CoCalc Public FilesBHLectures / sage / Kerr_equatorial_geodesics.ipynb
Author: Eric Gourgoulhon
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Compute Environment: Ubuntu 18.04 (Deprecated)

# Equatorial geodesics in Kerr spacetime

In [1]:
%display latex


The spacetime manifold and Boyer-Lindquist coordinates:

In [2]:
M = Manifold(4, 'M')
X.<t,r,th,ph> = M.chart(r"t r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi")
X

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

The spacetime metric:

In [3]:
g = M.lorentzian_metric('g')
var('m, a', domain='real')
rho2 = r^2 + (a*cos(th))^2
Delta = r^2 -2*m*r + a^2
g[0,0] = -(1-2*m*r/rho2)
g[0,3] = -2*a*m*r*sin(th)^2/rho2
g[1,1], g[2,2] = rho2/Delta, rho2
g[3,3] = (r^2+a^2+2*m*r*(a*sin(th))^2/rho2)*sin(th)^2
g.display()

$g = \left( \frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} - 1 \right) \mathrm{d} t\otimes \mathrm{d} t + \left( -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} t\otimes \mathrm{d} {\phi} + \left( \frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{a^{2} - 2 \, m r + r^{2}} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( a^{2} \cos\left({\theta}\right)^{2} + r^{2} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \left( -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} t + {\left(\frac{2 \, a^{2} m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} + a^{2} + r^{2}\right)} \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$
In [4]:
u = M.vector_field('u')
var('eps', latex_name=r'\varepsilon')
var('ell', latex_name=r'\ell')
u[0] = ((r^2 + a^2*(1+2*m/r))*eps - 2*a*m/r*ell)/Delta
u[1] = sqrt(eps^2 - 1 + 2*m/r - (ell^2-a^2*(eps^2-1))/r^2
+ 2*m/r^3*(ell-a*eps)^2)
u[3] = (2*a*m/r*eps + (1-2*m/r)*ell)/Delta
u.display_comp()

$\begin{array}{lcl} u_{\phantom{\, t}}^{ \, t } & = & \frac{{\left(a^{2} {\left(\frac{2 \, m}{r} + 1\right)} + r^{2}\right)} {\varepsilon} - \frac{2 \, a {\ell} m}{r}}{a^{2} - 2 \, m r + r^{2}} \\ u_{\phantom{\, r}}^{ \, r } & = & \sqrt{{\varepsilon}^{2} + \frac{2 \, {\left(a {\varepsilon} - {\ell}\right)}^{2} m}{r^{3}} + \frac{2 \, m}{r} + \frac{{\left({\varepsilon}^{2} - 1\right)} a^{2} - {\ell}^{2}}{r^{2}} - 1} \\ u_{\phantom{\, {\phi}}}^{ \, {\phi} } & = & \frac{\frac{2 \, a {\varepsilon} m}{r} - {\ell} {\left(\frac{2 \, m}{r} - 1\right)}}{a^{2} - 2 \, m r + r^{2}} \end{array}$
In [5]:
norm = g(u,u)
norm.coord_function()

$\frac{{\left(a^{2} {\varepsilon}^{2} - 2 \, a^{2} - {\ell}^{2}\right)} r^{3} \cos\left({\theta}\right)^{2} - r^{5} + {\left(2 \, {\left(a^{4} {\varepsilon}^{2} - 2 \, a^{3} {\ell} {\varepsilon} + a^{2} {\ell}^{2}\right)} m + {\left(a^{4} {\varepsilon}^{2} - a^{4} - a^{2} {\ell}^{2}\right)} r\right)} \cos\left({\theta}\right)^{4}}{a^{2} r^{3} \cos\left({\theta}\right)^{2} + r^{5}}$

Value of $g(u,u)$ in the equatorial plane ($\theta=\frac{\pi}{2}$):

In [6]:
norm.coord_function()(t,r,pi/2,ph)

