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5D Kerr-AdS spacetime with a Nambu-Goto string

Case b = 0

This SageMath notebook is relative to the article Holographic drag force in 5d Kerr-AdS black hole by Irina Ya. Aref'eva, Anastasia A. Golubtsova and Eric Gourgoulhon, arXiv:2004.12984.

The involved differential geometry computations are based on tools developed through the SageManifolds project.

NB: a version of SageMath at least equal to 8.2 is required to run this notebook:

In [1]:
version()
'SageMath version 9.0, Release Date: 2020-01-01'

First we set up the notebook to display mathematical objects using LaTeX rendering:

In [2]:
%display latex

Since some computations are quite long, we ask for running them in parallel on 8 cores:

In [3]:
Parallelism().set(nproc=1) # only nproc=1 works on CoCalc

Spacetime manifold

We declare the Kerr-AdS spacetime as a 5-dimensional Lorentzian manifold:

In [4]:
M = Manifold(5, 'M', r'\mathcal{M}', structure='Lorentzian', metric_name='G') print(M)
5-dimensional Lorentzian manifold M

Let us define Boyer-Lindquist-type coordinates (rational polynomial version) on M\mathcal{M}, via the method chart(), the argument of which is a string expressing the coordinates names, their ranges (the default is (,+)(-\infty,+\infty)) and their LaTeX symbols:

In [5]:
BL.<t,r,mu,ph,ps> = M.chart(r't r:(0,+oo) mu:(0,1):\mu ph:(0,2*pi):\phi ps:(0,2*pi):\psi') BL
(M,(t,r,μ,ϕ,ψ))\left(\mathcal{M},(t, r, {\mu}, {\phi}, {\psi})\right)

The coordinate μ\mu is related to the standard Boyer-Lindquist coordinate θ\theta by μ=cosθ \mu = \cos\theta

The coordinate ranges are

In [6]:
BL.coord_range()
t: (,+);r: (0,+);μ: (0,1);ϕ: (0,2π);ψ: (0,2π)t :\ \left( -\infty, +\infty \right) ;\quad r :\ \left( 0 , +\infty \right) ;\quad {\mu} :\ \left( 0 , 1 \right) ;\quad {\phi} :\ \left( 0 , 2 \, \pi \right) ;\quad {\psi} :\ \left( 0 , 2 \, \pi \right)

Note that contrary to the 4-dimensional case, the range of μ\mu is (0,1)(0,1) only (cf. Sec. 1.2 of R.C. Myers, arXiv:1111.1903 or Sec. 2 of G.W. Gibbons, H. Lüb, Don N. Page, C.N. Pope, J. Geom. Phys. 53, 49 (2005)). In other words, the range of θ\theta is (0,π2)\left(0, \frac{\pi}{2}\right) only.

Metric tensor

The 4 parameters mm, aa, bb and \ell of the Kerr-AdS spacetime are declared as symbolic variables, aa and bb being the two angular momentum parameters and \ell being related to the cosmological constant by Λ=62\Lambda = - 6 \ell^2:

In [7]:
var('m a b', domain='real')
(m,a,b)\left(m, a, b\right)
In [8]:
var('l', domain='real', latex_name=r'\ell')
{\ell}
In [9]:
# Particular cases # m = 0 # a = 0 b = 0 # b = a

Some auxiliary functions:

In [10]:
keep_Delta = False # change to False to provide explicit expression for Delta_r, Xi_a, etc...
In [11]:
sig = (1 + r^2*l^2)/r^2 costh2 = mu^2 sinth2 = 1 - mu^2 rho2 = r^2 + a^2*mu^2 + b^2*sinth2 if keep_Delta: Delta_r = var('Delta_r', latex_name=r'\Delta_r', domain='real') Delta_th = var('Delta_th', latex_name=r'\Delta_\theta', domain='real') if a == b: Xi_a = var('Xi', latex_name=r'\Xi', domain='real') Xi_b = Xi_a else: Xi_a = var('Xi_a', latex_name=r'\Xi_a', domain='real') Xi_b = var('Xi_b', latex_name=r'\Xi_b', domain='real') #Delta_th = 1 - a^2*l^2*mu^2 - b^2*l^2*sinth2 Xi_a = 1 - a^2*l^2 Xi_b = 1 - b^2*l^2 else: Delta_r = (r^2+a^2)*(r^2+b^2)*sig - 2*m Delta_th = 1 - a^2*l^2*mu^2 - b^2*l^2*sinth2 Xi_a = 1 - a^2*l^2 Xi_b = 1 - b^2*l^2

The metric is set by its components in the coordinate frame associated with the Boyer-Lindquist-type coordinates, which is the current manifold's default frame. These components can be deduced from Eq. (5.22) of the article S.W. Hawking, C.J. Hunter & M.M. Taylor-Robinson, Phys. Rev. D 59, 064005 (1999) (the check of agreement with this equation is performed below):

