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Author: Eric Gourgoulhon
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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
(M,(t,r,θ,ϕ))\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=(2mra2cos(θ)2+r21)dtdt+(2amrsin(θ)2a2cos(θ)2+r2)dtdϕ+(a2cos(θ)2+r2a22mr+r2)drdr+(a2cos(θ)2+r2)dθdθ+(2amrsin(θ)2a2cos(θ)2+r2)dϕdt+(2a2mrsin(θ)2a2cos(θ)2+r2+a2+r2)sin(θ)2dϕdϕg = \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()
utt=(a2(2mr+1)+r2)ε2amra22mr+r2urr=ε2+2(aε)2mr3+2mr+(ε21)a22r21uϕϕ=2aεmr(2mr1)a22mr+r2\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()
(a2ε22a22)r3cos(θ)2r5+(2(a4ε22a3ε+a22)m+(a4ε2a4a22)r)cos(θ)4a2r3cos(θ)2+r5\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)g(u,u) in the equatorial plane (θ=π2\theta=\frac{\pi}{2}):

In [6]:
norm.coord_function()(t,r,pi/2,ph)
1-1
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=uua = \nabla_{u}\, u:

In [8]:
Du = nabla(u) a = u.contract(0, Du, 1) a.set_name('a') a.display_comp()
att=2(((a4εa3)mr2+(a6εa5)m)cos(θ)4+3((a2εa)mr4+(a4εa3)mr2)cos(θ)2)(ε21)r3+2mr2+2(a2ε22aε+2)m+(a2ε2a22)r(a2r72mr8+r9+(a6r32a4mr4+a4r5)cos(θ)4+2(a4r52a2mr6+a2r7)cos(θ)2)rarr=((a6ε2+4a5ε4a65a42)mr4(a6ε2a6a42)r52(a8ε2a8a622(a6ε23a5ε+a6+2a42)m2)r32(2(a6ε22a5ε+a42)m3+(a8ε25a7ε+2a8+4a62)m)r23(a10ε22a9ε+a82)m(a10ε2a10a828(a8ε22a7ε+a62)m2)r)cos(θ)6+((a4ε2+6a3ε8a47a22)mr62(a4ε2a4a22)r74(a6ε2a6a42(a4ε22a3ε+2a4+a22)m2)r5+2(2(a4ε22a3ε+a22)m3(3a6ε210a5ε+4a6+7a42)m)r47(a8ε22a7ε+a62)mr22(a8ε2a8a626(a6ε22a5ε+a42)m2)r3)cos(θ)4(2(a2ε23aε+2a2+22)mr8+(a2ε2a22)r9+2(4a4ε29a3ε+2a4+5a22)mr6+2(a4ε2a4a222(2a2ε23aε+a2+2)m2)r7+6(a6ε22a5ε+a42)mr4+(a6ε2a6a4212(a4ε22a3ε+a22)m2)r5)cos(θ)2a2r102mr11+r12+(a8r42a6mr5+a6r6)cos(θ)6+3(a6r62a4mr7+a4r8)cos(θ)4+3(a4r82a2mr9+a2r10)cos(θ)2aθθ=((2(2a5εa62a42)mr2(a6ε2a6a42)r32(a8ε22a7ε+a62)m(a8ε2a8a624(a6ε22a5ε+a42)m2)r)cos(θ)5+2(2(2a3εa42a22)mr4(a4ε2a4a22)r52(a6ε22a5ε+a42)mr2(a6ε2a6a424(a4ε22a3ε+a22)m2)r3)cos(θ)3+(2(a2ε2a22)mr6(a2ε2a22)r7(a4ε2a4a22)r5)cos(θ))sin(θ)a2r92mr10+r11+(a8r32a6mr4+a6r5)cos(θ)6+3(a6r52a4mr6+a4r7)cos(θ)4+3(a4r72a2mr8+a2r9)cos(θ)2aϕϕ=2(3(a3εa2)mr2cos(θ)2+(a5εa4)mcos(θ)4)(ε21)r3+2mr2+2(a2ε22aε+2)m+(a2ε2a22)r(a2r72mr8+r9+(a6r32a4mr4+a4r5)cos(θ)4+2(a4r52a2mr6+a2r7)cos(θ)2)r\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 (θ=π2\theta=\frac{\pi}{2}):

