\, {\phi}} \, {\theta} \, {\phi} }^{ \, {\phi} \phantom{\, {\theta}} \phantom{\, {\phi}} } & = & \frac{\cos\left({\theta}\right)}{\sin\left({\theta}\right)} \end{array}$

Sig = Manifold(3, 'Sigma', r'\Sigma', start_index=1)
print(Sig)

3-dimensional differentiable manifold Sigma

Sigfr.<r,th,ph> = Sig.chart(r'r:[0,+oo) th:[0,pi) ph:[0,2*pi):\phi')
print(Sigfr) ; Sigfr

Chart (Sigma, (r, th, ph))

$\displaystyle \left(\Sigma,(r, {\theta}, {\phi})\right)$

Phi = Sig.diff_map(Man, [t_0, r, th, ph] ,
                  name='Phi', latex_name=r'\Phi')
print(Phi) ; Phi.display()

Differentiable map Phi from the 3-dimensional differentiable manifold Sigma to the 4-dimensional differentiable manifold M

$\displaystyle \begin{array}{llcl} \Phi:& \Sigma & \longrightarrow & \mathcal{M} \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & \left(t, r, {\theta}, {\phi}\right) = \left(t_{0}, r, {\theta}, {\phi}\right) \end{array}$

Phi.parent()

Phi.pullback(a).display()

gam3 = Sig.riemannian_metric('gam3', latex_name=r'\gamma')
gam3.set( Phi.pullback(g) )

var('a_0, Adot_0, A_0')
var('eta_0')
Adot_0   = Sig.scalar_field( name='Adot_0')
A_0   = Sig.scalar_field( name='A_0')
A_0 = A(t=t_0) #Phi.pullback(A)
a_0   = Sig.scalar_field( name='a_0')
a_0 = Phi.pullback(a) #a(t_0,r)
a_0.display()
eta_0 = Sig.scalar_field( name=r'eta_0')
eta_0 = eta(t=t_0) #Phi.pullback(eta) #eta(t_0,r)
Adot.category()
eta_0.show()
Adot.show()
Adot_0 = Adot(t=t_0) # Phi.pullback(Adot)
Adot_0.show()
Nsig = Sig.scalar_field(name='N_0')
Nsig = Phi.pullback(N)
Nsig.display()
#aSig = function('a') (x)
 #a(eta_0)
#a_0.expr().display()
#aSig
#rho = Sig.scalar_field(function('rho')(t), name='rho')

($\displaystyle a_{0}$, $\displaystyle \mathit{Adot}_{0}$, $\displaystyle A_{0}$)

$\displaystyle \eta_{0}$

$\displaystyle \begin{array}{llcl} \Phi_*a:& \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & a\left(t_{0}, r\right) \end{array}$

$\displaystyle \mathbf{Elt}_{\text{SR}}$

$\displaystyle -A\left(t_{0}\right) \cosh\left(r\right) + t_{0}$

$\displaystyle \frac{\partial}{\partial t}A\left(t\right)$

$\displaystyle \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)$

$\displaystyle \begin{array}{llcl} & \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & -\frac{{\left(a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - a\left(t_{0}, r\right) \cosh\left(r\right)\right)} B}{\sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)}} \end{array}$

gam3.display()

$\displaystyle \gamma = \left( B^{2} a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} \right) \mathrm{d} r\otimes \mathrm{d} r + B^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + B^{2} a\left(t_{0}, r\right)^{2} \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

O = Sig((0,0,0), chart=Sigfr, name='O')
dPhi_O = Phi.differential(O)

O_M = Phi(O); print(O_M) ; O_M ; O_M.coord()

gam3_inv = gam3.inverse()
gam3_inv[:]

$\displaystyle \left(\begin{array}{rrr} \frac{1}{B^{2} a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2}} & 0 & 0 \\ 0 & \frac{1}{B^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2}} & 0 \\ 0 & 0 & \frac{1}{B^{2} a\left(t_{0}, r\right)^{2} \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{2}} \end{array}\right)$

D = gam3.connection()
gam3.christoffel_symbols_display()

$\displaystyle \begin{array}{lcl} \Gamma_{ \phantom{\, r} \, r \, r }^{ \, r \phantom{\, r} \phantom{\, r} } & = & -\frac{A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right) + A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial r} - {\left(a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right) + \cosh\left(r\right)^{2} \frac{\partial\,a}{\partial r}\right)} B^{2}}{B^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)^{2}} \\ \Gamma_{ \phantom{\, r} \, {\theta} \, {\theta} }^{ \, r \phantom{\, {\theta}} \phantom{\, {\theta}} } & = & -\frac{{\left(a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right) + \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial r}\right)} B^{2}}{B^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)^{2}} \\ \Gamma_{ \phantom{\, r} \, {\phi} \, {\phi} }^{ \, r \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & -\frac{{\left(a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right) + \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial r}\right)} B^{2} \sin\left({\theta}\right)^{2}}{B^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)^{2}} \\ \Gamma_{ \phantom{\, {\theta}} \, r \, {\theta} }^{ \, {\theta} \phantom{\, r} \phantom{\, {\theta}} } & = & \frac{a\left(t_{0}, r\right) \cosh\left(r\right) + \sinh\left(r\right) \frac{\partial\,a}{\partial r}}{a\left(t_{0}, r\right) \sinh\left(r\right)} \\ \Gamma_{ \phantom{\, {\theta}} \, {\phi} \, {\phi} }^{ \, {\theta} \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & -\cos\left({\theta}\right) \sin\left({\theta}\right) \\ \Gamma_{ \phantom{\, {\phi}} \, r \, {\phi} }^{ \, {\phi} \phantom{\, r} \phantom{\, {\phi}} } & = & \frac{a\left(t_{0}, r\right) \cosh\left(r\right) + \sinh\left(r\right) \frac{\partial\,a}{\partial r}}{a\left(t_{0}, r\right) \sinh\left(r\right)} \\ \Gamma_{ \phantom{\, {\phi}} \, {\theta} \, {\phi} }^{ \, {\phi} \phantom{\, {\theta}} \phantom{\, {\phi}} } & = & \frac{\cos\left({\theta}\right)}{\sin\left({\theta}\right)} \end{array}$

bta = Sig.vector_field('beta', latex_name=r'\beta')
bta[1] =  a_0*(1 - Adot_0*cosh(r))*sinh(r)/(B^2*cosh(r)^2 - A_0^2*sinh(r)^2) # b(t_0)  #/(a*b)
bta.display()
# unset components are zero
bta1f = Sig.one_form('Beta', latex_name=r'\Beta')
bta1f = bta.down(gam3) # the 1-form associated to bta by metric duality

$\displaystyle \beta = \left( -\frac{{\left(\cosh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - 1\right)} a\left(t_{0}, r\right) \sinh\left(r\right)}{B^{2} \cosh\left(r\right)^{2} - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2}} \right) \frac{\partial}{\partial r }$

Ricci3 = D.ricci()
Ricci3.display()
Ricci3[:].siplify_full()
#Ricci3_scalar = Ricci3.trace()

$\displaystyle \mathrm{Ric}\left(\gamma\right) = \left( -\frac{2 \, {\left(A\left(t_{0}\right)^{2} \sinh\left(r\right)^{3} \left(\frac{\partial\,a}{\partial r}\right)^{2} - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)^{3} \frac{\partial^2\,a}{\partial r ^ 2} + A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) + {\left(a\left(t_{0}, r\right) \cosh\left(r\right) \frac{\partial\,a}{\partial r} - {\left(\sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} \left(\frac{\partial\,a}{\partial r}\right)^{2} + {\left(\sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} a\left(t_{0}, r\right) \frac{\partial^2\,a}{\partial r ^ 2}\right)} B^{2}\right)}}{B^{2} a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right) - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{3}} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( \frac{A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right)^{4} - {\left({\left(2 \, \cosh\left(r\right) \sinh\left(r\right)^{3} + 3 \, \cosh\left(r\right) \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r} + {\left(\sinh\left(r\right)^{4} + \sinh\left(r\right)^{2}\right)} \frac{\partial^2\,a}{\partial r ^ 2}\right)} B^{4} + {\left(2 \, A\left(t_{0}\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial\,a}{\partial r} + A\left(t_{0}\right)^{2} \sinh\left(r\right)^{4} \frac{\partial^2\,a}{\partial r ^ 2} - {\left(\sinh\left(r\right)^{4} + 2 \, \sinh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)\right)} B^{2}}{B^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{4} - 2 \, B^{2} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} + A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right)^{4}} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \left( \frac{A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{4} - {\left({\left(2 \, \cosh\left(r\right) \sinh\left(r\right)^{3} + 3 \, \cosh\left(r\right) \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r} + {\left(\sinh\left(r\right)^{4} + \sinh\left(r\right)^{2}\right)} \frac{\partial^2\,a}{\partial r ^ 2}\right)} B^{4} \sin\left({\theta}\right)^{2} + {\left(2 \, A\left(t_{0}\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial\,a}{\partial r} + A\left(t_{0}\right)^{2} \sinh\left(r\right)^{4} \frac{\partial^2\,a}{\partial r ^ 2} - {\left(\sinh\left(r\right)^{4} + 2 \, \sinh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)\right)} B^{2} \sin\left({\theta}\right)^{2}}{B^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{4} - 2 \, B^{2} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} + A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right)^{4}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

$\displaystyle \left(\begin{array}{rrr} -\frac{2 \, {\left(A\left(t_{0}\right)^{2} \sinh\left(r\right)^{3} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)^{3} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right) + A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) + {\left(a\left(t_{0}, r\right) \cosh\left(r\right) \frac{\partial}{\partial r}a\left(t_{0}, r\right) - {\left(\sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} + {\left(\sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} a\left(t_{0}, r\right) \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{2}\right)}}{B^{2} a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right) - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{3}} & 0 & 0 \\ 0 & \frac{A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right)^{4} - {\left({\left(2 \, \cosh\left(r\right) \sinh\left(r\right)^{3} + 3 \, \cosh\left(r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) + {\left(\sinh\left(r\right)^{4} + \sinh\left(r\right)^{2}\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{4} + {\left(2 \, A\left(t_{0}\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial}{\partial r}a\left(t_{0}, r\right) + A\left(t_{0}\right)^{2} \sinh\left(r\right)^{4} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right) - {\left(\sinh\left(r\right)^{4} + 2 \, \sinh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)\right)} B^{2}}{B^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{4} - 2 \, B^{2} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} + A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right)^{4}} & 0 \\ 0 & 0 & \frac{A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{4} - {\left({\left(2 \, \cosh\left(r\right) \sinh\left(r\right)^{3} + 3 \, \cosh\left(r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) + {\left(\sinh\left(r\right)^{4} + \sinh\left(r\right)^{2}\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{4} \sin\left({\theta}\right)^{2} + {\left(2 \, A\left(t_{0}\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial}{\partial r}a\left(t_{0}, r\right) + A\left(t_{0}\right)^{2} \sinh\left(r\right)^{4} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right) - {\left(\sinh\left(r\right)^{4} + 2 \, \sinh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)\right)} B^{2} \sin\left({\theta}\right)^{2}}{B^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{4} - 2 \, B^{2} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} + A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right)^{4}} \end{array}\right)$

Ricci3_scalar = gam3.ricci_scalar()
temp = gam3_inv['^ij']*Ricci3['_ij']
temp.expr().simplify_full().show()
Ricci3_scalar.display()

$\displaystyle \frac{2 \, {\left(A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{3} - {\left(2 \, {\left(\cosh\left(r\right) \sinh\left(r\right)^{2} + 2 \, \cosh\left(r\right)\right)} a\left(t_{0}, r\right) \frac{\partial}{\partial r}a\left(t_{0}, r\right) - {\left(\sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} + 2 \, {\left(\sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} a\left(t_{0}, r\right) \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{4} + {\left(2 \, A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial}{\partial r}a\left(t_{0}, r\right) - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{3} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} + 2 \, A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)^{3} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right) - {\left(\sinh\left(r\right)^{3} + 3 \, \sinh\left(r\right)\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2}\right)} B^{2}\right)}}{B^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{4} \sinh\left(r\right) - 2 \, B^{4} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{2} \sinh\left(r\right)^{3} + B^{2} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \sinh\left(r\right)^{5}}$

$\displaystyle \begin{array}{llcl} \mathrm{r}\left(\gamma\right):& \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & \frac{2 \, {\left(A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{3} - {\left(2 \, {\left(\cosh\left(r\right) \sinh\left(r\right)^{2} + 2 \, \cosh\left(r\right)\right)} a\left(t_{0}, r\right) \frac{\partial\,a}{\partial r} - {\left(\sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}^{2} + 2 \, {\left(\sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} a\left(t_{0}, r\right) \frac{\partial^2\,a}{\partial r ^ 2}\right)} B^{4} + {\left(2 \, A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial r} - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{3} \frac{\partial\,a}{\partial r}^{2} + 2 \, A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)^{3} \frac{\partial^2\,a}{\partial r ^ 2} - {\left(\sinh\left(r\right)^{3} + 3 \, \sinh\left(r\right)\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2}\right)} B^{2}\right)}}{B^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{4} \sinh\left(r\right) - 2 \, B^{4} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{2} \sinh\left(r\right)^{3} + B^{2} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \sinh\left(r\right)^{5}} \end{array}$

accel = nablaM(log(N))
accel.display()
var('n1f')
n1f = Man.one_form(name='n')
N.display() #category()
#%var Nfact
var('Nfact')
Nfact = Man.scalar_field( chart = fr,name='N_f')
Nfact = -a*(1 - Adot*cosh(r))*B*cosh(r) /sqrt(B*B*cosh(r)*cosh(r) - A*A*sinh(r)*sinh(r))
nablaM(Nfact).display()
n1f[0]= Nfact #N
n1f.display()
nablaM(n1f).display()
n1f[:]

$\displaystyle \left( -\frac{A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t}^{2} - A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial^2\,A}{\partial t ^ 2} - A\left(t\right) a\left(t, r\right) \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} + {\left(a\left(t, r\right) \cosh\left(r\right)^{3} \frac{\partial^2\,A}{\partial t ^ 2} + {\left(\cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial t}\right)} B^{2} - {\left(A\left(t\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial t}}{A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right)^{2} - {\left(a\left(t, r\right) \cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - a\left(t, r\right) \cosh\left(r\right)^{2}\right)} B^{2}} \right) \mathrm{d} t + \left( \frac{{\left(\cosh\left(r\right)^{3} - 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) - {\left(a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2}\right)} \frac{\partial\,a}{\partial r}}{A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} - {\left(a\left(t, r\right) \cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - a\left(t, r\right) \cosh\left(r\right)^{3}\right)} B^{2}} \right) \mathrm{d} r$

$\displaystyle \mathit{n1f}$

$\displaystyle \begin{array}{llcl} & \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & -\frac{{\left(\cosh\left(r\right) \frac{\partial\,A}{\partial t} - 1\right)} B a\left(t, r\right) \cosh\left(r\right)}{\sqrt{B^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} \sinh\left(r\right)^{2}}} \end{array}$

$\displaystyle \mathit{Nfact}$

$\displaystyle \left( \frac{{\left(a\left(t, r\right) \cosh\left(r\right)^{4} \frac{\partial^2\,A}{\partial t ^ 2} + {\left(\cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial t}\right)} B^{3} + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t}^{2} - A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial^2\,A}{\partial t ^ 2} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - {\left(A\left(t\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial t}\right)} B}{{\left(B^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} \sinh\left(r\right)^{2}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}} \right) \mathrm{d} t + \left( \frac{{\left({\left(a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B^{3} - {\left({\left(\cosh\left(r\right)^{3} - 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} B\right)} \sqrt{B^{2} \cosh\left(r\right)^{2} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{2}}}{B^{4} \cosh\left(r\right)^{4} - 2 \, B^{2} A\left(t\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} + A\left(t\right)^{4} \sinh\left(r\right)^{4}} \right) \mathrm{d} r$

$\displaystyle n = \left( \frac{{\left(\cosh\left(r\right) \frac{\partial\,A}{\partial t} - 1\right)} B a\left(t, r\right) \cosh\left(r\right)}{\sqrt{B^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} \sinh\left(r\right)^{2}}} \right) \mathrm{d} t$

$\displaystyle \nabla_{g} n = \left( \frac{{\left({\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} \left(\frac{\partial\,A}{\partial t}\right)^{2} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - {\left(A\left(t\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial t} + {\left(A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t}\right)^{2} - 2 \, A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right) \cosh\left(r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} + {\left(A\left(t\right)^{4} \cosh\left(r\right) \sinh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - A\left(t\right)^{4} \sinh\left(r\right)^{4}\right)} \frac{\partial\,a}{\partial t} - {\left(A\left(t\right)^{3} \cosh\left(r\right)^{2} \sinh\left(r\right)^{3} \left(\frac{\partial\,A}{\partial t}\right)^{2} - 2 \, A\left(t\right)^{3} \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t} + A\left(t\right)^{3} \sinh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}}{B^{5} \cosh\left(r\right)^{5} - 2 \, B^{3} A\left(t\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} + B A\left(t\right)^{4} \cosh\left(r\right) \sinh\left(r\right)^{4}} \right) \mathrm{d} t\otimes \mathrm{d} t + \left( -\frac{A\left(t\right)^{7} \sinh\left(r\right)^{7} \frac{\partial\,a}{\partial t} + {\left(a\left(t, r\right) \cosh\left(r\right)^{6} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{4} \sinh\left(r\right)^{3} + \cosh\left(r\right)^{4} \sinh\left(r\right)\right)} A\left(t\right) \frac{\partial\,a}{\partial t} + {\left(\cosh\left(r\right)^{7} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{6}\right)} \frac{\partial\,a}{\partial r}\right)} B^{6} - {\left(2 \, A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - {\left(2 \, \cosh\left(r\right)^{2} \sinh\left(r\right)^{5} + {\left(\cosh\left(r\right)^{4} + 2 \, \cosh\left(r\right)^{2}\right)} \sinh\left(r\right)^{3}\right)} A\left(t\right)^{3} \frac{\partial\,a}{\partial t} + 3 \, {\left(A\left(t\right)^{2} \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} B^{4} + {\left(A\left(t\right)^{4} a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{5} \frac{\partial\,A}{\partial t} - {\left(\sinh\left(r\right)^{7} + {\left(2 \, \cosh\left(r\right)^{2} + 1\right)} \sinh\left(r\right)^{5}\right)} A\left(t\right)^{5} \frac{\partial\,a}{\partial t} + 3 \, {\left(A\left(t\right)^{4} \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - A\left(t\right)^{4} \cosh\left(r\right)^{2} \sinh\left(r\right)^{4}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - {\left({\left(a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{5} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{4}\right)} \frac{\partial\,a}{\partial r}\right)} B^{4} - {\left({\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right) + {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right)^{2} \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2}\right)} \sqrt{B^{2} \cosh\left(r\right)^{2} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{2}} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} - {\left(A\left(t\right)^{6} \cosh\left(r\right) \sinh\left(r\right)^{6} \frac{\partial\,A}{\partial t} - A\left(t\right)^{6} \sinh\left(r\right)^{6}\right)} \frac{\partial\,a}{\partial r}}{{\left(B^{5} \cosh\left(r\right)^{5} - 2 \, B^{3} A\left(t\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} + B A\left(t\right)^{4} \cosh\left(r\right) \sinh\left(r\right)^{4}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}} \right) \mathrm{d} t\otimes \mathrm{d} r + \left( -\frac{A\left(t\right)^{3} \sinh\left(r\right)^{3} \frac{\partial\,a}{\partial t} + {\left(a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - {\left(\sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} A\left(t\right) \frac{\partial\,a}{\partial t} + {\left(\cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - {\left(A\left(t\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial r}}{\sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} B \cosh\left(r\right)} \right) \mathrm{d} r\otimes \mathrm{d} t + \left( -\frac{B^{4} \cosh\left(r\right)^{4} \frac{\partial\,a}{\partial t} + {\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t\right)^{4} \frac{\partial\,a}{\partial t} + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial\,a}{\partial t} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) + {\left(A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - A\left(t\right) \cosh\left(r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - {\left({\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}}{{\left(\cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} B} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( -\frac{B^{3} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial t} + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \sinh\left(r\right)^{4} \frac{\partial\,a}{\partial t} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} + {\left(A\left(t\right) \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - A\left(t\right) \sinh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B}{{\left(\cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \left( -\frac{B^{3} \cosh\left(r\right)^{2} \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial t} + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \sinh\left(r\right)^{4} \frac{\partial\,a}{\partial t} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} + {\left(A\left(t\right) \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - A\left(t\right) \sinh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B \sin\left({\theta}\right)^{2}}{{\left(\cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