In [12]:
G = M.metric() tmp = 1/rho2*( -Delta_r + Delta_th*(a^2*sinth2 + b^2*mu^2) + a^2*b^2*sig ) G[0,0] = tmp.simplify_full() tmp = a*sinth2/(rho2*Xi_a)*( Delta_r - (r^2+a^2)*(Delta_th + b^2*sig) ) G[0,3] = tmp.simplify_full() tmp = b*mu^2/(rho2*Xi_b)*( Delta_r - (r^2+b^2)*(Delta_th + a^2*sig) ) G[0,4] = tmp.simplify_full() G[1,1] = (rho2/Delta_r).simplify_full() G[2,2] = (rho2/Delta_th/(1-mu^2)).simplify_full() tmp = sinth2/(rho2*Xi_a^2)*( - Delta_r*a^2*sinth2 + (r^2+a^2)^2*(Delta_th + sig*b^2*sinth2) ) G[3,3] = tmp.simplify_full() tmp = a*b*sinth2*mu^2/(rho2*Xi_a*Xi_b)*( - Delta_r + sig*(r^2+a^2)*(r^2+b^2) ) G[3,4] = tmp.simplify_full() tmp = mu^2/(rho2*Xi_b^2)*( - Delta_r*b^2*mu^2 + (r^2+b^2)^2*(Delta_th + sig*a^2*mu^2) ) G[4,4] = tmp.simplify_full() G.display()
G=(a42μ42r4(a42+a2)μ2(a22+1)r2+2ma2μ2+r2)dtdt+(a52μ4+(a2μ2a2)r4(a52+2am)μ2+(a32μ4a32)r2+2am(a42a2)μ2+(a221)r2)dtdϕ+(a2μ2+r22r4+(a22+1)r2+a22m)drdr+(a2μ2+r2a22μ4(a22+1)μ2+1)dμdμ+(a52μ4+(a2μ2a2)r4(a52+2am)μ2+(a32μ4a32)r2+2am(a42a2)μ2+(a221)r2)dϕdt+((a62a4+2a2m)μ4(a22(a221)μ21)r4+2a2m(a62a4+4a2m)μ2(a42(a42a2)μ4a2)r2(a642a42+a2)μ2+(a442a22+1)r2)dϕdϕ+μ2r2dψdψG = \left( \frac{a^{4} {\ell}^{2} {\mu}^{4} - {\ell}^{2} r^{4} - {\left(a^{4} {\ell}^{2} + a^{2}\right)} {\mu}^{2} - {\left(a^{2} {\ell}^{2} + 1\right)} r^{2} + 2 \, m}{a^{2} {\mu}^{2} + r^{2}} \right) \mathrm{d} t\otimes \mathrm{d} t + \left( \frac{a^{5} {\ell}^{2} {\mu}^{4} + {\left(a {\ell}^{2} {\mu}^{2} - a {\ell}^{2}\right)} r^{4} - {\left(a^{5} {\ell}^{2} + 2 \, a m\right)} {\mu}^{2} + {\left(a^{3} {\ell}^{2} {\mu}^{4} - a^{3} {\ell}^{2}\right)} r^{2} + 2 \, a m}{{\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2}} \right) \mathrm{d} t\otimes \mathrm{d} {\phi} + \left( \frac{a^{2} {\mu}^{2} + r^{2}}{{\ell}^{2} r^{4} + {\left(a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2} - 2 \, m} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( \frac{a^{2} {\mu}^{2} + r^{2}}{a^{2} {\ell}^{2} {\mu}^{4} - {\left(a^{2} {\ell}^{2} + 1\right)} {\mu}^{2} + 1} \right) \mathrm{d} {\mu}\otimes \mathrm{d} {\mu} + \left( \frac{a^{5} {\ell}^{2} {\mu}^{4} + {\left(a {\ell}^{2} {\mu}^{2} - a {\ell}^{2}\right)} r^{4} - {\left(a^{5} {\ell}^{2} + 2 \, a m\right)} {\mu}^{2} + {\left(a^{3} {\ell}^{2} {\mu}^{4} - a^{3} {\ell}^{2}\right)} r^{2} + 2 \, a m}{{\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} t + \left( \frac{{\left(a^{6} {\ell}^{2} - a^{4} + 2 \, a^{2} m\right)} {\mu}^{4} - {\left(a^{2} {\ell}^{2} - {\left(a^{2} {\ell}^{2} - 1\right)} {\mu}^{2} - 1\right)} r^{4} + 2 \, a^{2} m - {\left(a^{6} {\ell}^{2} - a^{4} + 4 \, a^{2} m\right)} {\mu}^{2} - {\left(a^{4} {\ell}^{2} - {\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{4} - a^{2}\right)} r^{2}}{{\left(a^{6} {\ell}^{4} - 2 \, a^{4} {\ell}^{2} + a^{2}\right)} {\mu}^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\phi} + {\mu}^{2} r^{2} \mathrm{d} {\psi}\otimes \mathrm{d} {\psi}
In [13]:
G.display_comp(only_nonredundant=True)
Gtttt=a42μ42r4(a42+a2)μ2(a22+1)r2+2ma2μ2+r2Gtϕtϕ=a52μ4+(a2μ2a2)r4(a52+2am)μ2+(a32μ4a32)r2+2am(a42a2)μ2+(a221)r2Grrrr=a2μ2+r22r4+(a22+1)r2+a22mGμμμμ=a2μ2+r2a22μ4(a22+1)μ2+1Gϕϕϕϕ=(a62a4+2a2m)μ4(a22(a221)μ21)r4+2a2m(a62a4+4a2m)μ2(a42(a42a2)μ4a2)r2(a642a42+a2)μ2+(a442a22+1)r2Gψψψψ=μ2r2\begin{array}{lcl} G_{ \, t \, t }^{ \phantom{\, t}\phantom{\, t} } & = & \frac{a^{4} {\ell}^{2} {\mu}^{4} - {\ell}^{2} r^{4} - {\left(a^{4} {\ell}^{2} + a^{2}\right)} {\mu}^{2} - {\left(a^{2} {\ell}^{2} + 1\right)} r^{2} + 2 \, m}{a^{2} {\mu}^{2} + r^{2}} \\ G_{ \, t \, {\phi} }^{ \phantom{\, t}\phantom{\, {\phi}} } & = & \frac{a^{5} {\ell}^{2} {\mu}^{4} + {\left(a {\ell}^{2} {\mu}^{2} - a {\ell}^{2}\right)} r^{4} - {\left(a^{5} {\ell}^{2} + 2 \, a m\right)} {\mu}^{2} + {\left(a^{3} {\ell}^{2} {\mu}^{4} - a^{3} {\ell}^{2}\right)} r^{2} + 2 \, a m}{{\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2}} \\ G_{ \, r \, r }^{ \phantom{\, r}\phantom{\, r} } & = & \frac{a^{2} {\mu}^{2} + r^{2}}{{\ell}^{2} r^{4} + {\left(a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2} - 2 \, m} \\ G_{ \, {\mu} \, {\mu} }^{ \phantom{\, {\mu}}\phantom{\, {\mu}} } & = & \frac{a^{2} {\mu}^{2} + r^{2}}{a^{2} {\ell}^{2} {\mu}^{4} - {\left(a^{2} {\ell}^{2} + 1\right)} {\mu}^{2} + 1} \\ G_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & \frac{{\left(a^{6} {\ell}^{2} - a^{4} + 2 \, a^{2} m\right)} {\mu}^{4} - {\left(a^{2} {\ell}^{2} - {\left(a^{2} {\ell}^{2} - 1\right)} {\mu}^{2} - 1\right)} r^{4} + 2 \, a^{2} m - {\left(a^{6} {\ell}^{2} - a^{4} + 4 \, a^{2} m\right)} {\mu}^{2} - {\left(a^{4} {\ell}^{2} - {\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{4} - a^{2}\right)} r^{2}}{{\left(a^{6} {\ell}^{4} - 2 \, a^{4} {\ell}^{2} + a^{2}\right)} {\mu}^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2}} \\ G_{ \, {\psi} \, {\psi} }^{ \phantom{\, {\psi}}\phantom{\, {\psi}} } & = & {\mu}^{2} r^{2} \end{array}