In [9]:
a[0](t,r,pi/2,ph)
00
In [10]:
a[1](t,r,pi/2,ph)
00
In [11]:
a[2](t,r,pi/2,ph)
00
In [12]:
a[3](t,r,pi/2,ph)
00

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

In [13]:
g.christoffel_symbols_display()
Γttrttr=a2mr2+mr4(a4m+a2mr2)cos(θ)2a2r42mr5+r6+(a62a4mr+a4r2)cos(θ)4+2(a4r22a2mr3+a2r4)cos(θ)2Γttθttθ=2a2mrcos(θ)sin(θ)a4cos(θ)4+2a2r2cos(θ)2+r4Γtrϕtrϕ=(a3mr2+3amr4(a5ma3mr2)cos(θ)2)sin(θ)2a2r42mr5+r6+(a62a4mr+a4r2)cos(θ)4+2(a4r22a2mr3+a2r4)cos(θ)2Γtθϕtθϕ=2(a5mrcos(θ)sin(θ)5(a5mr+a3mr3)cos(θ)sin(θ)3)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γrttrtt=a2mr22m2r3+mr4(a4m2a2m2r+a2mr2)cos(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γrtϕrtϕ=(a3mr22am2r3+amr4(a5m2a3m2r+a3mr2)cos(θ)2)sin(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γrrrrrr=a2rmr2+(a2ma2r)cos(θ)2a2r22mr3+r4+(a42a2mr+a2r2)cos(θ)2Γrrθrrθ=a2cos(θ)sin(θ)a2cos(θ)2+r2Γrθθrθθ=a2r2mr2+r3a2cos(θ)2+r2Γrϕϕrϕϕ=(a4mr22a2m2r3+a2mr4(a6m2a4m2r+a4mr2)cos(θ)2)sin(θ)4(a2r52mr6+r7+(a6r2a4mr2+a4r3)cos(θ)4+2(a4r32a2mr4+a2r5)cos(θ)2)sin(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γθttθtt=2a2mrcos(θ)sin(θ)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γθtϕθtϕ=2(a3mr+amr3)cos(θ)sin(θ)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γθrrθrr=a2cos(θ)sin(θ)a2r22mr3+r4+(a42a2mr+a2r2)cos(θ)2Γθrθθrθ=ra2cos(θ)2+r2Γθθθθθθ=a2cos(θ)sin(θ)a2cos(θ)2+r2Γθϕϕθϕϕ=((a62a4mr+a4r2)cos(θ)5+2(a4r22a2mr3+a2r4)cos(θ)3+(2a4mr+4a2mr3+a2r4+r6)cos(θ))sin(θ)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γϕtrϕtr=a3mcos(θ)2amr2a2r42mr5+r6+(a62a4mr+a4r2)cos(θ)4+2(a4r22a2mr3+a2r4)cos(θ)2Γϕtθϕtθ=2amrcos(θ)(a4cos(θ)4+2a2r2cos(θ)2+r4)sin(θ)Γϕrϕϕrϕ=a2mr2+2mr4r5+(a4ma4r)cos(θ)4(a4ma2mr2+2a2r3)cos(θ)2a2r42mr5+r6+(a62a4mr+a4r2)cos(θ)4+2(a4r22a2mr3+a2r4)cos(θ)2Γϕθϕϕθϕ=a4cos(θ)sin(θ)42(a4a2mr+a2r2)cos(θ)sin(θ)2+(a4+2a2r2+r4)cos(θ)(a4cos(θ)4+2a2r2cos(θ)2+r4)sin(θ)\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}
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