[$\displaystyle \frac{{\left(\cosh\left(r\right) \frac{\partial\,A}{\partial t} - 1\right)} B a\left(t, r\right) \cosh\left(r\right)}{\sqrt{B^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} \sinh\left(r\right)^{2}}}$, $\displaystyle 0$, $\displaystyle 0$, $\displaystyle 0$]

nv = Man.vector_field('nv')
nv = n1f.up(g, 0)
print(nv) ; nv.display()
tmp = g(nv,nv)
tmp.display()

Vector field on the 4-dimensional differentiable manifold M

$\displaystyle \left( -\frac{\sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}}{{\left(a\left(t, r\right) \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - a\left(t, r\right) \cosh\left(r\right)\right)} B} \right) \frac{\partial}{\partial t } + \left( -\frac{A\left(t\right) \sinh\left(r\right)}{\sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} B a\left(t, r\right) \cosh\left(r\right)} \right) \frac{\partial}{\partial r }$

$\displaystyle \begin{array}{llcl} & \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & -1 \end{array}$

P =g.up(g,1)+n1f*nv
P.display()
P[:]

$\displaystyle \left( -\frac{A\left(t\right) \cosh\left(r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} - A\left(t\right) \sinh\left(r\right)}{B^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} \sinh\left(r\right)^{2}} \right) \frac{\partial}{\partial r }\otimes \mathrm{d} t +\frac{\partial}{\partial r }\otimes \mathrm{d} r +\frac{\partial}{\partial {\theta} }\otimes \mathrm{d} {\theta} +\frac{\partial}{\partial {\phi} }\otimes \mathrm{d} {\phi}$

$\displaystyle \left(\begin{array}{rrrr} 0 & 0 & 0 & 0 \\ -\frac{A\left(t\right) \cosh\left(r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right) \sinh\left(r\right)}{B^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} \sinh\left(r\right)^{2}} & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

gam0 = P['^i_a']*g['_ij']
gam0[:]
gam = gam0['_aj']*P['^j_b']
gam.display()
gam[:]

$\displaystyle \left(\begin{array}{rrrr} \frac{A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) + A\left(t\right)^{2} a\left(t, r\right)^{2} \sinh\left(r\right)^{2}}{B^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} \sinh\left(r\right)^{2}} & -A\left(t\right) a\left(t, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + A\left(t\right) a\left(t, r\right)^{2} \sinh\left(r\right) & 0 & 0 \\ -A\left(t\right) a\left(t, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + A\left(t\right) a\left(t, r\right)^{2} \sinh\left(r\right) & B^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} a\left(t, r\right)^{2} \sinh\left(r\right)^{2} & 0 & 0 \\ 0 & 0 & B^{2} a\left(t, r\right)^{2} \sinh\left(r\right)^{2} & 0 \\ 0 & 0 & 0 & B^{2} a\left(t, r\right)^{2} \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{2} \end{array}\right)$

$\displaystyle \left( \frac{A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \left(\frac{\partial\,A}{\partial t}\right)^{2} - 2 \, A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} + A\left(t\right)^{2} a\left(t, r\right)^{2} \sinh\left(r\right)^{2}}{B^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} \sinh\left(r\right)^{2}} \right) \mathrm{d} t\otimes \mathrm{d} t + \left( -A\left(t\right) a\left(t, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right) a\left(t, r\right)^{2} \sinh\left(r\right) \right) \mathrm{d} t\otimes \mathrm{d} r + \left( -A\left(t\right) a\left(t, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right) a\left(t, r\right)^{2} \sinh\left(r\right) \right) \mathrm{d} r\otimes \mathrm{d} t + \left( B^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} a\left(t, r\right)^{2} \sinh\left(r\right)^{2} \right) \mathrm{d} r\otimes \mathrm{d} r + B^{2} a\left(t, r\right)^{2} \sinh\left(r\right)^{2} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + B^{2} a\left(t, r\right)^{2} \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

$\displaystyle \left(\begin{array}{rrrr} \frac{A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) + A\left(t\right)^{2} a\left(t, r\right)^{2} \sinh\left(r\right)^{2}}{B^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} \sinh\left(r\right)^{2}} & -A\left(t\right) a\left(t, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + A\left(t\right) a\left(t, r\right)^{2} \sinh\left(r\right) & 0 & 0 \\ -A\left(t\right) a\left(t, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + A\left(t\right) a\left(t, r\right)^{2} \sinh\left(r\right) & B^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} a\left(t, r\right)^{2} \sinh\left(r\right)^{2} & 0 & 0 \\ 0 & 0 & B^{2} a\left(t, r\right)^{2} \sinh\left(r\right)^{2} & 0 \\ 0 & 0 & 0 & B^{2} a\left(t, r\right)^{2} \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{2} \end{array}\right)$

Dg = nablaM(g)
Dg == 0
DN = nablaM(N)
DN.display()
N.disp()
DN[:]

$\displaystyle \mathrm{True}$

$\displaystyle \left( -\frac{{\left(a\left(t, r\right) \cosh\left(r\right)^{4} \frac{\partial^2\,A}{\partial t ^ 2} + {\left(\cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial t}\right)} B^{3} + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t}^{2} - A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial^2\,A}{\partial t ^ 2} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - {\left(A\left(t\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial t}\right)} B}{{\left(B^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} \sinh\left(r\right)^{2}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}} \right) \mathrm{d} t + \left( -\frac{{\left({\left(a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B^{3} - {\left({\left(\cosh\left(r\right)^{3} - 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} B\right)} \sqrt{B^{2} \cosh\left(r\right)^{2} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{2}}}{B^{4} \cosh\left(r\right)^{4} - 2 \, B^{2} A\left(t\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} + A\left(t\right)^{4} \sinh\left(r\right)^{4}} \right) \mathrm{d} r$

$\displaystyle \begin{array}{llcl} & \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & -\frac{{\left(\cosh\left(r\right) \frac{\partial\,A}{\partial t} - 1\right)} B a\left(t, r\right) \cosh\left(r\right)}{\sqrt{B^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} \sinh\left(r\right)^{2}}} \end{array}$

[$\displaystyle -\frac{{\left(a\left(t, r\right) \cosh\left(r\right)^{4} \frac{\partial^2\,A}{\partial t ^ 2} + {\left(\cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial t}\right)} B^{3} + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t}^{2} - A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial^2\,A}{\partial t ^ 2} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - {\left(A\left(t\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial t}\right)} B}{{\left(B^{2} \cosh\left(r\right)^{2} - A\left(t\right)^{2} \sinh\left(r\right)^{2}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}}$, $\displaystyle -\frac{{\left({\left(a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B^{3} - {\left({\left(\cosh\left(r\right)^{3} - 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} B\right)} \sqrt{B^{2} \cosh\left(r\right)^{2} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{2}}}{B^{4} \cosh\left(r\right)^{4} - 2 \, B^{2} A\left(t\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} + A\left(t\right)^{4} \sinh\left(r\right)^{4}}$, $\displaystyle 0$, $\displaystyle 0$]

acc1f = Man.one_form('acc1f',latex_name=r'alpha')
Dacc = nablaM(n1f)
#temp = nablaM(nv)
acc1f = nablaM(n1f)
acc1f.display()
accel1f = Dacc['_ij']*nv['^j']
accel1f.display()
accel1f[:]

$\displaystyle \nabla_{g} n = \left( \frac{{\left({\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} \left(\frac{\partial\,A}{\partial t}\right)^{2} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - {\left(A\left(t\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial t} + {\left(A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t}\right)^{2} - 2 \, A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right) \cosh\left(r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} + {\left(A\left(t\right)^{4} \cosh\left(r\right) \sinh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - A\left(t\right)^{4} \sinh\left(r\right)^{4}\right)} \frac{\partial\,a}{\partial t} - {\left(A\left(t\right)^{3} \cosh\left(r\right)^{2} \sinh\left(r\right)^{3} \left(\frac{\partial\,A}{\partial t}\right)^{2} - 2 \, A\left(t\right)^{3} \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t} + A\left(t\right)^{3} \sinh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}}{B^{5} \cosh\left(r\right)^{5} - 2 \, B^{3} A\left(t\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} + B A\left(t\right)^{4} \cosh\left(r\right) \sinh\left(r\right)^{4}} \right) \mathrm{d} t\otimes \mathrm{d} t + \left( -\frac{A\left(t\right)^{7} \sinh\left(r\right)^{7} \frac{\partial\,a}{\partial t} + {\left(a\left(t, r\right) \cosh\left(r\right)^{6} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{4} \sinh\left(r\right)^{3} + \cosh\left(r\right)^{4} \sinh\left(r\right)\right)} A\left(t\right) \frac{\partial\,a}{\partial t} + {\left(\cosh\left(r\right)^{7} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{6}\right)} \frac{\partial\,a}{\partial r}\right)} B^{6} - {\left(2 \, A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - {\left(2 \, \cosh\left(r\right)^{2} \sinh\left(r\right)^{5} + {\left(\cosh\left(r\right)^{4} + 2 \, \cosh\left(r\right)^{2}\right)} \sinh\left(r\right)^{3}\right)} A\left(t\right)^{3} \frac{\partial\,a}{\partial t} + 3 \, {\left(A\left(t\right)^{2} \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} B^{4} + {\left(A\left(t\right)^{4} a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{5} \frac{\partial\,A}{\partial t} - {\left(\sinh\left(r\right)^{7} + {\left(2 \, \cosh\left(r\right)^{2} + 1\right)} \sinh\left(r\right)^{5}\right)} A\left(t\right)^{5} \frac{\partial\,a}{\partial t} + 3 \, {\left(A\left(t\right)^{4} \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - A\left(t\right)^{4} \cosh\left(r\right)^{2} \sinh\left(r\right)^{4}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - {\left({\left(a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{5} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{4}\right)} \frac{\partial\,a}{\partial r}\right)} B^{4} - {\left({\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right) + {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right)^{2} \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2}\right)} \sqrt{B^{2} \cosh\left(r\right)^{2} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{2}} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} - {\left(A\left(t\right)^{6} \cosh\left(r\right) \sinh\left(r\right)^{6} \frac{\partial\,A}{\partial t} - A\left(t\right)^{6} \sinh\left(r\right)^{6}\right)} \frac{\partial\,a}{\partial r}}{{\left(B^{5} \cosh\left(r\right)^{5} - 2 \, B^{3} A\left(t\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} + B A\left(t\right)^{4} \cosh\left(r\right) \sinh\left(r\right)^{4}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}} \right) \mathrm{d} t\otimes \mathrm{d} r + \left( -\frac{A\left(t\right)^{3} \sinh\left(r\right)^{3} \frac{\partial\,a}{\partial t} + {\left(a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - {\left(\sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} A\left(t\right) \frac{\partial\,a}{\partial t} + {\left(\cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - {\left(A\left(t\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial r}}{\sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} B \cosh\left(r\right)} \right) \mathrm{d} r\otimes \mathrm{d} t + \left( -\frac{B^{4} \cosh\left(r\right)^{4} \frac{\partial\,a}{\partial t} + {\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t\right)^{4} \frac{\partial\,a}{\partial t} + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial\,a}{\partial t} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) + {\left(A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - A\left(t\right) \cosh\left(r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - {\left({\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}}{{\left(\cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} B} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( -\frac{B^{3} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial t} + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \sinh\left(r\right)^{4} \frac{\partial\,a}{\partial t} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} + {\left(A\left(t\right) \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - A\left(t\right) \sinh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B}{{\left(\cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \left( -\frac{B^{3} \cosh\left(r\right)^{2} \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial t} + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \sinh\left(r\right)^{4} \frac{\partial\,a}{\partial t} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} + {\left(A\left(t\right) \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - A\left(t\right) \sinh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B \sin\left({\theta}\right)^{2}}{{\left(\cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

$\displaystyle \left( \frac{{\left({\left(\cosh\left(r\right)^{5} - 3 \, \cosh\left(r\right)^{3} + 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{3} a\left(t, r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} a\left(t, r\right) - {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial\,A}{\partial t} + {\left(A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} \sqrt{B^{2} \cosh\left(r\right)^{2} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{2}} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}}{B^{6} a\left(t, r\right) \cosh\left(r\right)^{7} - 3 \, B^{4} A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} + 3 \, B^{2} A\left(t\right)^{4} a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} - A\left(t\right)^{6} a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{6}} \right) \mathrm{d} t + \left( \frac{{\left(\cosh\left(r\right)^{3} - 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) - {\left(a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2}\right)} \frac{\partial\,a}{\partial r}}{A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} - {\left(a\left(t, r\right) \cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - a\left(t, r\right) \cosh\left(r\right)^{3}\right)} B^{2}} \right) \mathrm{d} r$

[$\displaystyle \frac{{\left({\left(\cosh\left(r\right)^{5} - 3 \, \cosh\left(r\right)^{3} + 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{3} a\left(t, r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} a\left(t, r\right) - {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial\,A}{\partial t} + {\left(A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} \sqrt{B^{2} \cosh\left(r\right)^{2} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{2}} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}}{B^{6} a\left(t, r\right) \cosh\left(r\right)^{7} - 3 \, B^{4} A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} + 3 \, B^{2} A\left(t\right)^{4} a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} - A\left(t\right)^{6} a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{6}}$, $\displaystyle \frac{{\left(\cosh\left(r\right)^{3} - 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) - {\left(a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2}\right)} \frac{\partial\,a}{\partial r}}{A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} - {\left(a\left(t, r\right) \cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - a\left(t, r\right) \cosh\left(r\right)^{3}\right)} B^{2}}$, $\displaystyle 0$, $\displaystyle 0$]

K = Man.sym_bilin_form_field('K')
K = - acc1f - accel1f*n1f
K.display()
K[:].simplify()

$\displaystyle \left( -\frac{{\left({\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} \left(\frac{\partial\,A}{\partial t}\right)^{2} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - {\left(A\left(t\right)^{2} \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial t} + {\left(A\left(t\right) \cosh\left(r\right)^{6} \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t}\right)^{2} - 2 \, A\left(t\right) \cosh\left(r\right)^{5} \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{4} - {\left(A\left(t\right)^{3} a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} \left(\frac{\partial\,A}{\partial t}\right)^{2} - A\left(t\right)^{3} a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - 2 \, {\left(A\left(t\right)^{4} \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - A\left(t\right)^{4} \cosh\left(r\right)^{2} \sinh\left(r\right)^{4}\right)} \frac{\partial\,a}{\partial t} + 2 \, {\left(A\left(t\right)^{3} \cosh\left(r\right)^{4} \sinh\left(r\right)^{3} \left(\frac{\partial\,A}{\partial t}\right)^{2} - 2 \, A\left(t\right)^{3} \cosh\left(r\right)^{3} \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t} + A\left(t\right)^{3} \cosh\left(r\right)^{2} \sinh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - {\left(A\left(t\right)^{6} \cosh\left(r\right) \sinh\left(r\right)^{6} \frac{\partial\,A}{\partial t} - A\left(t\right)^{6} \sinh\left(r\right)^{6}\right)} \frac{\partial\,a}{\partial t} + {\left(A\left(t\right)^{5} \cosh\left(r\right)^{2} \sinh\left(r\right)^{5} \left(\frac{\partial\,A}{\partial t}\right)^{2} - 2 \, A\left(t\right)^{5} \cosh\left(r\right) \sinh\left(r\right)^{5} \frac{\partial\,A}{\partial t} + A\left(t\right)^{5} \sinh\left(r\right)^{5}\right)} \frac{\partial\,a}{\partial r}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} - {\left({\left({\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t\right) a\left(t, r\right) \left(\frac{\partial\,A}{\partial t}\right)^{2} - {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial\,A}{\partial t} + {\left(A\left(t\right) \cosh\left(r\right)^{6} \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t}\right)^{2} - 2 \, A\left(t\right) \cosh\left(r\right)^{5} \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{4} - {\left({\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 2 \, \cosh\left(r\right)^{3}\right)} A\left(t\right)^{3} a\left(t, r\right) \left(\frac{\partial\,A}{\partial t}\right)^{2} - {\left(\cosh\left(r\right)^{6} - 4 \, \cosh\left(r\right)^{4} + 3 \, \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} a\left(t, r\right) \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} a\left(t, r\right) + {\left({\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t\right)^{3} \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{2}\right)} \sqrt{B^{2} \cosh\left(r\right)^{2} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{2}}}{B^{7} \cosh\left(r\right)^{7} - 3 \, B^{5} A\left(t\right)^{2} \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} + 3 \, B^{3} A\left(t\right)^{4} \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} - B A\left(t\right)^{6} \cosh\left(r\right) \sinh\left(r\right)^{6}} \right) \mathrm{d} t\otimes \mathrm{d} t + \left( \frac{A\left(t\right)^{7} \sinh\left(r\right)^{7} \frac{\partial\,a}{\partial t} + {\left(a\left(t, r\right) \cosh\left(r\right)^{6} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{4} \sinh\left(r\right)^{3} + \cosh\left(r\right)^{4} \sinh\left(r\right)\right)} A\left(t\right) \frac{\partial\,a}{\partial t} + {\left(\cosh\left(r\right)^{7} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{6}\right)} \frac{\partial\,a}{\partial r}\right)} B^{6} - {\left(2 \, A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - {\left(2 \, \cosh\left(r\right)^{2} \sinh\left(r\right)^{5} + {\left(\cosh\left(r\right)^{4} + 2 \, \cosh\left(r\right)^{2}\right)} \sinh\left(r\right)^{3}\right)} A\left(t\right)^{3} \frac{\partial\,a}{\partial t} + 3 \, {\left(A\left(t\right)^{2} \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} B^{4} + {\left(A\left(t\right)^{4} a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{5} \frac{\partial\,A}{\partial t} - {\left(\sinh\left(r\right)^{7} + {\left(2 \, \cosh\left(r\right)^{2} + 1\right)} \sinh\left(r\right)^{5}\right)} A\left(t\right)^{5} \frac{\partial\,a}{\partial t} + 3 \, {\left(A\left(t\right)^{4} \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - A\left(t\right)^{4} \cosh\left(r\right)^{2} \sinh\left(r\right)^{4}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - {\left({\left(a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{5} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{4}\right)} \frac{\partial\,a}{\partial r}\right)} B^{4} - {\left({\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right) + {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right)^{2} \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2}\right)} \sqrt{B^{2} \cosh\left(r\right)^{2} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{2}} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} - {\left(A\left(t\right)^{6} \cosh\left(r\right) \sinh\left(r\right)^{6} \frac{\partial\,A}{\partial t} - A\left(t\right)^{6} \sinh\left(r\right)^{6}\right)} \frac{\partial\,a}{\partial r}}{{\left(B^{5} \cosh\left(r\right)^{5} - 2 \, B^{3} A\left(t\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} + B A\left(t\right)^{4} \cosh\left(r\right) \sinh\left(r\right)^{4}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}} \right) \mathrm{d} t\otimes \mathrm{d} r + \left( -\frac{{\left(A\left(t\right)^{5} \sinh\left(r\right)^{5} \frac{\partial\,a}{\partial t} + {\left(\sinh\left(r\right)^{5} + 2 \, \sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} B^{4} A\left(t\right) \frac{\partial\,a}{\partial t} - {\left(A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right) - {\left(\sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \frac{\partial\,A}{\partial t} + 2 \, {\left(\sinh\left(r\right)^{5} + \sinh\left(r\right)^{3}\right)} A\left(t\right)^{3} \frac{\partial\,a}{\partial t} - {\left({\left(\cosh\left(r\right) \sinh\left(r\right)^{4} + \cosh\left(r\right) \sinh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial\,A}{\partial t} - {\left(\sinh\left(r\right)^{4} + \sinh\left(r\right)^{2}\right)} A\left(t\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - {\left(A\left(t\right)^{4} \cosh\left(r\right) \sinh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - A\left(t\right)^{4} \sinh\left(r\right)^{4}\right)} \frac{\partial\,a}{\partial r}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}}{B^{5} \cosh\left(r\right)^{5} - 2 \, B^{3} A\left(t\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} + B A\left(t\right)^{4} \cosh\left(r\right) \sinh\left(r\right)^{4}} \right) \mathrm{d} r\otimes \mathrm{d} t + \left( \frac{B^{4} \cosh\left(r\right)^{4} \frac{\partial\,a}{\partial t} + {\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t\right)^{4} \frac{\partial\,a}{\partial t} + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial\,a}{\partial t} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) + {\left(A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - A\left(t\right) \cosh\left(r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - {\left({\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}}{{\left(\cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} B} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( \frac{B^{3} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial t} + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \sinh\left(r\right)^{4} \frac{\partial\,a}{\partial t} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} + {\left(A\left(t\right) \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - A\left(t\right) \sinh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B}{{\left(\cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \left( \frac{B^{3} \cosh\left(r\right)^{2} \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial t} + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} \sinh\left(r\right)^{4} \frac{\partial\,a}{\partial t} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} + {\left(A\left(t\right) \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t} - A\left(t\right) \sinh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B \sin\left({\theta}\right)^{2}}{{\left(\cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