$-1$
In [7]:
nabla = g.connection()
print(nabla)

Levi-Civita connection nabla_g associated with the Lorentzian metric g on the 4-dimensional differentiable manifold M

The 4-acceleration vector $a = \nabla_{u}\, u$:

In [8]:
Du = nabla(u)
a = u.contract(0, Du, 1)
a.set_name('a')
a.display_comp()

$\begin{array}{lcl} a_{\phantom{\, t}}^{ \, t } & = & -\frac{2 \, {\left({\left({\left(a^{4} {\varepsilon} - a^{3} {\ell}\right)} m r^{2} + {\left(a^{6} {\varepsilon} - a^{5} {\ell}\right)} m\right)} \cos\left({\theta}\right)^{4} + 3 \, {\left({\left(a^{2} {\varepsilon} - a {\ell}\right)} m r^{4} + {\left(a^{4} {\varepsilon} - a^{3} {\ell}\right)} m r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sqrt{{\left({\varepsilon}^{2} - 1\right)} r^{3} + 2 \, m r^{2} + 2 \, {\left(a^{2} {\varepsilon}^{2} - 2 \, a {\ell} {\varepsilon} + {\ell}^{2}\right)} m + {\left(a^{2} {\varepsilon}^{2} - a^{2} - {\ell}^{2}\right)} r}}{{\left(a^{2} r^{7} - 2 \, m r^{8} + r^{9} + {\left(a^{6} r^{3} - 2 \, a^{4} m r^{4} + a^{4} r^{5}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{5} - 2 \, a^{2} m r^{6} + a^{2} r^{7}\right)} \cos\left({\theta}\right)^{2}\right)} \sqrt{r}} \\ a_{\phantom{\, r}}^{ \, r } & = & \frac{{\left({\left(a^{6} {\varepsilon}^{2} + 4 \, a^{5} {\ell} {\varepsilon} - 4 \, a^{6} - 5 \, a^{4} {\ell}^{2}\right)} m r^{4} - {\left(a^{6} {\varepsilon}^{2} - a^{6} - a^{4} {\ell}^{2}\right)} r^{5} - 2 \, {\left(a^{8} {\varepsilon}^{2} - a^{8} - a^{6} {\ell}^{2} - 2 \, {\left(a^{6} {\varepsilon}^{2} - 3 \, a^{5} {\ell} {\varepsilon} + a^{6} + 2 \, a^{4} {\ell}^{2}\right)} m^{2}\right)} r^{3} - 2 \, {\left(2 \, {\left(a^{6} {\varepsilon}^{2} - 2 \, a^{5} {\ell} {\varepsilon} + a^{4} {\ell}^{2}\right)} m^{3} + {\left(a^{8} {\varepsilon}^{2} - 5 \, a^{7} {\ell} {\varepsilon} + 2 \, a^{8} + 4 \, a^{6} {\ell}^{2}\right)} m\right)} r^{2} - 3 \, {\left(a^{10} {\varepsilon}^{2} - 2 \, a^{9} {\ell} {\varepsilon} + a^{8} {\ell}^{2}\right)} m - {\left(a^{10} {\varepsilon}^{2} - a^{10} - a^{8} {\ell}^{2} - 8 \, {\left(a^{8} {\varepsilon}^{2} - 2 \, a^{7} {\ell} {\varepsilon} + a^{6} {\ell}^{2}\right)} m^{2}\right)} r\right)} \cos\left({\theta}\right)^{6} + {\left({\left(a^{4} {\varepsilon}^{2} + 6 \, a^{3} {\ell} {\varepsilon} - 