Check of agreement with Eq. (5.22) of Hawking et al or Eq. (2.3) of 5dKerr-AdS-AGG-5-12

We need the 1-forms dt\mathrm{d}t, dr\mathrm{d}r, dμ\mathrm{d}\mu, dϕ\mathrm{d}\phi and dψ\mathrm{d}\psi:

In [14]:
dt, dr, dmu, dph, dps = (BL.coframe()[i] for i in M.irange()) dt, dr, dmu, dph, dps
(dt,dr,dμ,dϕ,dψ)\left(\mathrm{d} t, \mathrm{d} r, \mathrm{d} {\mu}, \mathrm{d} {\phi}, \mathrm{d} {\psi}\right)
In [15]:
print(dt)
1-form dt on the 5-dimensional Lorentzian manifold M

In agreement with μ=cosθ\mu = \cos\theta, we introduce the 1-form dθ=dμ/sinθ\mathrm{d}\theta = - \mathrm{d}\mu /\sin\theta , with sinθ=1μ2\sin\theta = \sqrt{1-\mu^2} since θ(0,π2)\theta\in\left(0, \frac{\pi}{2}\right):

In [16]:
dth = - 1/sqrt(1 - mu^2)*dmu
In [17]:
s1 = dt - a*sinth2/Xi_a*dph - b*costh2/Xi_b*dps s1.display()
dt+(aμ2aa221)dϕ\mathrm{d} t + \left( -\frac{a {\mu}^{2} - a}{a^{2} {\ell}^{2} - 1} \right) \mathrm{d} {\phi}
In [18]:
s2 = a*dt - (r^2 + a^2)/Xi_a*dph s2.display()
adt+(a2+r2a221)dϕa \mathrm{d} t + \left( \frac{a^{2} + r^{2}}{a^{2} {\ell}^{2} - 1} \right) \mathrm{d} {\phi}
In [19]:
s3 = b*dt - (r^2 + b^2)/Xi_b*dps s3.display()
r2dψ-r^{2} \mathrm{d} {\psi}
In [20]:
s4 = a*b*dt - b*(r^2 + a^2)*sinth2/Xi_a * dph - a*(r^2 + b^2)*costh2/Xi_b * dps s4.display()
aμ2r2dψ-a {\mu}^{2} r^{2} \mathrm{d} {\psi}
In [21]:
G0 = - Delta_r/rho2 * s1*s1 + Delta_th*sinth2/rho2 * s2*s2 + Delta_th*costh2/rho2 * s3*s3 \ + rho2/Delta_r * dr*dr + rho2/Delta_th * dth*dth + sig/rho2 * s4*s4 G0.display_comp(only_nonredundant=True)
Xtttt=a42μ42r4(a42+a2)μ2(a22+1)r2+2ma2μ2+r2Xtϕtϕ=a52μ4+(a2μ2a2)r4(a52+2am)μ2+(a32μ4a32)r2+2am(a42a2)μ2+(a221)r2Xrrrr=a2μ2+r22r4+(a22+1)r2+a22mXμμμμ=a2μ2+r2a22μ4(a22+1)μ2+1Xϕtϕt=a52μ4+(a2μ2a2)r4(a52+2am)μ2+(a32μ4a32)r2+2am(a42a2)μ2+(a221)r2Xϕϕϕϕ=(a62a4+2a2m)μ4(a22(a221)μ21)r4+2a2m(a62a4+4a2m)μ2(a42(a42a2)μ4a2)r2(a642a42+a2)μ2+(a442a22+1)r2Xψψψψ=μ2r2\begin{array}{lcl} X_{ \, t \, t }^{ \phantom{\, t}\phantom{\, t} } & = & \frac{a^{4} {\ell}^{2} {\mu}^{4} - {\ell}^{2} r^{4} - {\left(a^{4} {\ell}^{2} + a^{2}\right)} {\mu}^{2} - {\left(a^{2} {\ell}^{2} + 1\right)} r^{2} + 2 \, m}{a^{2} {\mu}^{2} + r^{2}} \\ X_{ \, t \, {\phi} }^{ \phantom{\, t}\phantom{\, {\phi}} } & = & \frac{a^{5} {\ell}^{2} {\mu}^{4} + {\left(a {\ell}^{2} {\mu}^{2} - a {\ell}^{2}\right)} r^{4} - {\left(a^{5} {\ell}^{2} + 2 \, a m\right)} {\mu}^{2} + {\left(a^{3} {\ell}^{2} {\mu}^{4} - a^{3} {\ell}^{2}\right)} r^{2} + 2 \, a m}{{\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2}} \\ X_{ \, r \, r }^{ \phantom{\, r}\phantom{\, r} } & = & \frac{a^{2} {\mu}^{2} + r^{2}}{{\ell}^{2} r^{4} + {\left(a^{2} {\ell}^{2} + 1\right)} r^{2} + a^{2} - 2 \, m} \\ X_{ \, {\mu} \, {\mu} }^{ \phantom{\, {\mu}}\phantom{\, {\mu}} } & = & \frac{a^{2} {\mu}^{2} + r^{2}}{a^{2} {\ell}^{2} {\mu}^{4} - {\left(a^{2} {\ell}^{2} + 1\right)} {\mu}^{2} + 1} \\ X_{ \, {\phi} \, t }^{ \phantom{\, {\phi}}\phantom{\, t} } & = & \frac{a^{5} {\ell}^{2} {\mu}^{4} + {\left(a {\ell}^{2} {\mu}^{2} - a {\ell}^{2}\right)} r^{4} - {\left(a^{5} {\ell}^{2} + 2 \, a m\right)} {\mu}^{2} + {\left(a^{3} {\ell}^{2} {\mu}^{4} - a^{3} {\ell}^{2}\right)} r^{2} + 2 \, a m}{{\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{2} + {\left(a^{2} {\ell}^{2} - 1\right)} r^{2}} \\ X_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & \frac{{\left(a^{6} {\ell}^{2} - a^{4} + 2 \, a^{2} m\right)} {\mu}^{4} - {\left(a^{2} {\ell}^{2} - {\left(a^{2} {\ell}^{2} - 1\right)} {\mu}^{2} - 1\right)} r^{4} + 2 \, a^{2} m - {\left(a^{6} {\ell}^{2} - a^{4} + 4 \, a^{2} m\right)} {\mu}^{2} - {\left(a^{4} {\ell}^{2} - {\left(a^{4} {\ell}^{2} - a^{2}\right)} {\mu}^{4} - a^{2}\right)} r^{2}}{{\left(a^{6} {\ell}^{4} - 2 \, a^{4} {\ell}^{2} + a^{2}\right)} {\mu}^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2}} \\ X_{ \, {\psi} \, {\psi} }^{ \phantom{\, {\psi}}\phantom{\, {\psi}} } & = & {\mu}^{2} r^{2} \end{array}
In [22]:
G0 == G
True\mathrm{True}