$\displaystyle \left(\begin{array}{rrrr} -\frac{{\left({\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right)^{2} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - {\left(A\left(t\right)^{2} \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right)^{2}\right)} \frac{\partial}{\partial t}a\left(t, r\right) + {\left(A\left(t\right) \cosh\left(r\right)^{6} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, A\left(t\right) \cosh\left(r\right)^{5} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B^{4} - {\left(A\left(t\right)^{3} a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right)^{2} - A\left(t\right)^{3} a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) - 2 \, {\left(A\left(t\right)^{4} \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right)^{4} \cosh\left(r\right)^{2} \sinh\left(r\right)^{4}\right)} \frac{\partial}{\partial t}a\left(t, r\right) + 2 \, {\left(A\left(t\right)^{3} \cosh\left(r\right)^{4} \sinh\left(r\right)^{3} \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, A\left(t\right)^{3} \cosh\left(r\right)^{3} \sinh\left(r\right)^{3} \frac{\partial}{\partial t}A\left(t\right) + A\left(t\right)^{3} \cosh\left(r\right)^{2} \sinh\left(r\right)^{3}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B^{2} - {\left(A\left(t\right)^{6} \cosh\left(r\right) \sinh\left(r\right)^{6} \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right)^{6} \sinh\left(r\right)^{6}\right)} \frac{\partial}{\partial t}a\left(t, r\right) + {\left(A\left(t\right)^{5} \cosh\left(r\right)^{2} \sinh\left(r\right)^{5} \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, A\left(t\right)^{5} \cosh\left(r\right) \sinh\left(r\right)^{5} \frac{\partial}{\partial t}A\left(t\right) + A\left(t\right)^{5} \sinh\left(r\right)^{5}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} - {\left({\left({\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} - {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(A\left(t\right) \cosh\left(r\right)^{6} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, A\left(t\right) \cosh\left(r\right)^{5} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B^{4} - {\left({\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 2 \, \cosh\left(r\right)^{3}\right)} A\left(t\right)^{3} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} - {\left(\cosh\left(r\right)^{6} - 4 \, \cosh\left(r\right)^{4} + 3 \, \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} a\left(t, r\right) + {\left({\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B^{2}\right)} \sqrt{B^{2} \cosh\left(r\right)^{2} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{2}}}{B^{7} \cosh\left(r\right)^{7} - 3 \, B^{5} A\left(t\right)^{2} \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} + 3 \, B^{3} A\left(t\right)^{4} \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} - B A\left(t\right)^{6} \cosh\left(r\right) \sinh\left(r\right)^{6}} & \frac{A\left(t\right)^{7} \sinh\left(r\right)^{7} \frac{\partial}{\partial t}a\left(t, r\right) + {\left(a\left(t, r\right) \cosh\left(r\right)^{6} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{4} \sinh\left(r\right)^{3} + \cosh\left(r\right)^{4} \sinh\left(r\right)\right)} A\left(t\right) \frac{\partial}{\partial t}a\left(t, r\right) + {\left(\cosh\left(r\right)^{7} \frac{\partial}{\partial t}A\left(t\right) - \cosh\left(r\right)^{6}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B^{6} - {\left(2 \, A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right)^{3} \frac{\partial}{\partial t}A\left(t\right) - {\left(2 \, \cosh\left(r\right)^{2} \sinh\left(r\right)^{5} + {\left(\cosh\left(r\right)^{4} + 2 \, \cosh\left(r\right)^{2}\right)} \sinh\left(r\right)^{3}\right)} A\left(t\right)^{3} \frac{\partial}{\partial t}a\left(t, r\right) + 3 \, {\left(A\left(t\right)^{2} \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right)^{2}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B^{4} + {\left(A\left(t\right)^{4} a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{5} \frac{\partial}{\partial t}A\left(t\right) - {\left(\sinh\left(r\right)^{7} + {\left(2 \, \cosh\left(r\right)^{2} + 1\right)} \sinh\left(r\right)^{5}\right)} A\left(t\right)^{5} \frac{\partial}{\partial t}a\left(t, r\right) + 3 \, {\left(A\left(t\right)^{4} \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right)^{4} \cosh\left(r\right)^{2} \sinh\left(r\right)^{4}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B^{2} - {\left({\left(a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{5} \frac{\partial}{\partial t}A\left(t\right) - \cosh\left(r\right)^{4}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B^{4} - {\left({\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right) + {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B^{2}\right)} \sqrt{B^{2} \cosh\left(r\right)^{2} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{2}} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} - {\left(A\left(t\right)^{6} \cosh\left(r\right) \sinh\left(r\right)^{6} \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right)^{6} \sinh\left(r\right)^{6}\right)} \frac{\partial}{\partial r}a\left(t, r\right)}{{\left(B^{5} \cosh\left(r\right)^{5} - 2 \, B^{3} A\left(t\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} + B A\left(t\right)^{4} \cosh\left(r\right) \sinh\left(r\right)^{4}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}} & 0 & 0 \\ -\frac{{\left(A\left(t\right)^{5} \sinh\left(r\right)^{5} \frac{\partial}{\partial t}a\left(t, r\right) + {\left(\sinh\left(r\right)^{5} + 2 \, \sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} B^{4} A\left(t\right) \frac{\partial}{\partial t}a\left(t, r\right) - {\left(A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right) - {\left(\sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + 2 \, {\left(\sinh\left(r\right)^{5} + \sinh\left(r\right)^{3}\right)} A\left(t\right)^{3} \frac{\partial}{\partial t}a\left(t, r\right) - {\left({\left(\cosh\left(r\right) \sinh\left(r\right)^{4} + \cosh\left(r\right) \sinh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - {\left(\sinh\left(r\right)^{4} + \sinh\left(r\right)^{2}\right)} A\left(t\right)^{2}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B^{2} - {\left(A\left(t\right)^{4} \cosh\left(r\right) \sinh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right)^{4} \sinh\left(r\right)^{4}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}}{B^{5} \cosh\left(r\right)^{5} - 2 \, B^{3} A\left(t\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} + B A\left(t\right)^{4} \cosh\left(r\right) \sinh\left(r\right)^{4}} & \frac{B^{4} \cosh\left(r\right)^{4} \frac{\partial}{\partial t}a\left(t, r\right) + {\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t\right)^{4} \frac{\partial}{\partial t}a\left(t, r\right) + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial}{\partial t}a\left(t, r\right) - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) + {\left(A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right) \cosh\left(r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B^{2} - {\left({\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)}{{\left(\cosh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} B} & 0 & 0 \\ 0 & 0 & \frac{B^{3} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}a\left(t, r\right) + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right)^{2} \sinh\left(r\right)^{4} \frac{\partial}{\partial t}a\left(t, r\right) - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} + {\left(A\left(t\right) \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right) \sinh\left(r\right)^{3}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B}{{\left(\cosh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}} & 0 \\ 0 & 0 & 0 & \frac{B^{3} \cosh\left(r\right)^{2} \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}a\left(t, r\right) + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right)^{2} \sinh\left(r\right)^{4} \frac{\partial}{\partial t}a\left(t, r\right) - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} + {\left(A\left(t\right) \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right) \sinh\left(r\right)^{3}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B \sin\left({\theta}\right)^{2}}{{\left(\cosh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}} \end{array}\right)$

K3 = Sig.sym_bilin_form_field('K3', latex_name='K')
K3 =  Phi.pullback(K)
K3.display()

$\displaystyle \left( \frac{B^{4} \cosh\left(r\right)^{4} \frac{\partial\,a}{\partial t_{0}} + {\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t_{0}\right)^{4} \frac{\partial\,a}{\partial t_{0}} + {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} \frac{\partial\,a}{\partial t_{0}} - A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right) + {\left(A\left(t_{0}\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - A\left(t_{0}\right) \cosh\left(r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - {\left({\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}}{{\left(\cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)} B} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( \frac{B^{3} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial t_{0}} + {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{4} \frac{\partial\,a}{\partial t_{0}} - A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} + {\left(A\left(t_{0}\right) \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t_{0}} - A\left(t_{0}\right) \sinh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B}{{\left(\cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)}} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \left( \frac{B^{3} \cosh\left(r\right)^{2} \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial t_{0}} + {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{4} \frac{\partial\,a}{\partial t_{0}} - A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right)^{2} + {\left(A\left(t_{0}\right) \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t_{0}} - A\left(t_{0}\right) \sinh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B \sin\left({\theta}\right)^{2}}{{\left(\cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

K3u = K3.up(gam3, 0)
print(K3u) ; K3u.display()
K3u[:]

Tensor field of type (1,1) on the 3-dimensional differentiable manifold Sigma

$\displaystyle \left( \frac{B^{4} \cosh\left(r\right)^{4} \frac{\partial\,a}{\partial t_{0}} + {\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t_{0}\right)^{4} \frac{\partial\,a}{\partial t_{0}} + {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} \frac{\partial\,a}{\partial t_{0}} - A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right) + {\left(A\left(t_{0}\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - A\left(t_{0}\right) \cosh\left(r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - {\left({\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}}{{\left({\left(a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t_{0}} - a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{3}\right)} B^{3} - {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2}\right)} B\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)}} \right) \frac{\partial}{\partial r }\otimes \mathrm{d} r + \left( \frac{A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} + B^{2} \cosh\left(r\right)^{2} \frac{\partial\,a}{\partial t_{0}} - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial t_{0}} - A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right) + {\left(A\left(t_{0}\right) \cosh\left(r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - A\left(t_{0}\right) \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}}{{\left(a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - a\left(t_{0}, r\right)^{2} \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)} B} \right) \frac{\partial}{\partial {\theta} }\otimes \mathrm{d} {\theta} + \left( \frac{A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} + B^{2} \cosh\left(r\right)^{2} \frac{\partial\,a}{\partial t_{0}} - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial t_{0}} - A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right) + {\left(A\left(t_{0}\right) \cosh\left(r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - A\left(t_{0}\right) \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}}{{\left(a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - a\left(t_{0}, r\right)^{2} \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)} B} \right) \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} {\phi}$

$\displaystyle \left(\begin{array}{rrr} \frac{B^{4} \cosh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) + {\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t_{0}\right)^{4} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) + {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right) + {\left(A\left(t_{0}\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right) \cosh\left(r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} B^{2} - {\left({\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)}{{\left({\left(a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{3}\right)} B^{3} - {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2}\right)} B\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)}} & 0 & 0 \\ 0 & \frac{A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + B^{2} \cosh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right) + {\left(A\left(t_{0}\right) \cosh\left(r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)}{{\left(a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - a\left(t_{0}, r\right)^{2} \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)} B} & 0 \\ 0 & 0 & \frac{A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + B^{2} \cosh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right) + {\left(A\left(t_{0}\right) \cosh\left(r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)}{{\left(a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - a\left(t_{0}, r\right)^{2} \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)} B} \end{array}\right)$

K3uu = K3u.up(gam3, 1)
trKK3 = K3['_ij']*K3uu['^ij']
print(trKK3) ; trKK3.display()

Scalar field on the 3-dimensional differentiable manifold Sigma

$\displaystyle \begin{array}{llcl} & \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & \frac{3 \, B^{8} \cosh\left(r\right)^{8} \left(\frac{\partial\,a}{\partial t_{0}}\right)^{2} + 2 \, {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{2} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} + 3 \, {\left(\cosh\left(r\right)^{8} - 4 \, \cosh\left(r\right)^{6} + 6 \, \cosh\left(r\right)^{4} - 4 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t_{0}\right)^{8} \left(\frac{\partial\,a}{\partial t_{0}}\right)^{2} - 4 \, {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{2} \frac{\partial\,A}{\partial t_{0}} + 2 \, {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{2} - 2 \, {\left(6 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} \left(\frac{\partial\,a}{\partial t_{0}}\right)^{2} - {\left({\left(2 \, \cosh\left(r\right)^{8} + \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \frac{\partial\,A}{\partial t_{0}} - {\left(2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) + 3 \, {\left(A\left(t_{0}\right) \cosh\left(r\right)^{7} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - A\left(t_{0}\right) \cosh\left(r\right)^{6} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} \frac{\partial\,a}{\partial t_{0}}\right)} B^{6} + {\left({\left(2 \, \cosh\left(r\right)^{8} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} + 18 \, {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} \left(\frac{\partial\,a}{\partial t_{0}}\right)^{2} - 2 \, {\left(2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \frac{\partial\,A}{\partial t_{0}} + {\left(2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} + 3 \, {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} \frac{\partial\,A}{\partial t_{0}} + {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2}\right)} \frac{\partial\,a}{\partial r}^{2} - 2 \, {\left(2 \, {\left(3 \, \cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \frac{\partial\,A}{\partial t_{0}} - 2 \, {\left(3 \, \cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) + 9 \, {\left({\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} \frac{\partial\,a}{\partial t_{0}} + 2 \, {\left({\left(2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + {\left(2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{4} - 2 \, {\left(2 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} + 6 \, {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} \left(\frac{\partial\,a}{\partial t_{0}}\right)^{2} - 4 \, {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \frac{\partial\,A}{\partial t_{0}} + 2 \, {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} + 3 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} \frac{\partial\,A}{\partial t_{0}} + {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4}\right)} \frac{\partial\,a}{\partial r}^{2} - {\left({\left(6 \, \cosh\left(r\right)^{8} - 11 \, \cosh\left(r\right)^{6} + 4 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \frac{\partial\,A}{\partial t_{0}} - {\left(6 \, \cosh\left(r\right)^{7} - 11 \, \cosh\left(r\right)^{5} + 4 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) + 9 \, {\left({\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{5} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{5} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} \frac{\partial\,a}{\partial t_{0}} + {\left({\left(4 \, \cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(4 \, \cosh\left(r\right)^{6} - 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + {\left(4 \, \cosh\left(r\right)^{5} - 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} + 3 \, {\left({\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{6} \frac{\partial\,A}{\partial t_{0}} + {\left(\cosh\left(r\right)^{6} - 3 \, \cosh\left(r\right)^{4} + 3 \, \cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{6}\right)} \frac{\partial\,a}{\partial r}^{2} - 2 \, {\left(2 \, {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{7} a\left(t_{0}, r\right) \frac{\partial\,A}{\partial t_{0}} - 2 \, {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{7} a\left(t_{0}, r\right) + 3 \, {\left({\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{7} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{6} - 3 \, \cosh\left(r\right)^{4} + 3 \, \cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{7} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} \frac{\partial\,a}{\partial t_{0}} + 4 \, {\left({\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + {\left(\cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}}{{\left(a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{10} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{9} \frac{\partial\,A}{\partial t_{0}} + a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{8}\right)} B^{8} - 3 \, {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{8} \sinh\left(r\right)^{2} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{7} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} + A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{6} \sinh\left(r\right)^{2}\right)} B^{6} + 3 \, {\left(A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{6} \sinh\left(r\right)^{4} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{5} \sinh\left(r\right)^{4} \frac{\partial\,A}{\partial t_{0}} + A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{4} \sinh\left(r\right)^{4}\right)} B^{4} - {\left(A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{4} \sinh\left(r\right)^{6} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{3} \sinh\left(r\right)^{6} \frac{\partial\,A}{\partial t_{0}} + A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{2} \sinh\left(r\right)^{6}\right)} B^{2}} \end{array}$

trK3 = K3u.trace()
trK3.display()

$\displaystyle \begin{array}{llcl} & \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & \frac{3 \, B^{4} \cosh\left(r\right)^{4} \frac{\partial\,a}{\partial t_{0}} - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \frac{\partial\,A}{\partial t_{0}} + 3 \, {\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t_{0}\right)^{4} \frac{\partial\,a}{\partial t_{0}} + 2 \, {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) + {\left({\left(2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \frac{\partial\,A}{\partial t_{0}} - 6 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} \frac{\partial\,a}{\partial t_{0}} - {\left(2 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) + 3 \, {\left(A\left(t_{0}\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - A\left(t_{0}\right) \cosh\left(r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - 3 \, {\left({\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}}{{\left({\left(a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t_{0}} - a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{3}\right)} B^{3} - {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2}\right)} B\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)}} \end{array}$

Ku = K.up(g, 0)
print(Ku) ; Ku.display()
Ku[:]

Tensor field of type (1,1) on the 4-dimensional differentiable manifold M

$\displaystyle \left( -\frac{{\left({\left(\cosh\left(r\right)^{5} - 3 \, \cosh\left(r\right)^{3} + 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{3} a\left(t, r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} a\left(t, r\right) - {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial\,A}{\partial t} + {\left(A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} - {\left({\left(\cosh\left(r\right)^{5} - 3 \, \cosh\left(r\right)^{3} + 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{3} a\left(t, r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} a\left(t, r\right) - {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial\,A}{\partial t} + {\left(A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} \sqrt{B^{2} \cosh\left(r\right)^{2} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{2}}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{7} \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{6}\right)} B^{5} - 2 \, {\left(A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right)^{2}\right)} B^{3} + {\left(A\left(t\right)^{4} a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - A\left(t\right)^{4} a\left(t, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{4}\right)} B} \right) \frac{\partial}{\partial t }\otimes \mathrm{d} t + \left( \frac{{\left({\left(\cosh\left(r\right)^{3} - 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) - {\left(a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} - {\left({\left(\cosh\left(r\right)^{3} - 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} + A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) - {\left(a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - \cosh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} \sqrt{B^{2} \cosh\left(r\right)^{2} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{2}}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{6} \left(\frac{\partial\,A}{\partial t}\right)^{2} - 2 \, a\left(t, r\right)^{2} \cosh\left(r\right)^{5} \frac{\partial\,A}{\partial t} + a\left(t, r\right)^{2} \cosh\left(r\right)^{4}\right)} B^{3} - {\left(A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right)^{2} \left(\frac{\partial\,A}{\partial t}\right)^{2} - 2 \, A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} + A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2}\right)} B} \right) \frac{\partial}{\partial t }\otimes \mathrm{d} r + \left( -\frac{{\left(A\left(t\right)^{5} \cosh\left(r\right) \sinh\left(r\right)^{5} \frac{\partial\,a}{\partial t} + A\left(t\right)^{4} a\left(t, r\right) \sinh\left(r\right)^{3} + {\left(\cosh\left(r\right) \sinh\left(r\right)^{5} - \cosh\left(r\right) \sinh\left(r\right)^{3}\right)} A\left(t\right)^{4} a\left(t, r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right) \sinh\left(r\right)^{5} + 2 \, \cosh\left(r\right) \sinh\left(r\right)^{3} + \cosh\left(r\right) \sinh\left(r\right)\right)} B^{4} A\left(t\right) \frac{\partial\,a}{\partial t} - {\left({\left(\cosh\left(r\right) \sinh\left(r\right)^{5} - \cosh\left(r\right) \sinh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \frac{\partial\,A}{\partial t} + 2 \, {\left(\cosh\left(r\right) \sinh\left(r\right)^{5} + \cosh\left(r\right) \sinh\left(r\right)^{3}\right)} A\left(t\right)^{3} \frac{\partial\,a}{\partial t} + {\left(\sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right)\right)} B^{2}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} - {\left(A\left(t\right)^{4} a\left(t, r\right) \sinh\left(r\right)^{3} + {\left(\cosh\left(r\right) \sinh\left(r\right)^{5} - \cosh\left(r\right) \sinh\left(r\right)^{3}\right)} A\left(t\right)^{4} a\left(t, r\right) \frac{\partial\,A}{\partial t} - {\left({\left(\cosh\left(r\right) \sinh\left(r\right)^{5} + \cosh\left(r\right) \sinh\left(r\right)^{3}\right)} A\left(t\right)^{2} a\left(t, r\right) \frac{\partial\,A}{\partial t} + {\left({\left(\sinh\left(r\right)^{6} + 2 \, \sinh\left(r\right)^{4} + \sinh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right) \sinh\left(r\right)^{4} + \cosh\left(r\right) \sinh\left(r\right)^{2}\right)} A\left(t\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - {\left(A\left(t\right)^{4} \cosh\left(r\right) \sinh\left(r\right)^{4} - {\left(\sinh\left(r\right)^{6} + \sinh\left(r\right)^{4}\right)} A\left(t\right)^{4} \frac{\partial\,A}{\partial t}\right)} \frac{\partial\,a}{\partial r}\right)} \sqrt{-A\left(t\right)^{2} \sinh\left(r\right)^{2} + {\left(\sinh\left(r\right)^{2} + 1\right)} B^{2}}}{B^{7} a\left(t, r\right)^{2} \cosh\left(r\right)^{8} - 3 \, B^{5} A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{6} \sinh\left(r\right)^{2} + 3 \, B^{3} A\left(t\right)^{4} a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right)^{4} - B A\left(t\right)^{6} a\left(t, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{6}} \right) \frac{\partial}{\partial r }\otimes \mathrm{d} t + \left( \frac{{\left(B^{4} \cosh\left(r\right)^{5} \frac{\partial\,a}{\partial t} + {\left(\cosh\left(r\right)^{5} - 3 \, \cosh\left(r\right)^{3} + 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{3} a\left(t, r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t\right)^{4} \frac{\partial\,a}{\partial t} + {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} a\left(t, r\right) - {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial\,A}{\partial t} + 2 \, {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right)^{2} \frac{\partial\,a}{\partial t}\right)} B^{2}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} - {\left({\left(\cosh\left(r\right)^{5} - 3 \, \cosh\left(r\right)^{3} + 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{3} a\left(t, r\right) \frac{\partial\,A}{\partial t} + {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} a\left(t, r\right) - {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial\,A}{\partial t} + {\left(A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} \sqrt{B^{2} \cosh\left(r\right)^{2} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{2}}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{7} \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{6}\right)} B^{5} - 2 \, {\left(A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right)^{2}\right)} B^{3} + {\left(A\left(t\right)^{4} a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - A\left(t\right)^{4} a\left(t, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{4}\right)} B} \right) \frac{\partial}{\partial r }\otimes \mathrm{d} r + \left( \frac{A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} + B^{2} \cosh\left(r\right)^{2} \frac{\partial\,a}{\partial t} - A\left(t\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial t} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) + {\left(A\left(t\right) \cosh\left(r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} - A\left(t\right) \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} B} \right) \frac{\partial}{\partial {\theta} }\otimes \mathrm{d} {\theta} + \left( \frac{A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} + B^{2} \cosh\left(r\right)^{2} \frac{\partial\,a}{\partial t} - A\left(t\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial t} - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) + {\left(A\left(t\right) \cosh\left(r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} - A\left(t\right) \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} B} \right) \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} {\phi}$