8 \, a^{4} - 7 \, a^{2} {\ell}^{2}\right)} m r^{6} - 2 \, {\left(a^{4} {\varepsilon}^{2} - a^{4} - a^{2} {\ell}^{2}\right)} r^{7} - 4 \, {\left(a^{6} {\varepsilon}^{2} - a^{6} - a^{4} {\ell}^{2} - {\left(a^{4} {\varepsilon}^{2} - 2 \, a^{3} {\ell} {\varepsilon} + 2 \, a^{4} + a^{2} {\ell}^{2}\right)} m^{2}\right)} r^{5} + 2 \, {\left(2 \, {\left(a^{4} {\varepsilon}^{2} - 2 \, a^{3} {\ell} {\varepsilon} + a^{2} {\ell}^{2}\right)} m^{3} - {\left(3 \, a^{6} {\varepsilon}^{2} - 10 \, a^{5} {\ell} {\varepsilon} + 4 \, a^{6} + 7 \, a^{4} {\ell}^{2}\right)} m\right)} r^{4} - 7 \, {\left(a^{8} {\varepsilon}^{2} - 2 \, a^{7} {\ell} {\varepsilon} + a^{6} {\ell}^{2}\right)} m r^{2} - 2 \, {\left(a^{8} {\varepsilon}^{2} - a^{8} - a^{6} {\ell}^{2} - 6 \, {\left(a^{6} {\varepsilon}^{2} - 2 \, a^{5} {\ell} {\varepsilon} + a^{4} {\ell}^{2}\right)} m^{2}\right)} r^{3}\right)} \cos\left({\theta}\right)^{4} - {\left(2 \, {\left(a^{2} {\varepsilon}^{2} - 3 \, a {\ell} {\varepsilon} + 2 \, a^{2} + 2 \, {\ell}^{2}\right)} m r^{8} + {\left(a^{2} {\varepsilon}^{2} - a^{2} - {\ell}^{2}\right)} r^{9} + 2 \, {\left(4 \, a^{4} {\varepsilon}^{2} - 9 \, a^{3} {\ell} {\varepsilon} + 2 \, a^{4} + 5 \, a^{2} {\ell}^{2}\right)} m r^{6} + 2 \, {\left(a^{4} {\varepsilon}^{2} - a^{4} - a^{2} {\ell}^{2} - 2 \, {\left(2 \, a^{2} {\varepsilon}^{2} - 3 \, a {\ell} {\varepsilon} + a^{2} + {\ell}^{2}\right)} m^{2}\right)} r^{7} + 6 \, {\left(a^{6} {\varepsilon}^{2} - 2 \, a^{5} {\ell} {\varepsilon} + a^{4} {\ell}^{2}\right)} m r^{4} + {\left(a^{6} {\varepsilon}^{2} - a^{6} - a^{4} {\ell}^{2} - 12 \, {\left(a^{4} {\varepsilon}^{2} - 2 \, a^{3} {\ell} {\varepsilon} + a^{2} {\ell}^{2}\right)} m^{2}\right)} r^{5}\right)} \cos\left({\theta}\right)^{2}}{a^{2} r^{10} - 2 \, m r^{11} + r^{12} + {\left(a^{8} r^{4} - 2 \, a^{6} m r^{5} + a^{6} r^{6}\right)} \cos\left({\theta}\right)^{6} + 3 \, {\left(a^{6} r^{6} - 2 \, a^{4} m r^{7} + a^{4} r^{8}\right)} \cos\left({\theta}\right)^{4} + 3 \, {\left(a^{4} r^{8} - 2 \, a^{2} m r^{9} + a^{2} r^{10}\right)} \cos\left({\theta}\right)^{2}} \\ a_{\phantom{\, {\theta}}}^{ \, {\theta} } & = & -\frac{{\left({\left(2 \, {\left(2 \, a^{5} {\ell} {\varepsilon} - a^{6} - 2 \, a^{4} {\ell}^{2}\right)} m