Einstein equation

The Ricci tensor of gg is

In [23]:
if not keep_Delta: # Ric = G.ricci() # print(Ric) pass
In [24]:
if not keep_Delta: # show(Ric.display_comp(only_nonredundant=True)) pass

Let us check that gg is a solution of the vacuum Einstein equation with the cosmological constant Λ=62\Lambda = - 6 \ell^2:

In [25]:
Lambda = -6*l^2 if not keep_Delta: # print(Ric == 2/3*Lambda*G) pass

Check of Eq. (2.10)

One must have a=ba=b and keep_Delta == False for the test to pass:

In [26]:
if a == b and not keep_Delta: G1 = - (1 + rho2*l^2 - 2*m/rho2) * dt*dt + rho2/Delta_r * dr*dr \ + rho2/Delta_th * dth*dth \ + sinth2/Xi_a^2*(rho2*Xi_a + 2*a^2*m/rho2*sinth2) * dph * dph \ + costh2/Xi_a^2*(rho2*Xi_a + 2*a^2*m/rho2*costh2) * dps * dps \ + a*sinth2/Xi_a*(rho2*l^2 - 2*m/rho2) * (dt*dph + dph*dt) \ + a*costh2/Xi_a*(rho2*l^2 - 2*m/rho2) * (dt*dps + dps*dt) \ + 2*m*a^2*sinth2*costh2/Xi_a^2/rho2 * (dph*dps + dps*dph) print(G1 == G)

Check of Eq. (2.14)

One must have b=0b=0 and keep_Delta == False for the test to pass:

In [27]:
if b == 0 and not keep_Delta: s1 = dt - a*sinth2/Xi_a*dph s2 = a*dt - (r^2 + a^2)/Xi_a*dph G2 = - Delta_r/rho2* s1*s1 + rho2/Delta_r * dr*dr + rho2/Delta_th * dth*dth \ + Delta_th*sinth2/rho2* s2*s2 + r^2*costh2 * dps*dps print(G2 == G)
True

String worldsheet

The string worldsheet as a 2-dimensional Lorentzian manifold:

In [28]:
W = Manifold(2, 'W', structure='Lorentzian') print(W)
2-dimensional Lorentzian manifold W

Let us assume that the string worldsheet is parametrized by (t,r)(t,r):

In [29]:
XW.<t,r> = W.chart(r't r:(0,+oo)') XW
(W,(t,r))\left(W,(t, r)\right)

The string embedding in Kerr-AdS spacetime, as an expansion about a straight string solution in AdS:

In [30]:
Mu0 = var('Mu0', latex_name=r'\mu_0', domain='real') Phi0 = var('Phi0', latex_name=r'\Phi_0', domain='real') Psi0 = var('Psi0', latex_name=r'\Psi_0', domain='real') beta1 = var('beta1', latex_name=r'\beta_1', domain='real') beta2 = var('beta2', latex_name=r'\beta_2', domain='real') cosTh0 = Mu0 sinTh0 = sqrt(1 - Mu0^2) mu_s = Mu0 + a^2*function('mu_1')(r) ph_s = Phi0 + beta1*a*l^2*t + beta1*a*function('phi_1')(r) ps_s = Psi0 + beta2*a*l^2*t + beta2*a*function('psi_1')(r) F = W.diff_map(M, {(XW, BL): [t, r, mu_s, ph_s, ps_s]}, name='F') F.display()
F:WM(t,r)(t,r,μ,ϕ,ψ)=(t,r,a2μ1(r)+μ0,aβ12t+aβ1ϕ1(r)+Φ0,aβ22t+aβ2ψ1(r)+Ψ0)\begin{array}{llcl} F:& W & \longrightarrow & \mathcal{M} \\ & \left(t, r\right) & \longmapsto & \left(t, r, {\mu}, {\phi}, {\psi}\right) = \left(t, r, a^{2} \mu_{1}\left(r\right) + {\mu_0}, a {\beta_1} {\ell}^{2} t + a {\beta_1} \phi_{1}\left(r\right) + {\Phi_0}, a {\beta_2} {\ell}^{2} t + a {\beta_2} \psi_{1}\left(r\right) + {\Psi_0}\right) \end{array}
In [31]:
F.jacobian_matrix()
(10010a2rμ1(r)aβ12aβ1rϕ1(r)aβ22aβ2rψ1(r))\left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \\ 0 & a^{2} \frac{\partial}{\partial r}\mu_{1}\left(r\right) \\ a {\beta_1} {\ell}^{2} & a {\beta_1} \frac{\partial}{\partial r}\phi_{1}\left(r\right) \\ a {\beta_2} {\ell}^{2} & a {\beta_2} \frac{\partial}{\partial r}\psi_{1}\left(r\right) \end{array}\right)

Induced metric on the string worldsheet

The string worldsheet metric is the metric gg induced by the spacetime metric GG, i.e. the pullback of GG by the embedding FF:

In [32]:
g = W.metric() g.set(F.pullback(G))
In [33]:
g[0,0]
(μ04+μ02)a42+((μ04μ02)a8β12+2(μ04μ02)a8β1+(μ04μ02)a8)6((μ04μ02)a6β12+2(μ04μ02)a6β1+(2μ04μ02)a6)4+(μ02a6β228(2μ02a4β22(μ021)a4β122(μ021)a4β1+a4)6+(μ02a2β22(μ021)a2β122(μ021)a2β1+2a2)42)r4+(2a12β124m+a122+(a16β12+2a16β1+a16)6(a14β12+2a14β1+2a14)4+(a16β228+(a14β122a14β22+2a14β1)6(a12β12a12β22+2a12β1)4)r2)μ1(r)4μ02a2+4(2μ0a10β124m+μ0a102+(μ0a14β12+2μ0a14β1+μ0a14)6(μ0a12β12+2μ0a12β1+2μ0a12)4+(μ0a14β228+(μ0a12β122μ0a12β22+2μ0a12β1)6(μ0a10β12μ0a10β22+2μ0a10β1)4)r2)μ1(r)3+(μ04a8β228(2μ04a6β22(μ041)a6β122(μ041)a6β1+a6)6+(μ04a4β22(μ041)a4β122(μ041)a4β1+a4)4+a221)r2+((6μ02+1)a82+((6μ021)a12β12+2(6μ021)a12β1+(6μ021)a12)6a6((6μ021)a10β12+2(6μ021)a10β1+(12μ021)a10)4+(a10β228+(a8β122a8β22+2a8β1)6(a6β12a6β22+2a6β1)4)r4+6(μ02a12β228+(μ02a10β122μ02a10β22+2μ02a10β1)6(μ02a8β12μ02a8β22+2μ02a8β1)4)r2+4(a6β12+((3μ021)a8β12a8β1)4)m)μ1(r)2+2(((μ042μ02+1)a4β122(μ021)a4β1+a4)4+2((μ021)a2β1a2)2+1)m+2((2μ03+μ0)a62+((2μ03μ0)a10β12+2(2μ03μ0)a10β1+(2μ03μ0)a10)6μ0a4((2μ03μ0)a8β12+2(2μ03μ0)a8β1+(4μ03μ0)a8)4+(μ0a8β228+(μ0a6β122μ0a6β22+2μ0a6β1)6(μ0a4β12μ0a4β22+2μ0a4β1)4)r4+2(μ03a10β228+(μ03a8β122μ03a8β22+2μ03a8β1)6(μ03a6β12μ03a6β22+2μ03a6β1)4)r2+4(μ0a4β12+((μ03μ0)a6β12μ0a6β1)4)m)μ1(r)μ02a642μ02a42+μ02a2+(a442a22+1)r2+(a1042a82+a6)μ1(r)2+2(μ0a842μ0a62+μ0a4)μ1(r)\frac{{\left({\mu_0}^{4} + {\mu_0}^{2}\right)} a^{4} {\ell}^{2} + {\left({\left({\mu_0}^{4} - {\mu_0}^{2}\right)} a^{8} {\beta_1}^{2} + 2 \, {\left({\mu_0}^{4} - {\mu_0}^{2}\right)} a^{8} {\beta_1} + {\left({\mu_0}^{4} - {\mu_0}^{2}\right)} a^{8}\right)} {\ell}^{6} - {\left({\left({\mu_0}^{4} - {\mu_0}^{2}\right)} a^{6} {\beta_1}^{2} + 2 \, {\left({\mu_0}^{4} - {\mu_0}^{2}\right)} a^{6} {\beta_1} + {\left(2 \, {\mu_0}^{4} - {\mu_0}^{2}\right)} a^{6}\right)} {\ell}^{4} + {\left({\mu_0}^{2} a^{6} {\beta_2}^{2} {\ell}^{8} - {\left(2 \, {\mu_0}^{2} a^{4} {\beta_2}^{2} - {\left({\mu_0}^{2} - 1\right)} a^{4} {\beta_1}^{2} - 2 \, {\left({\mu_0}^{2} - 1\right)} a^{4} {\beta_1} + a^{4}\right)} {\ell}^{6} + {\left({\mu_0}^{2} a^{2} {\beta_2}^{2} - {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} - 2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1} + 2 \, a^{2}\right)} {\ell}^{4} - {\ell}^{2}\right)} r^{4} + {\left(2 \, a^{12} {\beta_1}^{2} {\ell}^{4} m + a^{12} {\ell}^{2} + {\left(a^{16} {\beta_1}^{2} + 2 \, a^{16} {\beta_1} + a^{16}\right)} {\ell}^{6} - {\left(a^{14} {\beta_1}^{2} + 2 \, a^{14} {\beta_1} + 2 \, a^{14}\right)} {\ell}^{4} + {\left(a^{16} {\beta_2}^{2} {\ell}^{8} + {\left(a^{14} {\beta_1}^{2} - 2 \, a^{14} {\beta_2}^{2} + 2 \, a^{14} {\beta_1}\right)} {\ell}^{6} - {\left(a^{12} {\beta_1}^{2} - a^{12} {\beta_2}^{2} + 2 \, a^{12} {\beta_1}\right)} {\ell}^{4}\right)} r^{2}\right)} \mu_{1}\left(r\right)^{4} - {\mu_0}^{2} a^{2} + 4 \, {\left(2 \, {\mu_0} a^{10} {\beta_1}^{2} {\ell}^{4} m + {\mu_0} a^{10} {\ell}^{2} + {\left({\mu_0} a^{14} {\beta_1}^{2} + 2 \, {\mu_0} a^{14} {\beta_1} + {\mu_0} a^{14}\right)} {\ell}^{6} - {\left({\mu_0} a^{12} {\beta_1}^{2} + 2 \, {\mu_0} a^{12} {\beta_1} + 2 \, {\mu_0} a^{12}\right)} {\ell}^{4} + {\left({\mu_0} a^{14} {\beta_2}^{2} {\ell}^{8} + {\left({\mu_0} a^{12} {\beta_1}^{2} - 2 \, {\mu_0} a^{12} {\beta_2}^{2} + 2 \, {\mu_0} a^{12} {\beta_1}\right)} {\ell}^{6} - {\left({\mu_0} a^{10} {\beta_1}^{2} - {\mu_0} a^{10} {\beta_2}^{2} + 2 \, {\mu_0} a^{10} {\beta_1}\right)} {\ell}^{4}\right)} r^{2}\right)} \mu_{1}\left(r\right)^{3} + {\left({\mu_0}^{4} a^{8} {\beta_2}^{2} {\ell}^{8} - {\left(2 \, {\mu_0}^{4} a^{6} {\beta_2}^{2} - {\left({\mu_0}^{4} - 1\right)} a^{6} {\beta_1}^{2} - 2 \, {\left({\mu_0}^{4} - 1\right)} a^{6} {\beta_1} + a^{6}\right)} {\ell}^{6} + {\left({\mu_0}^{4} a^{4} {\beta_2}^{2} - {\left({\mu_0}^{4} - 1\right)} a^{4} {\beta_1}^{2} - 2 \, {\left({\mu_0}^{4} - 1\right)} a^{4} {\beta_1} + a^{4}\right)} {\ell}^{4} + a^{2} {\ell}^{2} - 1\right)} r^{2} + {\left({\left(6 \, {\mu_0}^{2} + 1\right)} a^{8} {\ell}^{2} + {\left({\left(6 \, {\mu_0}^{2} - 1\right)} a^{12} {\beta_1}^{2} + 2 \, {\left(6 \, {\mu_0}^{2} - 1\right)} a^{12} {\beta_1} + {\left(6 \, {\mu_0}^{2} - 1\right)} a^{12}\right)} {\ell}^{6} - a^{6} - {\left({\left(6 \, {\mu_0}^{2} - 1\right)} a^{10} {\beta_1}^{2} + 2 \, {\left(6 \, {\mu_0}^{2} - 1\right)} a^{10} {\beta_1} + {\left(12 \, {\mu_0}^{2} - 1\right)} a^{10}\right)} {\ell}^{4} + {\left(a^{10} {\beta_2}^{2} {\ell}^{8} + {\left(a^{8} {\beta_1}^{2} - 2 \, a^{8} {\beta_2}^{2} + 2 \, a^{8} {\beta_1}\right)} {\ell}^{6} - {\left(a^{6} {\beta_1}^{2} - a^{6} {\beta_2}^{2} + 2 \, a^{6} {\beta_1}\right)} {\ell}^{4}\right)} r^{4} + 6 \, {\left({\mu_0}^{2} a^{12} {\beta_2}^{2} {\ell}^{8} + {\left({\mu_0}^{2} a^{10} {\beta_1}^{2} - 2 \, {\mu_0}^{2} a^{10} {\beta_2}^{2} + 2 \, {\mu_0}^{2} a^{10} {\beta_1}\right)} {\ell}^{6} - {\left({\mu_0}^{2} a^{8} {\beta_1}^{2} - {\mu_0}^{2} a^{8} {\beta_2}^{2} + 2 \, {\mu_0}^{2} a^{8} {\beta_1}\right)} {\ell}^{4}\right)} r^{2} + 4 \, {\left(a^{6} {\beta_1} {\ell}^{2} + {\left({\left(3 \, {\mu_0}^{2} - 1\right)} a^{8} {\beta_1}^{2} - a^{8} {\beta_1}\right)} {\ell}^{4}\right)} m\right)} \mu_{1}\left(r\right)^{2} + 2 \, {\left({\left({\left({\mu_0}^{4} - 2 \, {\mu_0}^{2} + 1\right)} a^{4} {\beta_1}^{2} - 2 \, {\left({\mu_0}^{2} - 1\right)} a^{4} {\beta_1} + a^{4}\right)} {\ell}^{4} + 2 \, {\left({\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1} - a^{2}\right)} {\ell}^{2} + 1\right)} m + 2 \, {\left({\left(2 \, {\mu_0}^{3} + {\mu_0}\right)} a^{6} {\ell}^{2} + {\left({\left(2 \, {\mu_0}^{3} - {\mu_0}\right)} a^{10} {\beta_1}^{2} + 2 \, {\left(2 \, {\mu_0}^{3} - {\mu_0}\right)} a^{10} {\beta_1} + {\left(2 \, {\mu_0}^{3} - {\mu_0}\right)} a^{10}\right)} {\ell}^{6} - {\mu_0} a^{4} - {\left({\left(2 \, {\mu_0}^{3} - {\mu_0}\right)} a^{8} {\beta_1}^{2} + 2 \, {\left(2 \, {\mu_0}^{3} - {\mu_0}\right)} a^{8} {\beta_1} + {\left(4 \, {\mu_0}^{3} - {\mu_0}\right)} a^{8}\right)} {\ell}^{4} + {\left({\mu_0} a^{8} {\beta_2}^{2} {\ell}^{8} + {\left({\mu_0} a^{6} {\beta_1}^{2} - 2 \, {\mu_0} a^{6} {\beta_2}^{2} + 2 \, {\mu_0} a^{6} {\beta_1}\right)} {\ell}^{6} - {\left({\mu_0} a^{4} {\beta_1}^{2} - {\mu_0} a^{4} {\beta_2}^{2} + 2 \, {\mu_0} a^{4} {\beta_1}\right)} {\ell}^{4}\right)} r^{4} + 2 \, {\left({\mu_0}^{3} a^{10} {\beta_2}^{2} {\ell}^{8} + {\left({\mu_0}^{3} a^{8} {\beta_1}^{2} - 2 \, {\mu_0}^{3} a^{8} {\beta_2}^{2} + 2 \, {\mu_0}^{3} a^{8} {\beta_1}\right)} {\ell}^{6} - {\left({\mu_0}^{3} a^{6} {\beta_1}^{2} - {\mu_0}^{3} a^{6} {\beta_2}^{2} + 2 \, {\mu_0}^{3} a^{6} {\beta_1}\right)} {\ell}^{4}\right)} r^{2} + 4 \, {\left({\mu_0} a^{4} {\beta_1} {\ell}^{2} + {\left({\left({\mu_0}^{3} - {\mu_0}\right)} a^{6} {\beta_1}^{2} - {\mu_0} a^{6} {\beta_1}\right)} {\ell}^{4}\right)} m\right)} \mu_{1}\left(r\right)}{{\mu_0}^{2} a^{6} {\ell}^{4} - 2 \, {\mu_0}^{2} a^{4} {\ell}^{2} + {\mu_0}^{2} a^{2} + {\left(a^{4} {\ell}^{4} - 2 \, a^{2} {\ell}^{2} + 1\right)} r^{2} + {\left(a^{10} {\ell}^{4} - 2 \, a^{8} {\ell}^{2} + a^{6}\right)} \mu_{1}\left(r\right)^{2} + 2 \, {\left({\mu_0} a^{8} {\ell}^{4} - 2 \, {\mu_0} a^{6} {\ell}^{2} + {\mu_0} a^{4}\right)} \mu_{1}\left(r\right)}
In [34]:
# g[0,0].expr().factor()

Reduced form of gtrg_{tr}:

In [35]:
if b == 0 and not keep_Delta: gtr = a^2*sinth2/(rho2*Xi_a^2)*(beta1*Xi_a*(Delta_r - Delta_th*(r^2 + a^2)) \ + beta1^2*l^2*(Delta_th*(r^2 + a^2)^2 - a^2*Delta_r*sinth2)) \ * diff(phi_1(r), r) \ + a^2*l^2*beta2^2*r^2*costh2 * diff(psi_1(r), r) gtr = gtr.subs({mu: mu_s, ph: ph_s, ps: ps_s}) print(g[0,1] == gtr)
True

Reduced form of gttg_{tt}:

In [36]:
if b == 0 and not keep_Delta: gtt = (a^2*Delta_th*sinth2 - Delta_r)/rho2 \ + (2*beta1*a^2*l^2*sinth2*(Delta_r - Delta_th*(r^2 + a^2)))/(rho2*Xi_a) \ + beta1^2*a^2*l^4*sinth2/(rho2*Xi_a^2)*(Delta_th*(r^2 + a^2)^2 - a^2*Delta_r*sinth2) \ + beta2^2*a^2*l^4*r^2*costh2 gtt = gtt.subs({mu: mu_s, ph: ph_s, ps: ps_s}) print(g[0,0] == gtt)
True

Reduced form of grrg_{rr}:

In [37]:
if b == 0 and not keep_Delta: grr = rho2/Delta_r + a^2*beta1^2*sinth2/(Xi_a^2*rho2)*(Delta_th*(r^2 + a^2)^2 - Delta_r*a^2*sinth2)*diff(phi_1(r), r)^2 \ + a^4*rho2/Delta_th/sinth2*diff(mu_1(r), r)^2 \ + beta2^2*a^2*r^2*costh2*diff(psi_1(r), r)^2 grr = grr.subs({mu: mu_s, ph: ph_s, ps: ps_s}) print(g[1,1] == grr)
True

Nambu-Goto action

In [38]:
detg = g.determinant().expr()

Expanding at second order in aa:

In [39]:
detg_a2 = detg.series(a, 3).truncate().simplify_full() detg_a2
((μ02a2β22(μ021)a2β122(μ021)a2β1)42)r4(μ021)a2+((μ021)a2β124r8+2(μ021)a2β122r64(μ021)a2β12mr2+4(μ021)a2β12m2(4(μ021)a2β122m(μ021)a2β12)r4)rϕ1(r)2(μ02a2β224r8+2μ02a2β222r64μ02a2β22mr2+4μ02a2β22m2(4μ02a2β222mμ02a2β22)r4)rψ1(r)2+2(2(μ021)a2β12+1)mr22r4+r22m\frac{{\left({\left({\mu_0}^{2} a^{2} {\beta_2}^{2} - {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} - 2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}\right)} {\ell}^{4} - {\ell}^{2}\right)} r^{4} - {\left({\mu_0}^{2} - 1\right)} a^{2} + {\left({\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} {\ell}^{4} r^{8} + 2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} {\ell}^{2} r^{6} - 4 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} m r^{2} + 4 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} m^{2} - {\left(4 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} {\ell}^{2} m - {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2}\right)} r^{4}\right)} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - {\left({\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{4} r^{8} + 2 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{2} r^{6} - 4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} m r^{2} + 4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} m^{2} - {\left(4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{2} m - {\mu_0}^{2} a^{2} {\beta_2}^{2}\right)} r^{4}\right)} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} + 2 \, {\left(2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1} {\ell}^{2} + 1\right)} m - r^{2}}{{\ell}^{2} r^{4} + r^{2} - 2 \, m}