$\displaystyle \left(\begin{array}{rrrr} -\frac{{\left({\left(\cosh\left(r\right)^{5} - 3 \, \cosh\left(r\right)^{3} + 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{3} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} a\left(t, r\right) - {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} - {\left({\left(\cosh\left(r\right)^{5} - 3 \, \cosh\left(r\right)^{3} + 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{3} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} a\left(t, r\right) - {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} \sqrt{B^{2} \cosh\left(r\right)^{2} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{2}}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{7} \frac{\partial}{\partial t}A\left(t\right) - a\left(t, r\right)^{2} \cosh\left(r\right)^{6}\right)} B^{5} - 2 \, {\left(A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right)^{2}\right)} B^{3} + {\left(A\left(t\right)^{4} a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right)^{4} a\left(t, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{4}\right)} B} & \frac{{\left({\left(\cosh\left(r\right)^{3} - 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) - {\left(a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) - \cosh\left(r\right)^{3}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} - {\left({\left(\cosh\left(r\right)^{3} - 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) - {\left(a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) - \cosh\left(r\right)^{3}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} \sqrt{B^{2} \cosh\left(r\right)^{2} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{2}}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{6} \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, a\left(t, r\right)^{2} \cosh\left(r\right)^{5} \frac{\partial}{\partial t}A\left(t\right) + a\left(t, r\right)^{2} \cosh\left(r\right)^{4}\right)} B^{3} - {\left(A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) + A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2}\right)} B} & 0 & 0 \\ -\frac{{\left(A\left(t\right)^{5} \cosh\left(r\right) \sinh\left(r\right)^{5} \frac{\partial}{\partial t}a\left(t, r\right) + A\left(t\right)^{4} a\left(t, r\right) \sinh\left(r\right)^{3} + {\left(\cosh\left(r\right) \sinh\left(r\right)^{5} - \cosh\left(r\right) \sinh\left(r\right)^{3}\right)} A\left(t\right)^{4} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right) \sinh\left(r\right)^{5} + 2 \, \cosh\left(r\right) \sinh\left(r\right)^{3} + \cosh\left(r\right) \sinh\left(r\right)\right)} B^{4} A\left(t\right) \frac{\partial}{\partial t}a\left(t, r\right) - {\left({\left(\cosh\left(r\right) \sinh\left(r\right)^{5} - \cosh\left(r\right) \sinh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + 2 \, {\left(\cosh\left(r\right) \sinh\left(r\right)^{5} + \cosh\left(r\right) \sinh\left(r\right)^{3}\right)} A\left(t\right)^{3} \frac{\partial}{\partial t}a\left(t, r\right) + {\left(\sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right)\right)} B^{2}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} - {\left(A\left(t\right)^{4} a\left(t, r\right) \sinh\left(r\right)^{3} + {\left(\cosh\left(r\right) \sinh\left(r\right)^{5} - \cosh\left(r\right) \sinh\left(r\right)^{3}\right)} A\left(t\right)^{4} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left({\left(\cosh\left(r\right) \sinh\left(r\right)^{5} + \cosh\left(r\right) \sinh\left(r\right)^{3}\right)} A\left(t\right)^{2} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left({\left(\sinh\left(r\right)^{6} + 2 \, \sinh\left(r\right)^{4} + \sinh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right) \sinh\left(r\right)^{4} + \cosh\left(r\right) \sinh\left(r\right)^{2}\right)} A\left(t\right)^{2}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B^{2} - {\left(A\left(t\right)^{4} \cosh\left(r\right) \sinh\left(r\right)^{4} - {\left(\sinh\left(r\right)^{6} + \sinh\left(r\right)^{4}\right)} A\left(t\right)^{4} \frac{\partial}{\partial t}A\left(t\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} \sqrt{-A\left(t\right)^{2} \sinh\left(r\right)^{2} + {\left(\sinh\left(r\right)^{2} + 1\right)} B^{2}}}{B^{7} a\left(t, r\right)^{2} \cosh\left(r\right)^{8} - 3 \, B^{5} A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{6} \sinh\left(r\right)^{2} + 3 \, B^{3} A\left(t\right)^{4} a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right)^{4} - B A\left(t\right)^{6} a\left(t, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{6}} & \frac{{\left(B^{4} \cosh\left(r\right)^{5} \frac{\partial}{\partial t}a\left(t, r\right) + {\left(\cosh\left(r\right)^{5} - 3 \, \cosh\left(r\right)^{3} + 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{3} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t\right)^{4} \frac{\partial}{\partial t}a\left(t, r\right) + {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} a\left(t, r\right) - {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + 2 \, {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right)^{2} \frac{\partial}{\partial t}a\left(t, r\right)\right)} B^{2}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} - {\left({\left(\cosh\left(r\right)^{5} - 3 \, \cosh\left(r\right)^{3} + 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{3} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} a\left(t, r\right) - {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} \sqrt{B^{2} \cosh\left(r\right)^{2} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{2}}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{7} \frac{\partial}{\partial t}A\left(t\right) - a\left(t, r\right)^{2} \cosh\left(r\right)^{6}\right)} B^{5} - 2 \, {\left(A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right)^{2}\right)} B^{3} + {\left(A\left(t\right)^{4} a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right)^{4} a\left(t, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{4}\right)} B} & 0 & 0 \\ 0 & 0 & \frac{A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) + B^{2} \cosh\left(r\right)^{2} \frac{\partial}{\partial t}a\left(t, r\right) - A\left(t\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}a\left(t, r\right) - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) + {\left(A\left(t\right) \cosh\left(r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - a\left(t, r\right)^{2} \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} B} & 0 \\ 0 & 0 & 0 & \frac{A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) + B^{2} \cosh\left(r\right)^{2} \frac{\partial}{\partial t}a\left(t, r\right) - A\left(t\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}a\left(t, r\right) - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) + {\left(A\left(t\right) \cosh\left(r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - a\left(t, r\right)^{2} \cosh\left(r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)} B} \end{array}\right)$

Kuu = Ku.up(g, 1)
trKK = K['_ij']*Kuu['^ij']
print(trKK) ; trKK.display()

Scalar field on the 4-dimensional differentiable manifold M

ParseError: KaTeX parse error: Too many expansions: infinite loop or need to increase maxExpand setting

trK = Ku.trace()
print(trK)
trK.display()

Scalar field on the 4-dimensional differentiable manifold M

$\displaystyle \begin{array}{llcl} & \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & \frac{{\left(3 \, B^{4} \cosh\left(r\right)^{4} \frac{\partial\,a}{\partial t} - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} a\left(t, r\right) \frac{\partial\,A}{\partial t} + 3 \, {\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t\right)^{4} \frac{\partial\,a}{\partial t} + 2 \, {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} a\left(t, r\right) + {\left({\left(2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial\,A}{\partial t} - 6 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial\,a}{\partial t} - {\left(2 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t\right) a\left(t, r\right) + 3 \, {\left(A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - A\left(t\right) \cosh\left(r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} - 3 \, {\left({\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} \sqrt{B \cosh\left(r\right) + A\left(t\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t\right) \sinh\left(r\right)}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{6} \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{5}\right)} B^{5} - 2 \, {\left(A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t} - A\left(t\right)^{2} a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{2}\right)} B^{3} + {\left(A\left(t\right)^{4} a\left(t, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{4} \frac{\partial\,A}{\partial t} - A\left(t\right)^{4} a\left(t, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{4}\right)} B} \end{array}$

Ricci4 = nablaM.ricci()

Ricci4[:]

$\displaystyle \left(\begin{array}{rrrr} \frac{3 \, {\left(a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial^{2}}{(\partial t)^{2}}A\left(t\right) \frac{\partial}{\partial t}a\left(t, r\right) + {\left(\cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right)^{2} - {\left(a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial t)^{2}}a\left(t, r\right)\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right)^{2} + {\left(\cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - \cosh\left(r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)^{2} - {\left(3 \, A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial^{2}}{(\partial t)^{2}}A\left(t\right) - 2 \, {\left(\cosh\left(r\right)^{3} + 2 \, \cosh\left(r\right)\right)} A\left(t\right) a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t\right) a\left(t, r\right) \sinh\left(r\right) + 2 \, {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right) \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right) + {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial t)^{2}}a\left(t, r\right) - 2 \, {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right) a\left(t, r\right)\right)} \frac{\partial^{2}}{\partial t\partial r}a\left(t, r\right) + {\left({\left(\cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, {\left(\cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, {\left(\cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{2} + 1\right)} a\left(t, r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right) + {\left(a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t, r\right)}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} & -\frac{{\left({\left(A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right)^{2} - {\left(2 \, a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial^{2}}{(\partial t)^{2}}A\left(t\right) - 2 \, a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + 4 \, {\left(\cosh\left(r\right)^{5} \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, \cosh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) + \cosh\left(r\right)^{3}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right) + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial t)^{2}}a\left(t, r\right) + 2 \, {\left(a\left(t, r\right) \cosh\left(r\right)^{5} \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, a\left(t, r\right) \cosh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) + a\left(t, r\right) \cosh\left(r\right)^{3}\right)} \frac{\partial^{2}}{\partial t\partial r}a\left(t, r\right)\right)} B^{2} - {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right)^{2} - {\left(A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, A\left(t\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right) \cosh\left(r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)^{2} + {\left(3 \, A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} a\left(t, r\right) \sinh\left(r\right) \frac{\partial^{2}}{(\partial t)^{2}}A\left(t\right) - 2 \, {\left(\cosh\left(r\right)^{3} + 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) + 2 \, {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right)^{2} \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right) - {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} a\left(t, r\right) \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial t)^{2}}a\left(t, r\right) + 2 \, {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right)^{2} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right)\right)} \frac{\partial^{2}}{\partial t\partial r}a\left(t, r\right) - {\left({\left(\cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, {\left(\cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, {\left(\cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{2} + 1\right)} A\left(t\right) a\left(t, r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right) - {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t, r\right)}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{5} \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) + a\left(t, r\right)^{2} \cosh\left(r\right)^{3}\right)} B^{2}} & 0 & 0 \\ -\frac{{\left({\left(A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right)^{2} - {\left(2 \, a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial^{2}}{(\partial t)^{2}}A\left(t\right) - 2 \, a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + 4 \, {\left(\cosh\left(r\right)^{5} \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, \cosh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) + \cosh\left(r\right)^{3}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right) + {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial t)^{2}}a\left(t, r\right) + 2 \, {\left(a\left(t, r\right) \cosh\left(r\right)^{5} \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, a\left(t, r\right) \cosh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) + a\left(t, r\right) \cosh\left(r\right)^{3}\right)} \frac{\partial^{2}}{\partial t\partial r}a\left(t, r\right)\right)} B^{2} - {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right)^{2} - {\left(A\left(t\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, A\left(t\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right) \cosh\left(r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)^{2} + {\left(3 \, A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} a\left(t, r\right) \sinh\left(r\right) \frac{\partial^{2}}{(\partial t)^{2}}A\left(t\right) - 2 \, {\left(\cosh\left(r\right)^{3} + 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) + 2 \, {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right)^{2} \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right) - {\left({\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{3} a\left(t, r\right) \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial t)^{2}}a\left(t, r\right) + 2 \, {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right)^{2} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right)\right)} \frac{\partial^{2}}{\partial t\partial r}a\left(t, r\right) - {\left({\left(\cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, {\left(\cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, {\left(\cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{2} + 1\right)} A\left(t\right) a\left(t, r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right) - {\left(A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t, r\right)}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{5} \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) + a\left(t, r\right)^{2} \cosh\left(r\right)^{3}\right)} B^{2}} & -\frac{{\left(a\left(t, r\right) \cosh\left(r\right)^{6} \sinh\left(r\right) \frac{\partial^{2}}{(\partial t)^{2}}A\left(t\right) \frac{\partial}{\partial t}a\left(t, r\right) - {\left(\cosh\left(r\right)^{6} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - \cosh\left(r\right)^{5} \sinh\left(r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right)^{2} - {\left(a\left(t, r\right) \cosh\left(r\right)^{6} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - a\left(t, r\right) \cosh\left(r\right)^{5} \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial t)^{2}}a\left(t, r\right)\right)} B^{4} + {\left(2 \, {\left({\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right)^{2} - 3 \, {\left(\cosh\left(r\right)^{6} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, \cosh\left(r\right)^{5} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)^{2} - {\left(5 \, A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 2 \, {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial^{2}}{(\partial t)^{2}}A\left(t\right) - 2 \, {\left(\cosh\left(r\right)^{5} + 4 \, \cosh\left(r\right)^{3}\right)} A\left(t\right) a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(2 \, \cosh\left(r\right)^{4} + 3 \, \cosh\left(r\right)^{2}\right)} A\left(t\right) a\left(t, r\right) \sinh\left(r\right) + 2 \, {\left({\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t\right) \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right) + 2 \, {\left({\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial t)^{2}}a\left(t, r\right) - 2 \, {\left({\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right) a\left(t, r\right)\right)} \frac{\partial^{2}}{\partial t\partial r}a\left(t, r\right) - {\left({\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5}\right)} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, {\left(\cosh\left(r\right)^{6} - 3 \, \cosh\left(r\right)^{4}\right)} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, {\left(\cosh\left(r\right)^{5} - 3 \, \cosh\left(r\right)^{3}\right)} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{4} - 3 \, \cosh\left(r\right)^{2}\right)} a\left(t, r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right) + 3 \, {\left(a\left(t, r\right) \cosh\left(r\right)^{6} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, a\left(t, r\right) \cosh\left(r\right)^{5} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t, r\right)\right)} B^{2} - {\left({\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t\right)^{4} \sinh\left(r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right)^{2} - {\left({\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right)^{2} + {\left(3 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t\right)^{4} a\left(t, r\right) \sinh\left(r\right) \frac{\partial^{2}}{(\partial t)^{2}}A\left(t\right) - 2 \, {\left(\cosh\left(r\right)^{5} + \cosh\left(r\right)^{3} - 2 \, \cosh\left(r\right)\right)} A\left(t\right)^{3} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(2 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2} - 1\right)} A\left(t\right)^{3} a\left(t, r\right) \sinh\left(r\right) + 2 \, {\left({\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t\right)^{3} \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t\right)^{3}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right) - {\left({\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t\right)^{4} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t\right)^{4} a\left(t, r\right) \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial t)^{2}}a\left(t, r\right) + 2 \, {\left({\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t\right)^{3} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t\right)^{3} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t\right)^{3} a\left(t, r\right)\right)} \frac{\partial^{2}}{\partial t\partial r}a\left(t, r\right) - {\left({\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{3}\right)} A\left(t\right)^{2} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{4} - 1\right)} A\left(t\right)^{2} a\left(t, r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right) - {\left({\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right) \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t, r\right)}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{6} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, a\left(t, r\right)^{2} \cosh\left(r\right)^{5} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} & 0 & 0 \\ 0 & 0 & -\frac{{\left({\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} a\left(t, r\right) \frac{\partial^{2}}{(\partial t)^{2}}A\left(t\right) \frac{\partial}{\partial t}a\left(t, r\right) - {\left(\cosh\left(r\right)^{4} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - \cosh\left(r\right)^{3} \sinh\left(r\right)^{2}\right)} \frac{\partial}{\partial t}a\left(t, r\right)^{2} - {\left({\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} a\left(t, r\right)\right)} \frac{\partial^{2}}{(\partial t)^{2}}a\left(t, r\right)\right)} B^{2} + {\left(A\left(t\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{4}\right)} \frac{\partial}{\partial t}a\left(t, r\right)^{2} + {\left(\cosh\left(r\right)^{4} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - \cosh\left(r\right) \sinh\left(r\right)^{2}\right)} \frac{\partial}{\partial r}a\left(t, r\right)^{2} - {\left({\left(2 \, \cosh\left(r\right)^{4} \sinh\left(r\right)^{2} + 3 \, \cosh\left(r\right)^{4} - 3 \, \cosh\left(r\right)^{2}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} a\left(t, r\right) \frac{\partial^{2}}{(\partial t)^{2}}A\left(t\right) - 2 \, {\left(\cosh\left(r\right)^{5} + 2 \, \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} + \cosh\left(r\right)^{3} - 2 \, \cosh\left(r\right)\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(2 \, \cosh\left(r\right)^{4} + 2 \, \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} - \cosh\left(r\right)^{2} - 1\right)} A\left(t\right) a\left(t, r\right) + 2 \, {\left(A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right)^{3} \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, A\left(t\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{3} \frac{\partial}{\partial t}A\left(t\right) + A\left(t\right) \cosh\left(r\right) \sinh\left(r\right)^{3}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right) + {\left({\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right)\right)} \frac{\partial^{2}}{(\partial t)^{2}}a\left(t, r\right) - 2 \, {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right) a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right) a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right) a\left(t, r\right) \sinh\left(r\right)\right)} \frac{\partial^{2}}{\partial t\partial r}a\left(t, r\right) + {\left({\left(3 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, {\left(3 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, {\left(3 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(3 \, \cosh\left(r\right)^{2} + 1\right)} a\left(t, r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right) + {\left({\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} a\left(t, r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t, r\right)}{a\left(t, r\right)^{2} \cosh\left(r\right)^{6} \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, a\left(t, r\right)^{2} \cosh\left(r\right)^{5} \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) - a\left(t, r\right)^{2} \cosh\left(r\right)^{3}} & 0 \\ 0 & 0 & 0 & -\frac{{\left({\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} a\left(t, r\right) \frac{\partial^{2}}{(\partial t)^{2}}A\left(t\right) \frac{\partial}{\partial t}a\left(t, r\right) - {\left(\cosh\left(r\right)^{4} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - \cosh\left(r\right)^{3} \sinh\left(r\right)^{2}\right)} \frac{\partial}{\partial t}a\left(t, r\right)^{2} - {\left({\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} a\left(t, r\right)\right)} \frac{\partial^{2}}{(\partial t)^{2}}a\left(t, r\right)\right)} B^{2} \sin\left({\theta}\right)^{2} + {\left({\left(A\left(t\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) - A\left(t\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{4}\right)} \frac{\partial}{\partial t}a\left(t, r\right)^{2} + {\left(\cosh\left(r\right)^{4} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t}A\left(t\right) - \cosh\left(r\right) \sinh\left(r\right)^{2}\right)} \frac{\partial}{\partial r}a\left(t, r\right)^{2} - {\left({\left(2 \, \cosh\left(r\right)^{4} \sinh\left(r\right)^{2} + 3 \, \cosh\left(r\right)^{4} - 3 \, \cosh\left(r\right)^{2}\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} a\left(t, r\right) \frac{\partial^{2}}{(\partial t)^{2}}A\left(t\right) - 2 \, {\left(\cosh\left(r\right)^{5} + 2 \, \cosh\left(r\right)^{3} \sinh\left(r\right)^{2} + \cosh\left(r\right)^{3} - 2 \, \cosh\left(r\right)\right)} A\left(t\right) a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(2 \, \cosh\left(r\right)^{4} + 2 \, \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} - \cosh\left(r\right)^{2} - 1\right)} A\left(t\right) a\left(t, r\right) + 2 \, {\left(A\left(t\right) \cosh\left(r\right)^{3} \sinh\left(r\right)^{3} \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, A\left(t\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{3} \frac{\partial}{\partial t}A\left(t\right) + A\left(t\right) \cosh\left(r\right) \sinh\left(r\right)^{3}\right)} \frac{\partial}{\partial r}a\left(t, r\right)\right)} \frac{\partial}{\partial t}a\left(t, r\right) + {\left({\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t\right)^{2} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t\right)^{2} a\left(t, r\right)\right)} \frac{\partial^{2}}{(\partial t)^{2}}a\left(t, r\right) - 2 \, {\left({\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t\right) a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t\right) a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) + {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t\right) a\left(t, r\right) \sinh\left(r\right)\right)} \frac{\partial^{2}}{\partial t\partial r}a\left(t, r\right) + {\left({\left(3 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, {\left(3 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, {\left(3 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} a\left(t, r\right) \sinh\left(r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(3 \, \cosh\left(r\right)^{2} + 1\right)} a\left(t, r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t, r\right) + {\left({\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} a\left(t, r\right) \frac{\partial}{\partial t}A\left(t\right) - {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} a\left(t, r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t, r\right)\right)} \sin\left({\theta}\right)^{2}}{a\left(t, r\right)^{2} \cosh\left(r\right)^{6} \frac{\partial}{\partial t}A\left(t\right)^{3} - 3 \, a\left(t, r\right)^{2} \cosh\left(r\right)^{5} \frac{\partial}{\partial t}A\left(t\right)^{2} + 3 \, a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \frac{\partial}{\partial t}A\left(t\right) - a\left(t, r\right)^{2} \cosh\left(r\right)^{3}} \end{array}\right)$