r^{2} - {\left(a^{6} {\varepsilon}^{2} - a^{6} - a^{4} {\ell}^{2}\right)} r^{3} - 2 \, {\left(a^{8} {\varepsilon}^{2} - 2 \, a^{7} {\ell} {\varepsilon} + a^{6} {\ell}^{2}\right)} m - {\left(a^{8} {\varepsilon}^{2} - a^{8} - a^{6} {\ell}^{2} - 4 \, {\left(a^{6} {\varepsilon}^{2} - 2 \, a^{5} {\ell} {\varepsilon} + a^{4} {\ell}^{2}\right)} m^{2}\right)} r\right)} \cos\left({\theta}\right)^{5} + 2 \, {\left(2 \, {\left(2 \, a^{3} {\ell} {\varepsilon} - a^{4} - 2 \, a^{2} {\ell}^{2}\right)} m r^{4} - {\left(a^{4} {\varepsilon}^{2} - a^{4} - a^{2} {\ell}^{2}\right)} r^{5} - 2 \, {\left(a^{6} {\varepsilon}^{2} - 2 \, a^{5} {\ell} {\varepsilon} + a^{4} {\ell}^{2}\right)} m r^{2} - {\left(a^{6} {\varepsilon}^{2} - a^{6} - a^{4} {\ell}^{2} - 4 \, {\left(a^{4} {\varepsilon}^{2} - 2 \, a^{3} {\ell} {\varepsilon} + a^{2} {\ell}^{2}\right)} m^{2}\right)} r^{3}\right)} \cos\left({\theta}\right)^{3} + {\left(2 \, {\left(a^{2} {\varepsilon}^{2} - a^{2} - {\ell}^{2}\right)} m r^{6} - {\left(a^{2} {\varepsilon}^{2} - a^{2} - {\ell}^{2}\right)} r^{7} - {\left(a^{4} {\varepsilon}^{2} - a^{4} - a^{2} {\ell}^{2}\right)} r^{5}\right)} \cos\left({\theta}\right)\right)} \sin\left({\theta}\right)}{a^{2} r^{9} - 2 \, m r^{10} + r^{11} + {\left(a^{8} r^{3} - 2 \, a^{6} m r^{4} + a^{6} r^{5}\right)} \cos\left({\theta}\right)^{6} + 3 \, {\left(a^{6} r^{5} - 2 \, a^{4} m r^{6} + a^{4} r^{7}\right)} \cos\left({\theta}\right)^{4} + 3 \, {\left(a^{4} r^{7} - 2 \, a^{2} m r^{8} + a^{2} r^{9}\right)} \cos\left({\theta}\right)^{2}} \\ a_{\phantom{\, {\phi}}}^{ \, {\phi} } & = & -\frac{2 \, {\left(3 \, {\left(a^{3} {\varepsilon} - a^{2} {\ell}\right)} m r^{2} \cos\left({\theta}\right)^{2} + {\left(a^{5} {\varepsilon} - a^{4} {\ell}\right)} m \cos\left({\theta}\right)^{4}\right)} \sqrt{{\left({\varepsilon}^{2} - 1\right)} r^{3} + 2 \, m r^{2} + 2 \, {\left(a^{2} {\varepsilon}^{2} - 2 \, a {\ell} {\varepsilon} + {\ell}^{2}\right)} m + {\left(a^{2} {\varepsilon}^{2} - a^{2} - {\ell}^{2}\right)} r}}{{\left(a^{2} r^{7} - 2 \, m r^{8} + r^{9} + {\left(a^{6} r^{3} - 2 \, a^{4} m r^{4} + a^{4} r^{5}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{5} - 2 \, a^{2} m r^{6} + a^{2} r^{7}\right)} \cos\left({\theta}\right)^{2}\right)} \sqrt{r}} \end{array}$