The Nambu-Goto Lagrangian at second order in aa:

In [40]:
L_a2 = (sqrt(-detg_a2)).series(a, 3).truncate().simplify_full() L_a2
((μ02a2β22(μ021)a2β122(μ021)a2β1)422)r4(μ021)a2+((μ021)a2β124r8+2(μ021)a2β122r64(μ021)a2β12mr2+4(μ021)a2β12m2(4(μ021)a2β122m(μ021)a2β12)r4)rϕ1(r)2(μ02a2β224r8+2μ02a2β222r64μ02a2β22mr2+4μ02a2β22m2(4μ02a2β222mμ02a2β22)r4)rψ1(r)2+4((μ021)a2β12+1)m2r22(2r4+r22m)-\frac{{\left({\left({\mu_0}^{2} a^{2} {\beta_2}^{2} - {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} - 2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}\right)} {\ell}^{4} - 2 \, {\ell}^{2}\right)} r^{4} - {\left({\mu_0}^{2} - 1\right)} a^{2} + {\left({\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} {\ell}^{4} r^{8} + 2 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} {\ell}^{2} r^{6} - 4 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} m r^{2} + 4 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} m^{2} - {\left(4 \, {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2} {\ell}^{2} m - {\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1}^{2}\right)} r^{4}\right)} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - {\left({\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{4} r^{8} + 2 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{2} r^{6} - 4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} m r^{2} + 4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} m^{2} - {\left(4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{2} m - {\mu_0}^{2} a^{2} {\beta_2}^{2}\right)} r^{4}\right)} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} + 4 \, {\left({\left({\mu_0}^{2} - 1\right)} a^{2} {\beta_1} {\ell}^{2} + 1\right)} m - 2 \, r^{2}}{2 \, {\left({\ell}^{2} r^{4} + r^{2} - 2 \, m\right)}}
In [41]:
L_a2.numerator()
μ02a2β124r8rϕ1(r)2+μ02a2β224r8rψ1(r)2+a2β124r8rϕ1(r)22μ02a2β122r6rϕ1(r)2+2μ02a2β222r6rψ1(r)2+4μ02a2β122mr4rϕ1(r)24μ02a2β222mr4rψ1(r)2+μ02a2β124r4μ02a2β224r4+2a2β122r6rϕ1(r)2+2μ02a2β14r44a2β122mr4rϕ1(r)2a2β124r4μ02a2β12r4rϕ1(r)2+μ02a2β22r4rψ1(r)22a2β14r4+4μ02a2β12mr2rϕ1(r)24μ02a2β22mr2rψ1(r)24μ02a2β12m2rϕ1(r)2+a2β12r4rϕ1(r)2+4μ02a2β22m2rψ1(r)24a2β12mr2rϕ1(r)24μ02a2β12m+4a2β12m2rϕ1(r)2+4a2β12m+22r4+μ02a2a2+2r24m-{\mu_0}^{2} a^{2} {\beta_1}^{2} {\ell}^{4} r^{8} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{4} r^{8} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} + a^{2} {\beta_1}^{2} {\ell}^{4} r^{8} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - 2 \, {\mu_0}^{2} a^{2} {\beta_1}^{2} {\ell}^{2} r^{6} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + 2 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{2} r^{6} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} + 4 \, {\mu_0}^{2} a^{2} {\beta_1}^{2} {\ell}^{2} m r^{4} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - 4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{2} m r^{4} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} + {\mu_0}^{2} a^{2} {\beta_1}^{2} {\ell}^{4} r^{4} - {\mu_0}^{2} a^{2} {\beta_2}^{2} {\ell}^{4} r^{4} + 2 \, a^{2} {\beta_1}^{2} {\ell}^{2} r^{6} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + 2 \, {\mu_0}^{2} a^{2} {\beta_1} {\ell}^{4} r^{4} - 4 \, a^{2} {\beta_1}^{2} {\ell}^{2} m r^{4} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - a^{2} {\beta_1}^{2} {\ell}^{4} r^{4} - {\mu_0}^{2} a^{2} {\beta_1}^{2} r^{4} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + {\mu_0}^{2} a^{2} {\beta_2}^{2} r^{4} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} - 2 \, a^{2} {\beta_1} {\ell}^{4} r^{4} + 4 \, {\mu_0}^{2} a^{2} {\beta_1}^{2} m r^{2} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - 4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} m r^{2} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} - 4 \, {\mu_0}^{2} a^{2} {\beta_1}^{2} m^{2} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + a^{2} {\beta_1}^{2} r^{4} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + 4 \, {\mu_0}^{2} a^{2} {\beta_2}^{2} m^{2} \frac{\partial}{\partial r}\psi_{1}\left(r\right)^{2} - 4 \, a^{2} {\beta_1}^{2} m r^{2} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} - 4 \, {\mu_0}^{2} a^{2} {\beta_1} {\ell}^{2} m + 4 \, a^{2} {\beta_1}^{2} m^{2} \frac{\partial}{\partial r}\phi_{1}\left(r\right)^{2} + 4 \, a^{2} {\beta_1} {\ell}^{2} m + 2 \, {\ell}^{2} r^{4} + {\mu_0}^{2} a^{2} - a^{2} + 2 \, r^{2} - 4 \, m