Ricci4_scalar = g.ricci_scalar()

u = Man.vector_field('u')
g[0,0]
u[0] = 1/(a.expr()*gfactor.expr()) # 1/sqrt(-g[0,0].expr())
u.display()

$\displaystyle -{\left(\cosh\left(r\right) \frac{\partial\,A}{\partial t} - 1\right)}^{2} a\left(t, r\right)^{2}$

$\displaystyle u = -\frac{1}{{\left(\cosh\left(r\right) \frac{\partial\,A}{\partial t} - 1\right)} a\left(t, r\right)} \frac{\partial}{\partial t }$

g(u,u).expr()

$\displaystyle -1$

u_form = u.down(g) # the 1-form associated to u by metric duality
u_form.display()
pressure = 0
T = (rho+pressure)*(u_form*u_form) + pressure*g - Lambda*g/(8*pi)
S =Phi.pullback(T)
S.display()
trS = S['_ab']*gam3_inv['^ab']
trS.display()
T.set_name('T')
print(T)
T.display()
T(nv,nv).display()

$\displaystyle \left( a\left(t, r\right) \cosh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right) \right) \mathrm{d} t + A\left(t\right) a\left(t, r\right) \sinh\left(r\right) \mathrm{d} r$

$\displaystyle \left( \frac{8 \, \pi A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{2} - B^{2} \Lambda a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} + \Lambda A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2}}{8 \, \pi} \right) \mathrm{d} r\otimes \mathrm{d} r -\frac{B^{2} \Lambda a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2}}{8 \, \pi} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} -\frac{B^{2} \Lambda a\left(t_{0}, r\right)^{2} \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{2}}{8 \, \pi} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

$\displaystyle \begin{array}{llcl} & \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & \frac{8 \, \pi A\left(t_{0}\right)^{2} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{2} - 3 \, B^{2} \Lambda \cosh\left(r\right)^{2} + 3 \, \Lambda A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2}}{8 \, {\left(\pi B^{2} \cosh\left(r\right)^{2} - \pi A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2}\right)}} \end{array}$

Field of symmetric bilinear forms T on the 4-dimensional differentiable manifold M

$\displaystyle T = \left( \frac{8 \, \pi a\left(t, r\right)^{2} \rho\left(t, r\right) + \Lambda a\left(t, r\right)^{2} + {\left(8 \, \pi a\left(t, r\right)^{2} \cosh\left(r\right)^{2} \rho\left(t, r\right) + \Lambda a\left(t, r\right)^{2} \cosh\left(r\right)^{2}\right)} \left(\frac{\partial\,A}{\partial t}\right)^{2} - 2 \, {\left(8 \, \pi a\left(t, r\right)^{2} \cosh\left(r\right) \rho\left(t, r\right) + \Lambda a\left(t, r\right)^{2} \cosh\left(r\right)\right)} \frac{\partial\,A}{\partial t}}{8 \, \pi} \right) \mathrm{d} t\otimes \mathrm{d} t + \left( -\frac{8 \, \pi A\left(t\right) a\left(t, r\right)^{2} \rho\left(t, r\right) \sinh\left(r\right) + \Lambda A\left(t\right) a\left(t, r\right)^{2} \sinh\left(r\right) - {\left(8 \, \pi A\left(t\right) a\left(t, r\right)^{2} \cosh\left(r\right) \rho\left(t, r\right) \sinh\left(r\right) + \Lambda A\left(t\right) a\left(t, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t}}{8 \, \pi} \right) \mathrm{d} t\otimes \mathrm{d} r + \left( -\frac{8 \, \pi A\left(t\right) a\left(t, r\right)^{2} \rho\left(t, r\right) \sinh\left(r\right) + \Lambda A\left(t\right) a\left(t, r\right)^{2} \sinh\left(r\right) - {\left(8 \, \pi A\left(t\right) a\left(t, r\right)^{2} \cosh\left(r\right) \rho\left(t, r\right) \sinh\left(r\right) + \Lambda A\left(t\right) a\left(t, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t}}{8 \, \pi} \right) \mathrm{d} r\otimes \mathrm{d} t + \left( \frac{8 \, \pi A\left(t\right)^{2} a\left(t, r\right)^{2} \rho\left(t, r\right) \sinh\left(r\right)^{2} - B^{2} \Lambda a\left(t, r\right)^{2} \cosh\left(r\right)^{2} + \Lambda A\left(t\right)^{2} a\left(t, r\right)^{2} \sinh\left(r\right)^{2}}{8 \, \pi} \right) \mathrm{d} r\otimes \mathrm{d} r -\frac{B^{2} \Lambda a\left(t, r\right)^{2} \sinh\left(r\right)^{2}}{8 \, \pi} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} -\frac{B^{2} \Lambda a\left(t, r\right)^{2} \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{2}}{8 \, \pi} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

$\displaystyle \begin{array}{llcl} & \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & -\frac{\Lambda A\left(t\right)^{2} \sinh\left(r\right)^{2} - {\left(8 \, \pi \cosh\left(r\right)^{2} \rho\left(t, r\right) + \Lambda \cosh\left(r\right)^{2}\right)} B^{2}}{8 \, {\left(\pi B^{2} \cosh\left(r\right)^{2} - \pi A\left(t\right)^{2} \sinh\left(r\right)^{2}\right)}} \end{array}$

DT = nablaM(T)
divT = DT['_abc']*ginv['^bc']
#divT = divT.expr().simplify_full()

divT.display()

$\displaystyle \left( -\frac{3 \, \rho\left(t, r\right) \frac{\partial\,a}{\partial t} + a\left(t, r\right) \frac{\partial\,\rho}{\partial t}}{a\left(t, r\right)} \right) \mathrm{d} t + \left( -\frac{4 \, A\left(t\right) \rho\left(t, r\right) \sinh\left(r\right) \frac{\partial\,a}{\partial t} + A\left(t\right) a\left(t, r\right) \sinh\left(r\right) \frac{\partial\,\rho}{\partial t} - {\left(\cosh\left(r\right) \rho\left(t, r\right) \frac{\partial\,A}{\partial t} - \rho\left(t, r\right)\right)} \frac{\partial\,a}{\partial r}}{a\left(t, r\right) \cosh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)} \right) \mathrm{d} r$

E = Phi.pullback(T(nv,nv))
E.display()

$\displaystyle \begin{array}{llcl} & \Sigma & \longrightarrow & \mathbb{R} \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & -\frac{\Lambda A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2} - {\left(8 \, \pi \cosh\left(r\right)^{2} \rho\left(t_{0}, r\right) + \Lambda \cosh\left(r\right)^{2}\right)} B^{2}}{8 \, {\left(\pi B^{2} \cosh\left(r\right)^{2} - \pi A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2}\right)}} \end{array}$

Ham =  Ricci3_scalar + trK3^2 - trKK3 - 16*pi*E #Ricci_scalar (+ Ktrace^2 - K2 -16*pi*E)
#Ham = Ham.expr().simplify_full()

Ham.expr().simplify_full()

$\displaystyle \frac{2 \, {\left(3 \, {\left(\cosh\left(r\right)^{8} - 4 \, \cosh\left(r\right)^{6} + 6 \, \cosh\left(r\right)^{4} - 4 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t_{0}\right)^{8} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right)^{2} - {\left(8 \, \pi a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{8} \rho\left(t_{0}, r\right) \sinh\left(r\right) + \Lambda a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{8} \sinh\left(r\right) - 3 \, \cosh\left(r\right)^{8} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right)^{2} + {\left(8 \, \pi a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{10} \rho\left(t_{0}, r\right) \sinh\left(r\right) + \Lambda a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{10} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(8 \, \pi a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{9} \rho\left(t_{0}, r\right) \sinh\left(r\right) + \Lambda a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{9} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} B^{8} + {\left(16 \, \pi A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{6} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{3} + 3 \, \Lambda A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{6} \sinh\left(r\right)^{3} - 12 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right)^{2} + {\left(16 \, \pi A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{8} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{3} + 3 \, \Lambda A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{8} \sinh\left(r\right)^{3}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} + {\left(\cosh\left(r\right)^{8} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, \cosh\left(r\right)^{7} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + \cosh\left(r\right)^{6} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} - 2 \, {\left(16 \, \pi A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{7} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{3} + 3 \, \Lambda A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{7} \sinh\left(r\right)^{3}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + 2 \, {\left({\left(2 \, \cosh\left(r\right)^{8} + \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \sinh\left(r\right) + 3 \, {\left({\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - 2 \, {\left({\left(\cosh\left(r\right)^{9} + \cosh\left(r\right)^{7}\right)} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} + \cosh\left(r\right)^{6}\right)} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) - 2 \, {\left(a\left(t_{0}, r\right) \cosh\left(r\right)^{8} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, a\left(t_{0}, r\right) \cosh\left(r\right)^{7} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + a\left(t_{0}, r\right) \cosh\left(r\right)^{6} \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{6} - {\left(8 \, \pi A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{4} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{5} + 3 \, \Lambda A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{4} \sinh\left(r\right)^{5} - 18 \, {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right)^{2} + {\left(8 \, \pi A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{6} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{5} + 3 \, \Lambda A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{6} \sinh\left(r\right)^{5}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} - 2 \, {\left(8 \, \pi A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{5} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{5} + 3 \, \Lambda A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{5} \sinh\left(r\right)^{5}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + 2 \, {\left(2 \, {\left(3 \, \cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - 2 \, {\left(3 \, \cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) + 9 \, {\left({\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{3} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3}\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - 4 \, {\left({\left(2 \, \cosh\left(r\right)^{9} - \cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(2 \, \cosh\left(r\right)^{8} - \cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(2 \, \cosh\left(r\right)^{7} - \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) - 4 \, {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{4} + {\left(\Lambda A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{4} \sinh\left(r\right)^{7} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, \Lambda A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{3} \sinh\left(r\right)^{7} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + \Lambda A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{2} \sinh\left(r\right)^{7} - 12 \, {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right)^{2} - 5 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} + 2 \, {\left({\left(6 \, \cosh\left(r\right)^{8} - 11 \, \cosh\left(r\right)^{6} + 4 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(6 \, \cosh\left(r\right)^{7} - 11 \, \cosh\left(r\right)^{5} + 4 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \sinh\left(r\right) + 9 \, {\left({\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{5} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{5}\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - 2 \, {\left({\left(5 \, \cosh\left(r\right)^{9} - 9 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(5 \, \cosh\left(r\right)^{8} - 9 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(5 \, \cosh\left(r\right)^{7} - 9 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) - 2 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{2} + 3 \, {\left({\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{6} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{6} - 3 \, \cosh\left(r\right)^{4} + 3 \, \cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{6} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} - 2 \, {\left(2 \, {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{7} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - 2 \, {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{7} a\left(t_{0}, r\right) \sinh\left(r\right) + 3 \, {\left({\left(\cosh\left(r\right)^{9} - 4 \, \cosh\left(r\right)^{7} + 6 \, \cosh\left(r\right)^{5} - 4 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t_{0}\right)^{7} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{8} - 4 \, \cosh\left(r\right)^{6} + 6 \, \cosh\left(r\right)^{4} - 4 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t_{0}\right)^{7}\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) + 4 \, {\left({\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)}}{{\left(a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{10} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{9} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{8} \sinh\left(r\right)\right)} B^{8} - 3 \, {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{8} \sinh\left(r\right)^{3} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{7} \sinh\left(r\right)^{3} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{6} \sinh\left(r\right)^{3}\right)} B^{6} + 3 \, {\left(A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{6} \sinh\left(r\right)^{5} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{5} \sinh\left(r\right)^{5} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{4} \sinh\left(r\right)^{5}\right)} B^{4} - {\left(A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{4} \sinh\left(r\right)^{7} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{3} \sinh\left(r\right)^{7} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{2} \sinh\left(r\right)^{7}\right)} B^{2}}$

Dbeta1f = D(bta1f)
Dbeta1f.display()

$\displaystyle \left( \frac{A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{3} \sinh\left(r\right)^{4} \frac{\partial\,A}{\partial t_{0}} - {\left({\left(\sinh\left(r\right)^{4} + 2 \, \sinh\left(r\right)^{2} + 1\right)} a\left(t_{0}, r\right)^{3} \frac{\partial\,A}{\partial t_{0}} - a\left(t_{0}, r\right)^{3} \cosh\left(r\right) + 2 \, {\left({\left(\cosh\left(r\right) \sinh\left(r\right)^{3} + \cosh\left(r\right) \sinh\left(r\right)\right)} a\left(t_{0}, r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - {\left(\sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} a\left(t_{0}, r\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} + 2 \, {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t_{0}} - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}}{B^{2} \cosh\left(r\right)^{2} - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2}} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( -\frac{{\left(a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - a\left(t_{0}, r\right)^{3} \cosh\left(r\right) \sinh\left(r\right)^{2} + {\left(a\left(t_{0}, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t_{0}} - a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2}}{B^{2} \cosh\left(r\right)^{2} - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2}} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \left( -\frac{{\left(a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - a\left(t_{0}, r\right)^{3} \cosh\left(r\right) \sinh\left(r\right)^{2} + {\left(a\left(t_{0}, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t_{0}} - a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{3}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} \sin\left({\theta}\right)^{2}}{B^{2} \cosh\left(r\right)^{2} - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

# metric evolution
#Nsig = Sig.scalar_field(function('N')(r),name='N')
#Nsig = N_0 #Phi.pullback(N)
#N.variables()# (t=t_0,r)
gamma_dot = -2*Nsig*K3+2*Dbeta1f
gamma_dot.display()
gamma_dot[:]
gamma_dot[1,1].expr().simplify_full()

$\displaystyle \left( \frac{2 \, {\left(B^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{4} \frac{\partial\,a}{\partial t_{0}} + {\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{3} \frac{\partial\,A}{\partial t_{0}} + {\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \frac{\partial\,a}{\partial t_{0}} - {\left(2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \frac{\partial\,a}{\partial t_{0}} + A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \cosh\left(r\right) - a\left(t_{0}, r\right)^{3} \cosh\left(r\right) + {\left(a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{4} - A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2}\right)} \frac{\partial\,A}{\partial t_{0}} + {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) - 2 \, a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right) - {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) - 2 \, a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) - 2 \, {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - {\left({\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) - 2 \, {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \frac{\partial\,a}{\partial r}\right)}}{B^{2} \cosh\left(r\right)^{2} - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2}} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( \frac{2 \, {\left(B^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial t_{0}} - {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)^{4} \frac{\partial\,a}{\partial t_{0}} + A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2} - a\left(t_{0}, r\right)^{3} \cosh\left(r\right) \sinh\left(r\right)^{2} - {\left(A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} - a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial\,A}{\partial t_{0}} + {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \sinh\left(r\right)^{3} - a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{3} - {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right)^{3} - a\left(t_{0}, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{3}\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2}\right)}}{B^{2} \cosh\left(r\right)^{2} - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2}} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \left( \frac{2 \, {\left(B^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,a}{\partial t_{0}} - {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)^{4} \frac{\partial\,a}{\partial t_{0}} + A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2} - a\left(t_{0}, r\right)^{3} \cosh\left(r\right) \sinh\left(r\right)^{2} - {\left(A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} - a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial\,A}{\partial t_{0}} + {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \sinh\left(r\right)^{3} - a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{3} - {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right)^{3} - a\left(t_{0}, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{3}\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \frac{\partial\,a}{\partial r}\right)} B^{2} \sin\left({\theta}\right)^{2}\right)}}{B^{2} \cosh\left(r\right)^{2} - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

$\displaystyle \left(\begin{array}{rrr} \frac{2 \, {\left(B^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) + {\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{3} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - {\left(2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) + A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \cosh\left(r\right) - a\left(t_{0}, r\right)^{3} \cosh\left(r\right) + {\left(a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{4} - A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) - 2 \, a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right) - {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) - 2 \, a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) - 2 \, {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - {\left({\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) - 2 \, {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)}}{B^{2} \cosh\left(r\right)^{2} - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2}} & 0 & 0 \\ 0 & \frac{2 \, {\left(B^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) + A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2} - a\left(t_{0}, r\right)^{3} \cosh\left(r\right) \sinh\left(r\right)^{2} - {\left(A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} - a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \sinh\left(r\right)^{3} - a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{3} - {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right)^{3} - a\left(t_{0}, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{3}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} B^{2}\right)}}{B^{2} \cosh\left(r\right)^{2} - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2}} & 0 \\ 0 & 0 & \frac{2 \, {\left(B^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sin\left({\theta}\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) + A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2} - a\left(t_{0}, r\right)^{3} \cosh\left(r\right) \sinh\left(r\right)^{2} - {\left(A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} - a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \sinh\left(r\right)^{3} - a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{3} - {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right)^{3} - a\left(t_{0}, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{3}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} B^{2} \sin\left({\theta}\right)^{2}\right)}}{B^{2} \cosh\left(r\right)^{2} - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2}} \end{array}\right)$

$\displaystyle \frac{2 \, {\left(B^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) + {\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{3} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - {\left(2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) + A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \cosh\left(r\right) - a\left(t_{0}, r\right)^{3} \cosh\left(r\right) + {\left(a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{4} - A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) - 2 \, a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right) - {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) - 2 \, a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) - 2 \, {\left(\cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - {\left({\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) - 2 \, {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)}}{B^{2} \cosh\left(r\right)^{2} - A\left(t_{0}\right)^{2} \sinh\left(r\right)^{2}}$

#Nsig = Sig.scalar_field(cosh(r))
D2N = D(D(Nsig))
D2N[:]

$\displaystyle \left(\begin{array}{rrr} -\frac{{\left(a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + a\left(t_{0}, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial r}a\left(t_{0}, r\right) - {\left(\cosh\left(r\right)^{6} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - \cosh\left(r\right)^{5}\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} + {\left(a\left(t_{0}, r\right) \cosh\left(r\right)^{6} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - a\left(t_{0}, r\right) \cosh\left(r\right)^{5}\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{5} - {\left({\left(2 \, \cosh\left(r\right)^{4} - 3 \, \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(3 \, \cosh\left(r\right)^{3} - 4 \, \cosh\left(r\right)\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} - 2 \, {\left({\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2}\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} + 2 \, {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) + 2 \, {\left({\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{3} + {\left({\left(\cosh\left(r\right)^{4} + \cosh\left(r\right)^{2} - 2\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - 3 \, {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} - {\left({\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t_{0}\right)^{4}\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} - {\left(2 \, {\left(\cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{4} - 1\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) + {\left({\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B}{{\left(B^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{4} - 2 \, B^{2} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} + A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right)^{4}\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)}} & 0 & 0 \\ 0 & -\frac{{\left({\left(\sinh\left(r\right)^{6} + 2 \, \sinh\left(r\right)^{4} + \sinh\left(r\right)^{2}\right)} a\left(t_{0}, r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right) \sinh\left(r\right)^{4} + \cosh\left(r\right) \sinh\left(r\right)^{2} - {\left(\sinh\left(r\right)^{6} + 2 \, \sinh\left(r\right)^{4} + \sinh\left(r\right)^{2}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} + {\left({\left(2 \, \cosh\left(r\right) \sinh\left(r\right)^{5} + 3 \, \cosh\left(r\right) \sinh\left(r\right)^{3} + \cosh\left(r\right) \sinh\left(r\right)\right)} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\sinh\left(r\right)^{5} + 2 \, \sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} B^{5} - {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2} + {\left(\sinh\left(r\right)^{6} - \sinh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(A\left(t_{0}\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{4} - {\left(\sinh\left(r\right)^{6} + \sinh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} + {\left(2 \, A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right)^{5} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)^{5}\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} B^{3}}{{\left(B^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{4} - 2 \, B^{2} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} + A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right)^{4}\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)}} & 0 \\ 0 & 0 & -\frac{{\left({\left(\sinh\left(r\right)^{6} + 2 \, \sinh\left(r\right)^{4} + \sinh\left(r\right)^{2}\right)} a\left(t_{0}, r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right) \sinh\left(r\right)^{4} + \cosh\left(r\right) \sinh\left(r\right)^{2} - {\left(\sinh\left(r\right)^{6} + 2 \, \sinh\left(r\right)^{4} + \sinh\left(r\right)^{2}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} + {\left({\left(2 \, \cosh\left(r\right) \sinh\left(r\right)^{5} + 3 \, \cosh\left(r\right) \sinh\left(r\right)^{3} + \cosh\left(r\right) \sinh\left(r\right)\right)} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\sinh\left(r\right)^{5} + 2 \, \sinh\left(r\right)^{3} + \sinh\left(r\right)\right)} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} B^{5} \sin\left({\theta}\right)^{2} - {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{2} + {\left(\sinh\left(r\right)^{6} - \sinh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(A\left(t_{0}\right)^{2} \cosh\left(r\right) \sinh\left(r\right)^{4} - {\left(\sinh\left(r\right)^{6} + \sinh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} + {\left(2 \, A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right)^{5} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)^{5}\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} B^{3} \sin\left({\theta}\right)^{2}}{{\left(B^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{4} - 2 \, B^{2} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{2} + A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right)^{4}\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)}} \end{array}\right)$

LbetaK = K3.lie_derivative(bta)
LbetaK[:]

$\displaystyle \left(\begin{array}{rrr} \frac{{\left({\left({\left(2 \, a\left(t_{0}, r\right) \cosh\left(r\right)^{6} - {\left(\cosh\left(r\right)^{9} + \cosh\left(r\right)^{7}\right)} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - 2 \, {\left(\cosh\left(r\right)^{8} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - \cosh\left(r\right)^{7} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - {\left(a\left(t_{0}, r\right) \cosh\left(r\right)^{8} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - a\left(t_{0}, r\right) \cosh\left(r\right)^{7} \sinh\left(r\right)\right)} \frac{\partial^{2}}{\partial t_{0}\partial r}a\left(t_{0}, r\right)\right)} B^{6} + {\left({\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 4 \, {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(3 \, \cosh\left(r\right)^{5} - 5 \, \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} - 2 \, {\left({\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} + {\left({\left(3 \, \cosh\left(r\right)^{9} - \cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - 5 \, {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) + 6 \, {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) + 3 \, {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial^{2}}{\partial t_{0}\partial r}a\left(t_{0}, r\right) + {\left(12 \, A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{5} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{8} + 5 \, \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} + {\left(\cosh\left(r\right)^{6} - 7 \, \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) - {\left({\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{4} - {\left({\left(\cosh\left(r\right)^{7} + \cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(2 \, \cosh\left(r\right)^{6} - \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + 3 \, {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} - 4 \, {\left({\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{3} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3}\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} + {\left({\left(3 \, \cosh\left(r\right)^{9} - 5 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - 4 \, {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) + 6 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) + 3 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial^{2}}{\partial t_{0}\partial r}a\left(t_{0}, r\right) - 2 \, {\left({\left(\cosh\left(r\right)^{8} + \cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 6 \, {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{6} - 5 \, \cosh\left(r\right)^{4} + 4 \, \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) - 2 \, {\left({\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{2} - 2 \, {\left({\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{5} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{5} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{5}\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} + {\left({\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{6} - 3 \, \cosh\left(r\right)^{4} + 3 \, \cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) + 2 \, {\left({\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{6} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) + {\left({\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial^{2}}{\partial t_{0}\partial r}a\left(t_{0}, r\right) - {\left({\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - {\left(\cosh\left(r\right)^{6} - 3 \, \cosh\left(r\right)^{4} + 3 \, \cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) - {\left({\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)}}{{\left(\cosh\left(r\right)^{9} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - \cosh\left(r\right)^{8}\right)} B^{7} - 3 \, {\left(A\left(t_{0}\right)^{2} \cosh\left(r\right)^{7} \sinh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{2} \cosh\left(r\right)^{6} \sinh\left(r\right)^{2}\right)} B^{5} + 3 \, {\left(A\left(t_{0}\right)^{4} \cosh\left(r\right)^{5} \sinh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{4} \cosh\left(r\right)^{4} \sinh\left(r\right)^{4}\right)} B^{3} - {\left(A\left(t_{0}\right)^{6} \cosh\left(r\right)^{3} \sinh\left(r\right)^{6} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{6} \cosh\left(r\right)^{2} \sinh\left(r\right)^{6}\right)} B} & 0 & 0 \\ 0 & -\frac{{\left({\left({\left(2 \, a\left(t_{0}, r\right) \cosh\left(r\right)^{6} \sinh\left(r\right)^{2} - {\left(2 \, \cosh\left(r\right)^{7} \sinh\left(r\right)^{2} - \cosh\left(r\right)^{5} \sinh\left(r\right)^{4}\right)} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - {\left(a\left(t_{0}, r\right) \cosh\left(r\right)^{6} \sinh\left(r\right)^{3} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - a\left(t_{0}, r\right) \cosh\left(r\right)^{5} \sinh\left(r\right)^{3}\right)} \frac{\partial^{2}}{\partial t_{0}\partial r}a\left(t_{0}, r\right)\right)} B^{5} - {\left({\left(2 \, \cosh\left(r\right)^{7} \sinh\left(r\right)^{2} - \cosh\left(r\right)^{5} \sinh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(2 \, \cosh\left(r\right)^{6} \sinh\left(r\right)^{2} - \cosh\left(r\right)^{4} \sinh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(2 \, \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} - \cosh\left(r\right)^{3} \sinh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} - {\left({\left(5 \, \cosh\left(r\right)^{5} \sinh\left(r\right)^{4} - 3 \, \cosh\left(r\right)^{3} \sinh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(5 \, \cosh\left(r\right)^{4} \sinh\left(r\right)^{4} - \cosh\left(r\right)^{2} \sinh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - 2 \, {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right)^{5} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right)^{5}\right)} \frac{\partial^{2}}{\partial t_{0}\partial r}a\left(t_{0}, r\right) + 2 \, {\left({\left(2 \, \cosh\left(r\right)^{6} \sinh\left(r\right)^{3} - \cosh\left(r\right)^{4} \sinh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(2 \, \cosh\left(r\right)^{5} \sinh\left(r\right)^{3} - \cosh\left(r\right)^{3} \sinh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(2 \, \cosh\left(r\right)^{4} \sinh\left(r\right)^{3} - \cosh\left(r\right)^{2} \sinh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) + {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{5} \sinh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right)^{4}\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{3} + {\left(A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{5} \sinh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} - {\left({\left(3 \, \cosh\left(r\right)^{3} \sinh\left(r\right)^{6} - 2 \, \cosh\left(r\right) \sinh\left(r\right)^{8}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(3 \, \cosh\left(r\right)^{2} \sinh\left(r\right)^{6} - \sinh\left(r\right)^{8}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - {\left(A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{7} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right)^{7}\right)} \frac{\partial^{2}}{\partial t_{0}\partial r}a\left(t_{0}, r\right) + {\left({\left(3 \, \cosh\left(r\right)^{4} \sinh\left(r\right)^{5} - \cosh\left(r\right)^{2} \sinh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(3 \, \cosh\left(r\right)^{3} \sinh\left(r\right)^{5} - \cosh\left(r\right) \sinh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(3 \, \cosh\left(r\right)^{2} \sinh\left(r\right)^{5} - \sinh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) + {\left(A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right)^{6} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{6} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right)^{6}\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)}}{A\left(t_{0}\right)^{6} \cosh\left(r\right)^{3} \sinh\left(r\right)^{6} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{6} \cosh\left(r\right)^{2} \sinh\left(r\right)^{6} - {\left(\cosh\left(r\right)^{9} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - \cosh\left(r\right)^{8}\right)} B^{6} + 3 \, {\left(A\left(t_{0}\right)^{2} \cosh\left(r\right)^{7} \sinh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{2} \cosh\left(r\right)^{6} \sinh\left(r\right)^{2}\right)} B^{4} - 3 \, {\left(A\left(t_{0}\right)^{4} \cosh\left(r\right)^{5} \sinh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{4} \cosh\left(r\right)^{4} \sinh\left(r\right)^{4}\right)} B^{2}} & 0 \\ 0 & 0 & -\frac{{\left({\left({\left(2 \, a\left(t_{0}, r\right) \cosh\left(r\right)^{6} \sinh\left(r\right)^{2} - {\left(2 \, \cosh\left(r\right)^{7} \sinh\left(r\right)^{2} - \cosh\left(r\right)^{5} \sinh\left(r\right)^{4}\right)} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - {\left(a\left(t_{0}, r\right) \cosh\left(r\right)^{6} \sinh\left(r\right)^{3} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - a\left(t_{0}, r\right) \cosh\left(r\right)^{5} \sinh\left(r\right)^{3}\right)} \frac{\partial^{2}}{\partial t_{0}\partial r}a\left(t_{0}, r\right)\right)} B^{5} \sin\left({\theta}\right)^{2} - {\left({\left(2 \, \cosh\left(r\right)^{7} \sinh\left(r\right)^{2} - \cosh\left(r\right)^{5} \sinh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(2 \, \cosh\left(r\right)^{6} \sinh\left(r\right)^{2} - \cosh\left(r\right)^{4} \sinh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(2 \, \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} - \cosh\left(r\right)^{3} \sinh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} - {\left({\left(5 \, \cosh\left(r\right)^{5} \sinh\left(r\right)^{4} - 3 \, \cosh\left(r\right)^{3} \sinh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(5 \, \cosh\left(r\right)^{4} \sinh\left(r\right)^{4} - \cosh\left(r\right)^{2} \sinh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - 2 \, {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right)^{5} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right)^{5}\right)} \frac{\partial^{2}}{\partial t_{0}\partial r}a\left(t_{0}, r\right) + 2 \, {\left({\left(2 \, \cosh\left(r\right)^{6} \sinh\left(r\right)^{3} - \cosh\left(r\right)^{4} \sinh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(2 \, \cosh\left(r\right)^{5} \sinh\left(r\right)^{3} - \cosh\left(r\right)^{3} \sinh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(2 \, \cosh\left(r\right)^{4} \sinh\left(r\right)^{3} - \cosh\left(r\right)^{2} \sinh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) + {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{5} \sinh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right)^{4}\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{3} \sin\left({\theta}\right)^{2} + {\left(A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{5} \sinh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} - {\left({\left(3 \, \cosh\left(r\right)^{3} \sinh\left(r\right)^{6} - 2 \, \cosh\left(r\right) \sinh\left(r\right)^{8}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(3 \, \cosh\left(r\right)^{2} \sinh\left(r\right)^{6} - \sinh\left(r\right)^{8}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - {\left(A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{7} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right)^{7}\right)} \frac{\partial^{2}}{\partial t_{0}\partial r}a\left(t_{0}, r\right) + {\left({\left(3 \, \cosh\left(r\right)^{4} \sinh\left(r\right)^{5} - \cosh\left(r\right)^{2} \sinh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(3 \, \cosh\left(r\right)^{3} \sinh\left(r\right)^{5} - \cosh\left(r\right) \sinh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(3 \, \cosh\left(r\right)^{2} \sinh\left(r\right)^{5} - \sinh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) + {\left(A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right)^{6} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{6} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \cosh\left(r\right) \sinh\left(r\right)^{6}\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B \sin\left({\theta}\right)^{2}\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)}}{A\left(t_{0}\right)^{6} \cosh\left(r\right)^{3} \sinh\left(r\right)^{6} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{6} \cosh\left(r\right)^{2} \sinh\left(r\right)^{6} - {\left(\cosh\left(r\right)^{9} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - \cosh\left(r\right)^{8}\right)} B^{6} + 3 \, {\left(A\left(t_{0}\right)^{2} \cosh\left(r\right)^{7} \sinh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{2} \cosh\left(r\right)^{6} \sinh\left(r\right)^{2}\right)} B^{4} - 3 \, {\left(A\left(t_{0}\right)^{4} \cosh\left(r\right)^{5} \sinh\left(r\right)^{4} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{4} \cosh\left(r\right)^{4} \sinh\left(r\right)^{4}\right)} B^{2}} \end{array}\right)$

#KK = [[sum(Kij3[i][k]*sum(g3inv[k][p]*Kij3[p][j] for p in R3) for k in R3) for j in R3] for i in R3]
#RK = [[(Ricci30[i][j] + Ktrace*Kij3[i][j] + KK[i][j] ) for j in R3] for i in R3]  # -4*pi*E*g3[i][j]RK = 
KK = K3u['^a_b']*K3['_ca']
RK = Ricci3 + K3u.trace()*K3 - 2*KK +4*pi*((trS - E)*gam3 - 2*S)

K3_dot = LbetaK - D2N + Nsig*RK
K3_dot.display()
K3_dot[:]
K3_dotu = K3_dot.up(gam3,1)
K3_dotu.display()

ParseError: KaTeX parse error: Unexpected end of input in a macro argument, expected '}' at end of input: …\, \cosh\left(r...]

WARNING: Output: 61738 truncated by MAX_HTML_SIZE to 40000. Type 'smc?' to learn how to raise the output limit.

$\displaystyle \left(\begin{array}{rrr} -\frac{{\left({\left(\cosh\left(r\right)^{9} - 4 \, \cosh\left(r\right)^{7} + 6 \, \cosh\left(r\right)^{5} - 4 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t_{0}\right)^{8} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right)^{2} - {\left(4 \, \pi a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{9} \rho\left(t_{0}, r\right) \sinh\left(r\right) + \Lambda a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{9} \sinh\left(r\right) - \cosh\left(r\right)^{9} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right)^{2} + {\left(4 \, \pi a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{11} \rho\left(t_{0}, r\right) \sinh\left(r\right) + \Lambda a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{11} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(4 \, \pi a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{10} \rho\left(t_{0}, r\right) \sinh\left(r\right) + \Lambda a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{10} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} B^{8} + {\left(4 \, \pi A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{7} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{3} + 3 \, \Lambda A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{7} \sinh\left(r\right)^{3} - 4 \, {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right)^{2} + {\left(4 \, \pi A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{9} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{3} + 3 \, \Lambda A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{9} \sinh\left(r\right)^{3} - a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{7} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} + 3 \, {\left(\cosh\left(r\right)^{9} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, \cosh\left(r\right)^{8} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + \cosh\left(r\right)^{7} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} - {\left(8 \, \pi A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{8} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{3} + 6 \, \Lambda A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{8} \sinh\left(r\right)^{3} - a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{6} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(2 \, A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{8} \sinh\left(r\right) + 2 \, a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{6} \sinh\left(r\right) - {\left(2 \, A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{9} \sinh\left(r\right) + {\left(\cosh\left(r\right)^{9} + \cosh\left(r\right)^{7}\right)} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + 2 \, {\left(a\left(t_{0}, r\right) \cosh\left(r\right)^{7} \sinh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right) - {\left(a\left(t_{0}, r\right) \cosh\left(r\right)^{8} \sinh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{10} - \cosh\left(r\right)^{8}\right)} A\left(t_{0}\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) + {\left(a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{8} \sinh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{7} \sinh\left(r\right)^{2}\right)} \frac{\partial^{2}}{\partial t_{0}\partial r}a\left(t_{0}, r\right) - {\left(2 \, a\left(t_{0}, r\right) \cosh\left(r\right)^{8} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} + {\left(\cosh\left(r\right)^{7} \sinh\left(r\right)^{2} - 4 \, \cosh\left(r\right)^{7}\right)} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{6} \sinh\left(r\right)^{2} - 2 \, \cosh\left(r\right)^{6}\right)} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) - 3 \, {\left(a\left(t_{0}, r\right) \cosh\left(r\right)^{9} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, a\left(t_{0}, r\right) \cosh\left(r\right)^{8} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + a\left(t_{0}, r\right) \cosh\left(r\right)^{7} \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{6} + {\left(4 \, \pi A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{5} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{5} - 3 \, \Lambda A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{5} \sinh\left(r\right)^{5} + 6 \, {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{4} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right)^{2} + {\left(3 \, \cosh\left(r\right)^{5} - 5 \, \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - {\left(3 \, \cosh\left(r\right)^{5} - 5 \, \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{3} \sinh\left(r\right) + {\left(4 \, \pi A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{7} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{5} - 3 \, \Lambda A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{7} \sinh\left(r\right)^{5} + 2 \, {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - {\left(5 \, {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} \sinh\left(r\right) - 2 \, {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \sinh\left(r\right) + {\left(5 \, {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{2} \sinh\left(r\right) - 2 \, {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(5 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} \sinh\left(r\right) - 2 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} - {\left(8 \, \pi A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{6} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{5} - 6 \, \Lambda A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{6} \sinh\left(r\right)^{5} + {\left(5 \, \cosh\left(r\right)^{6} - 9 \, \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - 4 \, {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(6 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) + 5 \, {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - {\left(6 \, {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left(3 \, \cosh\left(r\right)^{9} - \cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + 6 \, {\left({\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{3} - {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{10} - 2 \, \cosh\left(r\right)^{8} + \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{3}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - 3 \, {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial^{2}}{\partial t_{0}\partial r}a\left(t_{0}, r\right) - {\left({\left(\cosh\left(r\right)^{6} - 7 \, \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6} \sinh\left(r\right)^{2} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) - {\left({\left(\cosh\left(r\right)^{8} + 5 \, \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} + 2 \, {\left(\cosh\left(r\right)^{10} - \cosh\left(r\right)^{6} \sinh\left(r\right)^{2} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} + 2 \, {\left(6 \, A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} + {\left(2 \, \cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{5} - {\left(\cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} \sinh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) + {\left(6 \, {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \sinh\left(r\right) + {\left(6 \, {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(6 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{4} - {\left(4 \, \pi A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{3} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{7} - \Lambda A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{3} \sinh\left(r\right)^{7} + 4 \, {\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{6} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right)^{2} + 3 \, {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - 3 \, {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{3} \sinh\left(r\right) + {\left(4 \, \pi A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{5} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{7} - \Lambda A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{5} \sinh\left(r\right)^{7} + {\left(\cosh\left(r\right)^{7} + \cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - {\left(\cosh\left(r\right)^{7} + \cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - {\left({\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} \sinh\left(r\right) - 4 \, {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left({\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{4} \sinh\left(r\right) - 4 \, {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} \sinh\left(r\right) - 4 \, {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} - 2 \, {\left(4 \, \pi A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{4} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{7} - \Lambda A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{4} \sinh\left(r\right)^{7} + {\left(2 \, \cosh\left(r\right)^{6} - \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - {\left(2 \, \cosh\left(r\right)^{6} - \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(6 \, {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \sinh\left(r\right) + 4 \, {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - {\left(6 \, {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left(3 \, \cosh\left(r\right)^{9} - 5 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + 6 \, {\left({\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{5} - {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{10} - 3 \, \cosh\left(r\right)^{8} + 3 \, \cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{5}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - 3 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial^{2}}{\partial t_{0}\partial r}a\left(t_{0}, r\right) - {\left(2 \, {\left(\cosh\left(r\right)^{6} - 5 \, \cosh\left(r\right)^{4} + 4 \, \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} - {\left(4 \, \cosh\left(r\right)^{8} - 8 \, \cosh\left(r\right)^{6} + 4 \, \cosh\left(r\right)^{4} - {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{2}\right)} \sinh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) - 2 \, {\left({\left(\cosh\left(r\right)^{8} + \cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} + {\left(2 \, \cosh\left(r\right)^{10} - 4 \, \cosh\left(r\right)^{8} + 2 \, \cosh\left(r\right)^{6} - {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} \sinh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} + {\left(12 \, {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} + {\left(8 \, \cosh\left(r\right)^{9} - 16 \, \cosh\left(r\right)^{7} + 8 \, \cosh\left(r\right)^{5} - {\left(\cosh\left(r\right)^{7} + 2 \, \cosh\left(r\right)^{5} - 3 \, \cosh\left(r\right)^{3}\right)} \sinh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) + {\left(3 \, {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right) + 2 \, {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) + {\left(3 \, {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right) + 2 \, {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(3 \, {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right) + 2 \, {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{6} \sinh\left(r\right) + 2 \, {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left({\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{6} \sinh\left(r\right) + 2 \, {\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left({\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} \sinh\left(r\right) + 2 \, {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} + {\left(2 \, {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{7} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left(\cosh\left(r\right)^{6} - 3 \, \cosh\left(r\right)^{4} + 3 \, \cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - {\left(2 \, {\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{7} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + 2 \, {\left({\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \sinh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{9} - 4 \, \cosh\left(r\right)^{7} + 6 \, \cosh\left(r\right)^{5} - 4 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t_{0}\right)^{7} - {\left({\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \sinh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{10} - 4 \, \cosh\left(r\right)^{8} + 6 \, \cosh\left(r\right)^{6} - 4 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{7}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) - {\left({\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial^{2}}{\partial t_{0}\partial r}a\left(t_{0}, r\right) - {\left({\left(\cosh\left(r\right)^{6} - 3 \, \cosh\left(r\right)^{4} + 3 \, \cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} + 4 \, {\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - 2 \, {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) - {\left({\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} + 2 \, {\left(\cosh\left(r\right)^{10} - 3 \, \cosh\left(r\right)^{8} + 3 \, \cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2}\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) + {\left({\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)}}{{\left(a\left(t_{0}, r\right) \cosh\left(r\right)^{9} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - a\left(t_{0}, r\right) \cosh\left(r\right)^{8} \sinh\left(r\right)\right)} B^{7} - 3 \, {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{7} \sinh\left(r\right)^{3} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{6} \sinh\left(r\right)^{3}\right)} B^{5} + 3 \, {\left(A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{5} \sinh\left(r\right)^{5} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right)^{5}\right)} B^{3} - {\left(A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right)^{7} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right)^{7}\right)} B} & 0 & 0 \\ 0 & -\frac{{\left({\left(4 \, {\left(\pi \cosh\left(r\right)^{9} - \pi \cosh\left(r\right)^{7}\right)} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + {\left(\Lambda \cosh\left(r\right)^{9} - \Lambda \cosh\left(r\right)^{7}\right)} a\left(t_{0}, r\right)^{4} + {\left(4 \, {\left(\pi \cosh\left(r\right)^{11} - \pi \cosh\left(r\right)^{9}\right)} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + {\left(\Lambda \cosh\left(r\right)^{11} - \Lambda \cosh\left(r\right)^{9}\right)} a\left(t_{0}, r\right)^{4}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right)^{2} - 2 \, {\left(4 \, {\left(\pi \cosh\left(r\right)^{10} - \pi \cosh\left(r\right)^{8}\right)} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + {\left(\Lambda \cosh\left(r\right)^{10} - \Lambda \cosh\left(r\right)^{8}\right)} a\left(t_{0}, r\right)^{4}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} B^{7} - {\left(8 \, {\left(\pi \cosh\left(r\right)^{9} - 2 \, \pi \cosh\left(r\right)^{7} + \pi \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + 2 \, {\left(\Lambda \cosh\left(r\right)^{9} - 2 \, \Lambda \cosh\left(r\right)^{7} + \Lambda \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} - 3 \, {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right)^{2} + {\left(8 \, {\left(\pi \cosh\left(r\right)^{11} - 2 \, \pi \cosh\left(r\right)^{9} + \pi \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + 2 \, {\left(\Lambda \cosh\left(r\right)^{11} - 2 \, \Lambda \cosh\left(r\right)^{9} + \Lambda \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} - {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} a\left(t_{0}, r\right)^{2}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5} + {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} - {\left(16 \, {\left(\pi \cosh\left(r\right)^{10} - 2 \, \pi \cosh\left(r\right)^{8} + \pi \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + 4 \, {\left(\Lambda \cosh\left(r\right)^{10} - 2 \, \Lambda \cosh\left(r\right)^{8} + \Lambda \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} - {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} a\left(t_{0}, r\right)^{2}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) + 2 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} a\left(t_{0}, r\right)^{2} - {\left({\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) + {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{5}\right)} a\left(t_{0}, r\right)^{2}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - 2 \, {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) + {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial^{2}}{\partial t_{0}\partial r}a\left(t_{0}, r\right) - {\left(4 \, a\left(t_{0}, r\right) \cosh\left(r\right)^{8} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - {\left(7 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(3 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) - {\left({\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} a\left(t_{0}, r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} a\left(t_{0}, r\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{5} + {\left(4 \, {\left(\pi \cosh\left(r\right)^{9} - 3 \, \pi \cosh\left(r\right)^{7} + 3 \, \pi \cosh\left(r\right)^{5} - \pi \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + {\left(\Lambda \cosh\left(r\right)^{9} - 3 \, \Lambda \cosh\left(r\right)^{7} + 3 \, \Lambda \cosh\left(r\right)^{5} - \Lambda \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} - 3 \, {\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right)^{2} + {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} - {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{3} + {\left(4 \, {\left(\pi \cosh\left(r\right)^{11} - 3 \, \pi \cosh\left(r\right)^{9} + 3 \, \pi \cosh\left(r\right)^{7} - \pi \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + {\left(\Lambda \cosh\left(r\right)^{11} - 3 \, \Lambda \cosh\left(r\right)^{9} + 3 \, \Lambda \cosh\left(r\right)^{7} - \Lambda \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} + 2 \, {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} - {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{3}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left({\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) + {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2}\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)^{2} - {\left(8 \, {\left(\pi \cosh\left(r\right)^{10} - 3 \, \pi \cosh\left(r\right)^{8} + 3 \, \pi \cosh\left(r\right)^{6} - \pi \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + 2 \, {\left(\Lambda \cosh\left(r\right)^{10} - 3 \, \Lambda \cosh\left(r\right)^{8} + 3 \, \Lambda \cosh\left(r\right)^{6} - \Lambda \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} + {\left(\cosh\left(r\right)^{8} + 2 \, \cosh\left(r\right)^{6} - 3 \, \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{3}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left({\left(2 \, \cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) + {\left(4 \, \cosh\left(r\right)^{8} - 7 \, \cosh\left(r\right)^{6} + 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} - {\left({\left(2 \, \cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) + {\left(2 \, \cosh\left(r\right)^{9} - \cosh\left(r\right)^{7} - 4 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - 4 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} \sinh\left(r\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right)\right)} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right) + 2 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) \frac{\partial}{\partial t_{0}}A\left(t_{0}\right) - {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial^{2}}{\partial t_{0}\partial r}a\left(t_{0}, r\right) - {\left(4 \, {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) + 2 \, {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \sinh\left(r\right) + {\left({\left(5 \, \cosh\left(r\right)^{8} - 6 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) + 2 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - {\left({\left(9 \, \cosh\left(r\right)^{7} - 10 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) + 4 \, {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial}{\partial r}a\left(t_{0}, r\right) - {\left({\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) + {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} + {\left({\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) + {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) + {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)\right)} \frac{\partial^{2}}{(\partial r)^{2}}a\left(t_{0}, r\right)\right)} B^{3} + {\left({\left(\cosh\left(r\right)^{9} - 4 \, \cosh\left(r\right)^{7} + 6 \, \cosh\left(r\right)^{5} - 4 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t_{0}\right)^{6} \frac{\partial}{\partial t_{0}}a\left(t_{0}, r\right)^{2} - {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} + {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{3} - {\left({\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} - {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{3}\right)} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} + {\left({\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} \frac{\partial}{\partial t_{0}}A\left(t_{0}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} \fra[...]

WARNING: Output: 64620 truncated by MAX_HTML_SIZE to 40000. Type 'smc?' to learn how to raise the output limit.

$\displaystyle \left( -\frac{{\left({\left(\cosh\left(r\right)^{9} - 4 \, \cosh\left(r\right)^{7} + 6 \, \cosh\left(r\right)^{5} - 4 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t_{0}\right)^{8} \sinh\left(r\right) \left(\frac{\partial\,a}{\partial t_{0}}\right)^{2} - {\left(4 \, \pi a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{9} \rho\left(t_{0}, r\right) \sinh\left(r\right) + \Lambda a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{9} \sinh\left(r\right) - \cosh\left(r\right)^{9} \sinh\left(r\right) \left(\frac{\partial\,a}{\partial t_{0}}\right)^{2} + {\left(4 \, \pi a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{11} \rho\left(t_{0}, r\right) \sinh\left(r\right) + \Lambda a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{11} \sinh\left(r\right)\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(4 \, \pi a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{10} \rho\left(t_{0}, r\right) \sinh\left(r\right) + \Lambda a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{10} \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t_{0}}\right)} B^{8} + {\left(4 \, \pi A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{7} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{3} + 3 \, \Lambda A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{7} \sinh\left(r\right)^{3} - 4 \, {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{2} \sinh\left(r\right) \left(\frac{\partial\,a}{\partial t_{0}}\right)^{2} + {\left(4 \, \pi A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{9} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{3} + 3 \, \Lambda A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{9} \sinh\left(r\right)^{3} - a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{7} \sinh\left(r\right)\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} + 3 \, {\left(\cosh\left(r\right)^{9} \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, \cosh\left(r\right)^{8} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + \cosh\left(r\right)^{7} \sinh\left(r\right)\right)} \left(\frac{\partial\,a}{\partial r}\right)^{2} - {\left(8 \, \pi A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{8} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{3} + 6 \, \Lambda A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{8} \sinh\left(r\right)^{3} - a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{6} \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t_{0}} - {\left(2 \, A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{8} \sinh\left(r\right) + 2 \, a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{6} \sinh\left(r\right) - {\left(2 \, A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{9} \sinh\left(r\right) + {\left(\cosh\left(r\right)^{9} + \cosh\left(r\right)^{7}\right)} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t_{0}} + 2 \, {\left(a\left(t_{0}, r\right) \cosh\left(r\right)^{7} \sinh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right) - {\left(a\left(t_{0}, r\right) \cosh\left(r\right)^{8} \sinh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{10} - \cosh\left(r\right)^{8}\right)} A\left(t_{0}\right)\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \frac{\partial\,a}{\partial r}\right)} \frac{\partial\,a}{\partial t_{0}} + {\left(a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{8} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{7} \sinh\left(r\right)^{2}\right)} \frac{\partial^2\,a}{\partial t_{0}\partial r} - {\left(2 \, a\left(t_{0}, r\right) \cosh\left(r\right)^{8} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} + {\left(\cosh\left(r\right)^{7} \sinh\left(r\right)^{2} - 4 \, \cosh\left(r\right)^{7}\right)} a\left(t_{0}, r\right) \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{6} \sinh\left(r\right)^{2} - 2 \, \cosh\left(r\right)^{6}\right)} a\left(t_{0}, r\right)\right)} \frac{\partial\,a}{\partial r} - 3 \, {\left(a\left(t_{0}, r\right) \cosh\left(r\right)^{9} \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, a\left(t_{0}, r\right) \cosh\left(r\right)^{8} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + a\left(t_{0}, r\right) \cosh\left(r\right)^{7} \sinh\left(r\right)\right)} \frac{\partial^2\,a}{\partial r ^ 2}\right)} B^{6} + {\left(4 \, \pi A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{5} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{5} - 3 \, \Lambda A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{5} \sinh\left(r\right)^{5} + 6 \, {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{4} \sinh\left(r\right) \left(\frac{\partial\,a}{\partial t_{0}}\right)^{2} + {\left(3 \, \cosh\left(r\right)^{5} - 5 \, \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - {\left(3 \, \cosh\left(r\right)^{5} - 5 \, \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{3} \sinh\left(r\right) + {\left(4 \, \pi A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{7} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{5} - 3 \, \Lambda A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{7} \sinh\left(r\right)^{5} + 2 \, {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{3} \sinh\left(r\right)\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - {\left(5 \, {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} \sinh\left(r\right) - 2 \, {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \sinh\left(r\right) + {\left(5 \, {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{2} \sinh\left(r\right) - 2 \, {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(5 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} \sinh\left(r\right) - 2 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \left(\frac{\partial\,a}{\partial r}\right)^{2} - {\left(8 \, \pi A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{6} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{5} - 6 \, \Lambda A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{6} \sinh\left(r\right)^{5} + {\left(5 \, \cosh\left(r\right)^{6} - 9 \, \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - 4 \, {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t_{0}} + {\left(6 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) + 5 \, {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - {\left(6 \, {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left(3 \, \cosh\left(r\right)^{9} - \cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t_{0}} + 6 \, {\left({\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{3} - {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{10} - 2 \, \cosh\left(r\right)^{8} + \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{3}\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \frac{\partial\,a}{\partial r}\right)} \frac{\partial\,a}{\partial t_{0}} - 3 \, {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial^2\,a}{\partial t_{0}\partial r} - {\left({\left(\cosh\left(r\right)^{6} - 7 \, \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6} \sinh\left(r\right)^{2} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) - {\left({\left(\cosh\left(r\right)^{8} + 5 \, \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} + 2 \, {\left(\cosh\left(r\right)^{10} - \cosh\left(r\right)^{6} \sinh\left(r\right)^{2} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} + 2 \, {\left(6 \, A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} + {\left(2 \, \cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{5} - {\left(\cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} \sinh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \frac{\partial\,a}{\partial r} + {\left(6 \, {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \sinh\left(r\right) + {\left(6 \, {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(6 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \frac{\partial^2\,a}{\partial r ^ 2}\right)} B^{4} - {\left(4 \, \pi A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{3} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{7} - \Lambda A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{3} \sinh\left(r\right)^{7} + 4 \, {\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{6} \sinh\left(r\right) \left(\frac{\partial\,a}{\partial t_{0}}\right)^{2} + 3 \, {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - 3 \, {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{3} \sinh\left(r\right) + {\left(4 \, \pi A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{5} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{7} - \Lambda A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{5} \sinh\left(r\right)^{7} + {\left(\cosh\left(r\right)^{7} + \cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - {\left(\cosh\left(r\right)^{7} + \cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{3} \sinh\left(r\right)\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - {\left({\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} \sinh\left(r\right) - 4 \, {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left({\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{4} \sinh\left(r\right) - 4 \, {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} \sinh\left(r\right) - 4 \, {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \left(\frac{\partial\,a}{\partial r}\right)^{2} - 2 \, {\left(4 \, \pi A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{4} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{7} - \Lambda A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{4} \sinh\left(r\right)^{7} + {\left(2 \, \cosh\left(r\right)^{6} - \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - {\left(2 \, \cosh\left(r\right)^{6} - \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t_{0}} + {\left(6 \, {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \sinh\left(r\right) + 4 \, {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - {\left(6 \, {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left(3 \, \cosh\left(r\right)^{9} - 5 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t_{0}} + 6 \, {\left({\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{5} - {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{10} - 3 \, \cosh\left(r\right)^{8} + 3 \, \cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{5}\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \frac{\partial\,a}{\partial r}\right)} \frac{\partial\,a}{\partial t_{0}} - 3 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial^2\,a}{\partial t_{0}\partial r} - {\left(2 \, {\left(\cosh\left(r\right)^{6} - 5 \, \cosh\left(r\right)^{4} + 4 \, \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} - {\left(4 \, \cosh\left(r\right)^{8} - 8 \, \cosh\left(r\right)^{6} + 4 \, \cosh\left(r\right)^{4} - {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{2}\right)} \sinh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) - 2 \, {\left({\left(\cosh\left(r\right)^{8} + \cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} + {\left(2 \, \cosh\left(r\right)^{10} - 4 \, \cosh\left(r\right)^{8} + 2 \, \cosh\left(r\right)^{6} - {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} \sinh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} + {\left(12 \, {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} + {\left(8 \, \cosh\left(r\right)^{9} - 16 \, \cosh\left(r\right)^{7} + 8 \, \cosh\left(r\right)^{5} - {\left(\cosh\left(r\right)^{7} + 2 \, \cosh\left(r\right)^{5} - 3 \, \cosh\left(r\right)^{3}\right)} \sinh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \frac{\partial\,a}{\partial r} + {\left(3 \, {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right) + 2 \, {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) + {\left(3 \, {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right) + 2 \, {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(3 \, {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right) + 2 \, {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \frac{\partial^2\,a}{\partial r ^ 2}\right)} B^{2} + {\left({\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{6} \sinh\left(r\right) + 2 \, {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left({\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{6} \sinh\left(r\right) + 2 \, {\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left({\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} \sinh\left(r\right) + 2 \, {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \left(\frac{\partial\,a}{\partial r}\right)^{2} + {\left(2 \, {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{7} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left(\cosh\left(r\right)^{6} - 3 \, \cosh\left(r\right)^{4} + 3 \, \cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) - {\left(2 \, {\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{7} a\left(t_{0}, r\right) \sinh\left(r\right) + {\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t_{0}} + 2 \, {\left({\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \sinh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{9} - 4 \, \cosh\left(r\right)^{7} + 6 \, \cosh\left(r\right)^{5} - 4 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t_{0}\right)^{7} - {\left({\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \sinh\left(r\right)^{2} + {\left(\cosh\left(r\right)^{10} - 4 \, \cosh\left(r\right)^{8} + 6 \, \cosh\left(r\right)^{6} - 4 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{7}\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \frac{\partial\,a}{\partial r}\right)} \frac{\partial\,a}{\partial t_{0}} - {\left({\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2}\right)} \frac{\partial^2\,a}{\partial t_{0}\partial r} - {\left({\left(\cosh\left(r\right)^{6} - 3 \, \cosh\left(r\right)^{4} + 3 \, \cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} + 4 \, {\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \frac{\partial\,A}{\partial t_{0}} - 2 \, {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) - {\left({\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)^{2} + 2 \, {\left(\cosh\left(r\right)^{10} - 3 \, \cosh\left(r\right)^{8} + 3 \, \cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2}\right)} \frac{\partial\,a}{\partial r} + {\left({\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial^2\,a}{\partial r ^ 2}\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)}}{{\left(a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{11} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{10} \sinh\left(r\right)\right)} B^{9} - 4 \, {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{9} \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t_{0}} - A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{8} \sinh\left(r\right)^{3}\right)} B^{7} + 6 \, {\left(A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{7} \sinh\left(r\right)^{5} \frac{\partial\,A}{\partial t_{0}} - A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{6} \sinh\left(r\right)^{5}\right)} B^{5} - 4 \, {\left(A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{5} \sinh\left(r\right)^{7} \frac{\partial\,A}{\partial t_{0}} - A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{4} \sinh\left(r\right)^{7}\right)} B^{3} + {\left(A\left(t_{0}\right)^{8} a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{3} \sinh\left(r\right)^{9} \frac{\partial\,A}{\partial t_{0}} - A\left(t_{0}\right)^{8} a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{2} \sinh\left(r\right)^{9}\right)} B} \right) \frac{\partial}{\partial r }\otimes \mathrm{d} r + \left( \frac{{\left({\left(\cosh\left(r\right)^{9} - 4 \, \cosh\left(r\right)^{7} + 6 \, \cosh\left(r\right)^{5} - 4 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t_{0}\right)^{6} \left(\frac{\partial\,a}{\partial t_{0}}\right)^{2} + {\left(4 \, {\left(\pi \cosh\left(r\right)^{9} - \pi \cosh\left(r\right)^{7}\right)} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + {\left(\Lambda \cosh\left(r\right)^{9} - \Lambda \cosh\left(r\right)^{7}\right)} a\left(t_{0}, r\right)^{4} + {\left(4 \, {\left(\pi \cosh\left(r\right)^{11} - \pi \cosh\left(r\right)^{9}\right)} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + {\left(\Lambda \cosh\left(r\right)^{11} - \Lambda \cosh\left(r\right)^{9}\right)} a\left(t_{0}, r\right)^{4}\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} \left(\frac{\partial\,a}{\partial t_{0}}\right)^{2} - 2 \, {\left(4 \, {\left(\pi \cosh\left(r\right)^{10} - \pi \cosh\left(r\right)^{8}\right)} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + {\left(\Lambda \cosh\left(r\right)^{10} - \Lambda \cosh\left(r\right)^{8}\right)} a\left(t_{0}, r\right)^{4}\right)} \frac{\partial\,A}{\partial t_{0}}\right)} B^{6} - {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} + {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{3} - {\left(8 \, {\left(\pi \cosh\left(r\right)^{9} - 2 \, \pi \cosh\left(r\right)^{7} + \pi \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + 2 \, {\left(\Lambda \cosh\left(r\right)^{9} - 2 \, \Lambda \cosh\left(r\right)^{7} + \Lambda \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} - 3 \, {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} \left(\frac{\partial\,a}{\partial t_{0}}\right)^{2} + {\left(8 \, {\left(\pi \cosh\left(r\right)^{11} - 2 \, \pi \cosh\left(r\right)^{9} + \pi \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + 2 \, {\left(\Lambda \cosh\left(r\right)^{11} - 2 \, \Lambda \cosh\left(r\right)^{9} + \Lambda \cosh\left(r\right)^{7}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} - {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} a\left(t_{0}, r\right)^{2}\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5} + {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \left(\frac{\partial\,a}{\partial r}\right)^{2} - {\left(16 \, {\left(\pi \cosh\left(r\right)^{10} - 2 \, \pi \cosh\left(r\right)^{8} + \pi \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + 4 \, {\left(\Lambda \cosh\left(r\right)^{10} - 2 \, \Lambda \cosh\left(r\right)^{8} + \Lambda \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{4} - {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} a\left(t_{0}, r\right)^{2}\right)} \frac{\partial\,A}{\partial t_{0}} - {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) + 2 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} a\left(t_{0}, r\right)^{2} - {\left({\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) + {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{5}\right)} a\left(t_{0}, r\right)^{2}\right)} \frac{\partial\,A}{\partial t_{0}} - 2 \, {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} \frac{\partial\,a}{\partial t_{0}} + {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial^2\,a}{\partial t_{0}\partial r} - {\left(4 \, a\left(t_{0}, r\right) \cosh\left(r\right)^{8} \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - {\left(7 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + {\left(3 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r} - {\left({\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7}\right)} a\left(t_{0}, r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} a\left(t_{0}, r\right) \frac{\partial\,A}{\partial t_{0}} + {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} a\left(t_{0}, r\right)\right)} \frac{\partial^2\,a}{\partial r ^ 2}\right)} B^{4} + {\left(4 \, {\left(\pi \cosh\left(r\right)^{9} - 3 \, \pi \cosh\left(r\right)^{7} + 3 \, \pi \cosh\left(r\right)^{5} - \pi \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + {\left(\Lambda \cosh\left(r\right)^{9} - 3 \, \Lambda \cosh\left(r\right)^{7} + 3 \, \Lambda \cosh\left(r\right)^{5} - \Lambda \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} - 3 \, {\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} \left(\frac{\partial\,a}{\partial t_{0}}\right)^{2} + {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} - {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{3} + {\left(4 \, {\left(\pi \cosh\left(r\right)^{11} - 3 \, \pi \cosh\left(r\right)^{9} + 3 \, \pi \cosh\left(r\right)^{7} - \pi \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + {\left(\Lambda \cosh\left(r\right)^{11} - 3 \, \Lambda \cosh\left(r\right)^{9} + 3 \, \Lambda \cosh\left(r\right)^{7} - \Lambda \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} + 2 \, {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} - {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{3}\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left({\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} \frac{\partial\,A}{\partial t_{0}} + {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2}\right)} \left(\frac{\partial\,a}{\partial r}\right)^{2} - {\left(8 \, {\left(\pi \cosh\left(r\right)^{10} - 3 \, \pi \cosh\left(r\right)^{8} + 3 \, \pi \cosh\left(r\right)^{6} - \pi \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} \rho\left(t_{0}, r\right) + 2 \, {\left(\Lambda \cosh\left(r\right)^{10} - 3 \, \Lambda \cosh\left(r\right)^{8} + 3 \, \Lambda \cosh\left(r\right)^{6} - \Lambda \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{4} + {\left(\cosh\left(r\right)^{8} + 2 \, \cosh\left(r\right)^{6} - 3 \, \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{3}\right)} \frac{\partial\,A}{\partial t_{0}} - {\left({\left(2 \, \cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) + {\left(4 \, \cosh\left(r\right)^{8} - 7 \, \cosh\left(r\right)^{6} + 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} - {\left({\left(2 \, \cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) + {\left(2 \, \cosh\left(r\right)^{9} - \cosh\left(r\right)^{7} - 4 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2}\right)} \frac{\partial\,A}{\partial t_{0}} - 4 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}\right)} \frac{\partial\,a}{\partial t_{0}} + 2 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial^2\,a}{\partial t_{0}\partial r} - {\left(4 \, {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) + 2 \, {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \sinh\left(r\right) + {\left({\left(5 \, \cosh\left(r\right)^{8} - 6 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) + 2 \, {\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - {\left({\left(9 \, \cosh\left(r\right)^{7} - 10 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) + 4 \, {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} \sinh\left(r\right)\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \frac{\partial\,a}{\partial r} - {\left({\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) + {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2} + {\left({\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) + {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2}\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) + {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)^{2}\right)} \frac{\partial\,A}{\partial t_{0}}\right)} \frac{\partial^2\,a}{\partial r ^ 2}\right)} B^{2} - {\left({\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} - {\left(\cosh\left(r\right)^{9} - 2 \, \cosh\left(r\right)^{7} + \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{3}\right)} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} + {\left({\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} \frac{\partial\,A}{\partial t_{0}} + {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{4}\right)} \left(\frac{\partial\,a}{\partial r}\right)^{2} + 2 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} - {\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{3}\right)} \frac{\partial\,A}{\partial t_{0}} + {\left({\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) + {\left(2 \, \cosh\left(r\right)^{8} - 5 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{2} - {\left({\left(\cosh\left(r\right)^{9} - 3 \, \cosh\left(r\right)^{7} + 3 \, \cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) + {\left(\cosh\left(r\right)^{9} - \cosh\left(r\right)^{7} - 3 \, \cosh\lef[...]

WARNING: Output: 62459 truncated by MAX_HTML_SIZE to 40000. Type 'smc?' to learn how to raise the output limit.

#Momentum Constraint
p =  -Phi.pullback(T['_ab']*nv['^b'] + T(nv,nv)*n1f)
Tp = T['_ab']*nv['^a']
p2 = -Phi.pullback( (Tp['_b']*P['^b_c']))
p.display()
#p2.display()
D_K3u = D(K3u)
#D_K3u.components()
Pconstraint = D_K3u.trace(0,2) -D(trK3)-8*pi*p
Pconstraint.display()

$\displaystyle \left( \frac{B A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right) \rho\left(t_{0}, r\right) \sinh\left(r\right)}{\sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)}} \right) \mathrm{d} r$

$\displaystyle \left( -\frac{2 \, {\left({\left(4 \, \pi A\left(t_{0}\right) a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{9} \rho\left(t_{0}, r\right) \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 8 \, \pi A\left(t_{0}\right) a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{8} \rho\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + 4 \, \pi A\left(t_{0}\right) a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{7} \rho\left(t_{0}, r\right) \sinh\left(r\right) - {\left(a\left(t_{0}, r\right) \cosh\left(r\right)^{7} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + 2 \, {\left(\cosh\left(r\right)^{8} \frac{\partial\,A}{\partial t_{0}} - \cosh\left(r\right)^{7}\right)} \frac{\partial\,a}{\partial r}\right)} \frac{\partial\,a}{\partial t_{0}} + {\left(a\left(t_{0}, r\right) \cosh\left(r\right)^{8} \frac{\partial\,A}{\partial t_{0}} - a\left(t_{0}, r\right) \cosh\left(r\right)^{7}\right)} \frac{\partial^2\,a}{\partial t_{0}\partial r}\right)} B^{6} - {\left(8 \, \pi A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{7} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{3} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 16 \, \pi A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{6} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{3} \frac{\partial\,A}{\partial t_{0}} + 8 \, \pi A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{5} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{3} + 2 \, {\left(A\left(t_{0}\right) \cosh\left(r\right)^{7} \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, A\left(t_{0}\right) \cosh\left(r\right)^{6} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + A\left(t_{0}\right) \cosh\left(r\right)^{5} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}^{2} - {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) + {\left(3 \, \cosh\left(r\right)^{7} - 4 \, \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + 6 \, {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2}\right)} \frac{\partial\,a}{\partial r}\right)} \frac{\partial\,a}{\partial t_{0}} + 3 \, {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right) \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)\right)} \frac{\partial^2\,a}{\partial t_{0}\partial r} + {\left({\left(\cosh\left(r\right)^{8} - \cosh\left(r\right)^{6}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right) \frac{\partial\,A}{\partial t_{0}} + {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right) a\left(t_{0}, r\right)\right)} \frac{\partial\,a}{\partial r} - {\left(A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{7} \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{6} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + A\left(t_{0}\right) a\left(t_{0}, r\right) \cosh\left(r\right)^{5} \sinh\left(r\right)\right)} \frac{\partial^2\,a}{\partial r ^ 2}\right)} B^{4} + {\left(4 \, \pi A\left(t_{0}\right)^{5} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{5} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{5} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 8 \, \pi A\left(t_{0}\right)^{5} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{4} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{5} \frac{\partial\,A}{\partial t_{0}} + 4 \, \pi A\left(t_{0}\right)^{5} a\left(t_{0}, r\right)^{4} \cosh\left(r\right)^{3} \rho\left(t_{0}, r\right) \sinh\left(r\right)^{5} + 4 \, {\left({\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{3} \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}^{2} - {\left({\left(3 \, \cosh\left(r\right)^{7} - 8 \, \cosh\left(r\right)^{5} + 5 \, \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + 2 \, {\left(\cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \sinh\left(r\right) + 6 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4}\right)} \frac{\partial\,a}{\partial r}\right)} \frac{\partial\,a}{\partial t_{0}} + 3 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right) \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)\right)} \frac{\partial^2\,a}{\partial t_{0}\partial r} + 2 \, {\left({\left(\cosh\left(r\right)^{8} - 2 \, \cosh\left(r\right)^{6} + \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \frac{\partial\,A}{\partial t_{0}} + {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right)\right)} \frac{\partial\,a}{\partial r} - 2 \, {\left({\left(\cosh\left(r\right)^{7} - \cosh\left(r\right)^{5}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{6} - \cosh\left(r\right)^{4}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + {\left(\cosh\left(r\right)^{5} - \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{3} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial^2\,a}{\partial r ^ 2}\right)} B^{2} - 2 \, {\left({\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{5} \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{5} \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + {\left(\cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t_{0}\right)^{5} \sinh\left(r\right)\right)} \frac{\partial\,a}{\partial r}^{2} + {\left({\left(\cosh\left(r\right)^{7} - 4 \, \cosh\left(r\right)^{5} + 5 \, \cosh\left(r\right)^{3} - 2 \, \cosh\left(r\right)\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + {\left(\cosh\left(r\right)^{4} - 2 \, \cosh\left(r\right)^{2} + 1\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \sinh\left(r\right) + 2 \, {\left({\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{6}\right)} \frac{\partial\,a}{\partial r}\right)} \frac{\partial\,a}{\partial t_{0}} - {\left({\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right) \frac{\partial\,A}{\partial t_{0}} - {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{6} a\left(t_{0}, r\right)\right)} \frac{\partial^2\,a}{\partial t_{0}\partial r} - {\left({\left(\cosh\left(r\right)^{8} - 3 \, \cosh\left(r\right)^{6} + 3 \, \cosh\left(r\right)^{4} - \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{7} - 3 \, \cosh\left(r\right)^{5} + 3 \, \cosh\left(r\right)^{3} - \cosh\left(r\right)\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \frac{\partial\,A}{\partial t_{0}} + {\left(\cosh\left(r\right)^{6} - 3 \, \cosh\left(r\right)^{4} + 3 \, \cosh\left(r\right)^{2} - 1\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right)\right)} \frac{\partial\,a}{\partial r} + {\left({\left(\cosh\left(r\right)^{7} - 2 \, \cosh\left(r\right)^{5} + \cosh\left(r\right)^{3}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \sinh\left(r\right) \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, {\left(\cosh\left(r\right)^{6} - 2 \, \cosh\left(r\right)^{4} + \cosh\left(r\right)^{2}\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t_{0}} + {\left(\cosh\left(r\right)^{5} - 2 \, \cosh\left(r\right)^{3} + \cosh\left(r\right)\right)} A\left(t_{0}\right)^{5} a\left(t_{0}, r\right) \sinh\left(r\right)\right)} \frac{\partial^2\,a}{\partial r ^ 2}\right)}}{{\left({\left(a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{8} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{7} \frac{\partial\,A}{\partial t_{0}} + a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{6}\right)} B^{5} - 2 \, {\left(A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{6} \sinh\left(r\right)^{2} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{5} \sinh\left(r\right)^{2} \frac{\partial\,A}{\partial t_{0}} + A\left(t_{0}\right)^{2} a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{4} \sinh\left(r\right)^{2}\right)} B^{3} + {\left(A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{4} \sinh\left(r\right)^{4} \left(\frac{\partial\,A}{\partial t_{0}}\right)^{2} - 2 \, A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{3} \sinh\left(r\right)^{4} \frac{\partial\,A}{\partial t_{0}} + A\left(t_{0}\right)^{4} a\left(t_{0}, r\right)^{3} \cosh\left(r\right)^{2} \sinh\left(r\right)^{4}\right)} B\right)} \sqrt{B \cosh\left(r\right) + A\left(t_{0}\right) \sinh\left(r\right)} \sqrt{B \cosh\left(r\right) - A\left(t_{0}\right) \sinh\left(r\right)}} \right) \mathrm{d} r$

Ttrace = g.inverse()['^ab']*T['_ab']
Ttrace.display()

$\displaystyle \begin{array}{llcl} & \mathcal{M} & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & -\frac{2 \, \pi \rho\left(t, r\right) + \Lambda}{2 \, \pi} \end{array}$

E1 = Ricci4 - Ricci4_scalar/2*g - (8*pi)*T  #cosmological constant included in T as exotic stress
print("First Friedmann equation:\n")
E1[0,0].expand()

Error in lines 1-1 Traceback (most recent call last): File "/usr/local/lib/python2.7/dist-packages/smc_sagews/sage_server.py", line 982, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'Ricci4' is not defined

E2 = Ricci4  - (8*pi)*(T - Ttrace/2*g)  #cosmological constant included in T as exotic stress
print("Second Friedmann equation:\n")
E2[0,0].expand()

Second Friedmann equation:

$\displaystyle -\frac{4 \, \pi a\left(t, r\right)^{4} \cosh\left(r\right)^{6} \rho\left(t, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t}^{3}}{a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)} + \frac{\Lambda a\left(t, r\right)^{4} \cosh\left(r\right)^{6} \sinh\left(r\right) \frac{\partial\,A}{\partial t}^{3}}{a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)} + \frac{12 \, \pi a\left(t, r\right)^{4} \cosh\left(r\right)^{5} \rho\left(t, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t}^{2}}{a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)} - \frac{3 \, \Lambda a\left(t, r\right)^{4} \cosh\left(r\right)^{5} \sinh\left(r\right) \frac{\partial\,A}{\partial t}^{2}}{a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)} - \frac{12 \, \pi a\left(t, r\right)^{4} \cosh\left(r\right)^{4} \rho\left(t, r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t}}{a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)} + \frac{3 \, \Lambda a\left(t, r\right)^{4} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t}}{a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)} + \frac{4 \, \pi a\left(t, r\right)^{4} \cosh\left(r\right)^{3} \rho\left(t, r\right) \sinh\left(r\right)}{a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)} - \frac{\Lambda a\left(t, r\right)^{4} \cosh\left(r\right)^{3} \sinh\left(r\right)}{a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)} + \frac{3 \, a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial^2\,A}{\partial t ^ 2} \frac{\partial\,a}{\partial t}}{a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)} - \frac{A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial^2\,A}{\partial t ^ 2} \frac{\partial\,a}{\partial t}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{3 \, \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} \frac{\partial\,a}{\partial t}^{2}}{a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)} + \frac{A\left(t\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} \frac{\partial\,a}{\partial t}^{2}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{3 \, a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} \frac{\partial^2\,a}{\partial t ^ 2}}{a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)} + \frac{A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} \frac{\partial^2\,a}{\partial t ^ 2}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{2 \, A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{5} \frac{\partial\,A}{\partial t}^{2} \frac{\partial^2\,a}{\partial t\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{a\left(t, r\right) \cosh\left(r\right)^{5} \frac{\partial\,A}{\partial t}^{3} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{2 \, A\left(t\right) \cosh\left(r\right)^{5} \frac{\partial\,A}{\partial t}^{2} \frac{\partial\,a}{\partial t} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{\cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t}^{3} \frac{\partial\,a}{\partial r}^{2}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{a\left(t, r\right) \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t}^{3} \frac{\partial^2\,a}{\partial r ^ 2}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{2 \, A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t} \frac{\partial\,a}{\partial t}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{3 \, A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial\,A}{\partial t}^{2} \frac{\partial\,a}{\partial t}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial^2\,A}{\partial t ^ 2} \frac{\partial\,a}{\partial t}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{3 \, \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,a}{\partial t}^{2}}{a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)} - \frac{A\left(t\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,a}{\partial t}^{2}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{A\left(t\right)^{2} \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial\,A}{\partial t} \frac{\partial\,a}{\partial t}^{2}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{3 \, a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial^2\,a}{\partial t ^ 2}}{a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)} - \frac{A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial^2\,a}{\partial t ^ 2}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial\,A}{\partial t} \frac{\partial^2\,a}{\partial t ^ 2}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{4 \, A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t} \frac{\partial^2\,a}{\partial t\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{2 \, A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t}^{2} \frac{\partial^2\,a}{\partial t\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{3 \, a\left(t, r\right) \cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t}^{2} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{a\left(t, r\right) \cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t}^{3} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{4 \, A\left(t\right) \cosh\left(r\right)^{4} \frac{\partial\,A}{\partial t} \frac{\partial\,a}{\partial t} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{2 \, A\left(t\right) \cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t}^{2} \frac{\partial\,a}{\partial t} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{3 \, \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t}^{2} \frac{\partial\,a}{\partial r}^{2}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{3 \, a\left(t, r\right) \cosh\left(r\right)^{3} \sinh\left(r\right) \frac{\partial\,A}{\partial t}^{2} \frac{\partial^2\,a}{\partial r ^ 2}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{2 \, A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial\,a}{\partial t}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{4 \, A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right) \frac{\partial\,A}{\partial t} \frac{\partial\,a}{\partial t}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{A\left(t\right)^{2} \cosh\left(r\right) \sinh\left(r\right) \frac{\partial\,a}{\partial t}^{2}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{A\left(t\right)^{2} a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right) \frac{\partial^2\,a}{\partial t ^ 2}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{2 \, A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{3} \frac{\partial^2\,a}{\partial t\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{4 \, A\left(t\right) a\left(t, r\right) \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} \frac{\partial^2\,a}{\partial t\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{3 \, a\left(t, r\right) \cosh\left(r\right)^{3} \frac{\partial\,A}{\partial t} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{3 \, a\left(t, r\right) \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t}^{2} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{2 \, A\left(t\right) \cosh\left(r\right)^{3} \frac{\partial\,a}{\partial t} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{4 \, A\left(t\right) \cosh\left(r\right)^{2} \frac{\partial\,A}{\partial t} \frac{\partial\,a}{\partial t} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{3 \, \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial\,A}{\partial t} \frac{\partial\,a}{\partial r}^{2}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{3 \, a\left(t, r\right) \cosh\left(r\right)^{2} \sinh\left(r\right) \frac{\partial\,A}{\partial t} \frac{\partial^2\,a}{\partial r ^ 2}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{A\left(t\right) a\left(t, r\right) \sinh\left(r\right) \frac{\partial\,a}{\partial t}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{2 \, A\left(t\right) a\left(t, r\right) \cosh\left(r\right) \frac{\partial^2\,a}{\partial t\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{a\left(t, r\right) \cosh\left(r\right)^{2} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{3 \, a\left(t, r\right) \cosh\left(r\right) \frac{\partial\,A}{\partial t} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} + \frac{2 \, A\left(t\right) \cosh\left(r\right) \frac{\partial\,a}{\partial t} \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{\cosh\left(r\right) \sinh\left(r\right) \frac{\partial\,a}{\partial r}^{2}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{a\left(t, r\right) \cosh\left(r\right) \sinh\left(r\right) \frac{\partial^2\,a}{\partial r ^ 2}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}} - \frac{a\left(t, r\right) \frac{\partial\,a}{\partial r}}{{\left(a\left(t, r\right)^{2} \cosh\left(r\right)^{4} \sinh\left(r\right) \frac{\partial\,A}{\partial t} - a\left(t, r\right)^{2} \cosh\left(r\right)^{3} \sinh\left(r\right)\right)} B^{2}}$

E1[1,1].display()

Error in lines 1-1 Traceback (most recent call last): File "/usr/local/lib/python2.7/dist-packages/smc_sagews/sage_server.py", line 982, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'E1' is not defined

E1[2,2].display()

E1[3,3].display()

E1[0,1].display()

E1[0,2].display()