Values of the 4-acceleration in the equatorial plane ($\theta=\frac{\pi}{2}$):

In [9]:
a[0](t,r,pi/2,ph)

$0$
In [10]:
a[1](t,r,pi/2,ph)

$0$
In [11]:
a[2](t,r,pi/2,ph)

$0$
In [12]:
a[3](t,r,pi/2,ph)

$0$

The (non-zero and non-redundant) Christoffel symbols in Boyer-Lindquist coordinates:

In [13]:
g.christoffel_symbols_display()

$\begin{array}{lcl} \Gamma_{ \phantom{\, t} \, t \, r }^{ \, t \phantom{\, t} \phantom{\, r} } & = & \frac{a^{2} m r^{2} + m r^{4} - {\left(a^{4} m + a^{2} m r^{2}\right)} \cos\left({\theta}\right)^{2}}{a^{2} r^{4} - 2 \, m r^{5} + r^{6} + {\left(a^{6} - 2 \, a^{4} m r + a^{4} r^{2}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{2} - 2 \, a^{2} m r^{3} + a^{2} r^{4}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, t} \, t \, {\theta} }^{ \, t \phantom{\, t} \phantom{\, {\theta}} } & = & -\frac{2 \, a^{2} m r \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \\ \Gamma_{ \phantom{\, t} \, r \, {\phi} }^{ \, t \phantom{\, r} \phantom{\, {\phi}} } & = & -\frac{{\left(a^{3} m r^{2} + 3 \, a m r^{4} - {\left(a^{5} m - a^{3} m r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} r^{4} - 2 \, m r^{5} + r^{6} + {\left(a^{6} - 2 \, a^{4} m r + a^{4} r^{2}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{2} - 2 \, a^{2} m r^{3} + a^{2} r^{4}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, t} \, {\theta} \, {\phi} }^{ \, t \phantom{\, {\theta}} \phantom{\, {\phi}} } & = & -\frac{2 \, {\left(a^{5} m r \cos\left({\theta}\right) \sin\left({\theta}\right)^{5} - {\left(a^{5} m r + a^{3} m r^{3}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)^{3}\right)}}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, r} \, t \, t }^{ \, r \phantom{\, t} \phantom{\, t} } & = & \frac{a^{2} m r^{2} - 2 \, m^{2} r^{3} + m r^{4} - {\left(a^{4} m - 2 \, a^{2} m^{2} r + a^{2} m r^{2}\right)} \cos\left({\theta}\right)^{2}}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, r} \, t \, {\phi} }^{ \, r \phantom{\, t} \phantom{\, {\phi}} } & = & -\frac{{\left(a^{3} m r^{2} - 2 \, a m^{2} r^{3} + a m r^{4} - {\left(a^{5} m - 2 \, a^{3} m^{2} r + a^{3} m r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, r} \, r \, r }^{ \, r \phantom{\, r} \phantom{\, r} } & = & \frac{a^{2} r - m r^{2} + {\left(a^{2} m - a^{2} r\right)} \cos\left({\theta}\right)^{2}}{a^{2} r^{2} - 2 \, m r^{3} + r^{4} + {\left(a^{4} - 2 \, a^{2} m r + a^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, r} \, r \, {\theta} }^{ \, r \phantom{\, r} \phantom{\, {\theta}} } & = & -\frac{a^{2} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, r} \, {\theta} \, {\theta} }^{ \, r \phantom{\, {\theta}} \phantom{\, {\theta}} } & = & -\frac{a^{2} r - 2 \, m r^{2} + r^{3}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, r} \, {\phi} \, {\phi} }^{ \, r \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & \frac{{\left(a^{4} m r^{2} - 2 \, a^{2} m^{2} r^{3} + a^{2} m r^{4} - {\left(a^{6} m - 2 \, a^{4} m^{2} r + a^{4} m r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{4} - {\left(a^{2} r^{5} - 2 \, m r^{6} + r^{7} + {\left(a^{6} r - 2 \, a^{4} m r^{2} + a^{4} r^{3}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{3} - 2 \, a^{2} m r^{4} + a^{2} r^{5}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, {\theta}} \, t \, t }^{ \, {\theta} \phantom{\, t} \phantom{\, t} } & = & -\frac{2 \, a^{2} m r \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, {\theta}} \, t \, {\phi} }^{ \, {\theta} \phantom{\, t} \phantom{\, {\phi}} } & = & \frac{2 \, {\left(a^{3} m r + a m r^{3}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, {\theta}} \, r \, r }^{ \, {\theta} \phantom{\, r} \phantom{\, r} } & = & \frac{a^{2} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{2} r^{2} - 2 \, m r^{3} + r^{4} + {\left(a^{4} - 2 \, a^{2} m r + a^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, {\theta}} \, r \, {\theta} }^{ \, {\theta} \phantom{\, r} \phantom{\, {\theta}} } & = & \frac{r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, {\theta}} \, {\theta} \, {\theta} }^{ \, {\theta} \phantom{\, {\theta}} \phantom{\, {\theta}} } & = & -\frac{a^{2} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, {\theta}} \, {\phi} \, {\phi} }^{ \, {\theta} \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & -\frac{{\left({\left(a^{6} - 2 \, a^{4} m r + a^{4} r^{2}\right)} \cos\left({\theta}\right)^{5} + 2 \, {\left(a^{4} r^{2} - 2 \, a^{2} m r^{3} + a^{2} r^{4}\right)} \cos\left({\theta}\right)^{3} + {\left(2 \, a^{4} m r + 4 \, a^{2} m r^{3} + a^{2} r^{4} + r^{6}\right)} \cos\left({\theta}\right)\right)} \sin\left({\theta}\right)}{a^{6} \cos\left({\theta}\right)^{6} + 3 \, a^{4} r^{2} \cos\left({\theta}\right)^{4} + 3 \, a^{2} r^{4} \cos\left({\theta}\right)^{2} + r^{6}} \\ \Gamma_{ \phantom{\, {\phi}} \, t \, r }^{ \, {\phi} \phantom{\, t} \phantom{\, r} } & = & -\frac{a^{3} m \cos\left({\theta}\right)^{2} - a m r^{2}}{a^{2} r^{4} - 2 \, m r^{5} + r^{6} + {\left(a^{6} - 2 \, a^{4} m r + a^{4} r^{2}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{2} - 2 \, a^{2} m r^{3} + a^{2} r^{4}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, {\phi}} \, t \, {\theta} }^{ \, {\phi} \phantom{\, t} \phantom{\, {\theta}} } & = & -\frac{2 \, a m r \cos\left({\theta}\right)}{{\left(a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}\right)} \sin\left({\theta}\right)} \\ \Gamma_{ \phantom{\, {\phi}} \, r \, {\phi} }^{ \, {\phi} \phantom{\, r} \phantom{\, {\phi}} } & = & -\frac{a^{2} m r^{2} + 2 \, m r^{4} - r^{5} + {\left(a^{4} m - a^{4} r\right)} \cos\left({\theta}\right)^{4} - {\left(a^{4} m - a^{2} m r^{2} + 2 \, a^{2} r^{3}\right)} \cos\left({\theta}\right)^{2}}{a^{2} r^{4} - 2 \, m r^{5} + r^{6} + {\left(a^{6} - 2 \, a^{4} m r + a^{4} r^{2}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{2} - 2 \, a^{2} m r^{3} + a^{2} r^{4}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, {\phi}} \, {\theta} \, {\phi} }^{ \, {\phi} \phantom{\, {\theta}} \phantom{\, {\phi}} } & = & \frac{a^{4} \cos\left({\theta}\right) \sin\left({\theta}\right)^{4} - 2 \, {\left(a^{4} - a^{2} m r + a^{2} r^{2}\right)} \cos\left({\theta}\right) \sin\left({\theta}\right)^{2} + {\left(a^{4} + 2 \, a^{2} r^{2} + r^{4}\right)} \cos\left({\theta}\right)}{{\left(a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}\right)} \sin\left({\theta}\right)} \end{array}$
In [ ]: