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SageMath notebooks associated to the Black Hole Lectures (https://luth.obspm.fr/~luthier/gourgoulhon/bh16)

Project: BHLectures
Views: 20094
Kernel: SageMath 8.9.beta4

Kerr spacetime in Kerr coordinates

This Jupyter/SageMath worksheet is relative to the lectures Geometry and physics of black holes

These 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 worksheet:

version()
'SageMath version 8.9.beta4, Release Date: 2019-07-28'

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

%display latex

To speed up computations, we ask for running them in parallel on 8 threads:

Parallelism().set(nproc=8)

Spacetime

We declare the spacetime manifold MM:

M = Manifold(4, 'M', structure='Lorentzian') print(M)
4-dimensional Lorentzian manifold M

and the Kerr coordinates (v,r,θ,ϕ)(v,r,\theta,\phi) as a chart on MM:

X.<v,r,th,ph> = M.chart(r'v r th:(0,pi):\theta ph:(0,2*pi):\phi') X
(M,(v,r,θ,ϕ))\renewcommand{\Bold}[1]{\mathbf{#1}}\left(M,(v, r, {\theta}, {\phi})\right)
X.coord_range()
v: (,+);r: (,+);θ: (0,π);ϕ: (0,2π)\renewcommand{\Bold}[1]{\mathbf{#1}}v :\ \left( -\infty, +\infty \right) ;\quad r :\ \left( -\infty, +\infty \right) ;\quad {\theta} :\ \left( 0 , \pi \right) ;\quad {\phi} :\ \left( 0 , 2 \, \pi \right)

The Kerr parameters mm and aa:

m = var('m', domain='real') assume(m>0) a = var('a', domain='real') assume(a>=0)

Kerr metric

We define the metric gg by its components w.r.t. the Kerr coordinates:

g = M.metric() rho2 = r^2 + (a*cos(th))^2 g[0,0] = -(1 - 2*m*r/rho2) g[0,1] = 1 g[0,3] = -2*a*m*r*sin(th)^2/rho2 g[1,3] = -a*sin(th)^2 g[2,2] = 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)dvdv+dvdr+(2amrsin(θ)2a2cos(θ)2+r2)dvdϕ+drdvasin(θ)2drdϕ+(a2cos(θ)2+r2)dθdθ+(2amrsin(θ)2a2cos(θ)2+r2)dϕdvasin(θ)2dϕdr+(2a2mrsin(θ)2a2cos(θ)2+r2+a2+r2)sin(θ)2dϕdϕ\renewcommand{\Bold}[1]{\mathbf{#1}}g = \left( \frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} - 1 \right) \mathrm{d} v\otimes \mathrm{d} v +\mathrm{d} v\otimes \mathrm{d} r + \left( -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} v\otimes \mathrm{d} {\phi} +\mathrm{d} r\otimes \mathrm{d} v -a \sin\left({\theta}\right)^{2} \mathrm{d} r\otimes \mathrm{d} {\phi} + \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} v -a \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} r + {\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}
g.display_comp()
gvvvv=2mra2cos(θ)2+r21gvrvr=1gvϕvϕ=2amrsin(θ)2a2cos(θ)2+r2grvrv=1grϕrϕ=asin(θ)2gθθθθ=a2cos(θ)2+r2gϕvϕv=2amrsin(θ)2a2cos(θ)2+r2gϕrϕr=asin(θ)2gϕϕϕϕ=(2a2mrsin(θ)2a2cos(θ)2+r2+a2+r2)sin(θ)2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} g_{ \, v \, v }^{ \phantom{\, v}\phantom{\, v} } & = & \frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} - 1 \\ g_{ \, v \, r }^{ \phantom{\, v}\phantom{\, r} } & = & 1 \\ g_{ \, v \, {\phi} }^{ \phantom{\, v}\phantom{\, {\phi}} } & = & -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ g_{ \, r \, v }^{ \phantom{\, r}\phantom{\, v} } & = & 1 \\ g_{ \, r \, {\phi} }^{ \phantom{\, r}\phantom{\, {\phi}} } & = & -a \sin\left({\theta}\right)^{2} \\ g_{ \, {\theta} \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, {\theta}} } & = & a^{2} \cos\left({\theta}\right)^{2} + r^{2} \\ g_{ \, {\phi} \, v }^{ \phantom{\, {\phi}}\phantom{\, v} } & = & -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ g_{ \, {\phi} \, r }^{ \phantom{\, {\phi}}\phantom{\, r} } & = & -a \sin\left({\theta}\right)^{2} \\ g_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi}}\phantom{\, {\phi}} } & = & {\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} \end{array}

The inverse metric is pretty simple:

g.inverse()[:]
(a2sin(θ)2a2cos(θ)2+r2a2+r2a2cos(θ)2+r20aa2cos(θ)2+r2a2+r2a2cos(θ)2+r2a22mr+r2a2cos(θ)2+r20aa2cos(θ)2+r2001a2cos(θ)2+r20aa2cos(θ)2+r2aa2cos(θ)2+r201a2sin(θ)4(a2+r2)sin(θ)2)\renewcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} \frac{a^{2} \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} & \frac{a^{2} + r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} & 0 & \frac{a}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \frac{a^{2} + r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} & \frac{a^{2} - 2 \, m r + r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} & 0 & \frac{a}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ 0 & 0 & \frac{1}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} & 0 \\ \frac{a}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} & \frac{a}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} & 0 & -\frac{1}{a^{2} \sin\left({\theta}\right)^{4} - {\left(a^{2} + r^{2}\right)} \sin\left({\theta}\right)^{2}} \end{array}\right)

as well as the determinant w.r.t. to the Kerr coordinates:

g.determinant().display()
MR(v,r,θ,ϕ)a4cos(θ)6(a42a2r2)cos(θ)4r4(2a2r2r4)cos(θ)2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(v, r, {\theta}, {\phi}\right) & \longmapsto & a^{4} \cos\left({\theta}\right)^{6} - {\left(a^{4} - 2 \, a^{2} r^{2}\right)} \cos\left({\theta}\right)^{4} - r^{4} - {\left(2 \, a^{2} r^{2} - r^{4}\right)} \cos\left({\theta}\right)^{2} \end{array}
g.determinant() == - (rho2*sin(th))^2
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}

Let us check that we are dealing with a solution of the Einstein equation in vacuum:

g.ricci().display()
Ric(g)=0\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{Ric}\left(g\right) = 0

The Christoffel symbols w.r.t. the Kerr coordinates:

g.ricci()[0,0]
0\renewcommand{\Bold}[1]{\mathbf{#1}}0
g.christoffel_symbols_display()
Γvvvvvv=a4mmr4(a4m+a2mr2)sin(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γvvθvvθ=2a2mrcos(θ)sin(θ)a4cos(θ)4+2a2r2cos(θ)2+r4Γvvϕvvϕ=(a5m+a3mr2)sin(θ)4(a5mamr4)sin(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γvrθvrθ=a2cos(θ)sin(θ)a2cos(θ)2+r2Γvrϕvrϕ=arsin(θ)2a2cos(θ)2+r2Γvθθvθθ=a2r+r3a2cos(θ)2+r2Γvθϕvθϕ=2(a5mrcos(θ)sin(θ)5(a5mr+a3mr3)cos(θ)sin(θ)3)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γvϕϕvϕϕ=(a4mr2+a2mr4(a6m+a4mr2)cos(θ)2)sin(θ)4(a2r5+r7+(a6r+a4r3)cos(θ)4+2(a4r3+a2r5)cos(θ)2)sin(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γrvvrvv=a2mr22m2r3+mr4(a4m2a2m2r+a2mr2)cos(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γrvrrvr=a2mcos(θ)2mr2a4cos(θ)4+2a2r2cos(θ)2+r4Γrvϕrvϕ=(a3mr22am2r3+amr4(a5m2a3m2r+a3mr2)cos(θ)2)sin(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γrrθrrθ=a2cos(θ)sin(θ)a2cos(θ)2+r2Γrrϕrrϕ=(a5mcos(θ)2+a5m)sin(θ)4+(a5rcos(θ)4+2a3r3cos(θ)2a5m+amr4+ar5)sin(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γrθθrθθ=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+r6Γθvvθvv=2a2mrcos(θ)sin(θ)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γθvϕθvϕ=2(a3mr+amr3)cos(θ)sin(θ)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γθrθθrθ=ra2cos(θ)2+r2Γθrϕθrϕ=acos(θ)sin(θ)a2cos(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+r6Γϕvvϕvv=a3mcos(θ)2amr2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γϕvθϕvθ=2amrcos(θ)(a4cos(θ)4+2a2r2cos(θ)2+r4)sin(θ)Γϕvϕϕvϕ=(a4mcos(θ)2a2mr2)sin(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γϕrθϕrθ=acos(θ)(a2cos(θ)2+r2)sin(θ)Γϕrϕϕrϕ=ra2cos(θ)2+r2Γϕθθϕθθ=ara2cos(θ)2+r2Γϕθϕϕθϕ=a4cos(θ)52(a2mra2r2)cos(θ)3+(2a2mr+r4)cos(θ)(a4cos(θ)4+2a2r2cos(θ)2+r4)sin(θ)Γϕϕϕϕϕϕ=(a5mcos(θ)2a3mr2)sin(θ)4+(a5rcos(θ)4+2a3r3cos(θ)2+ar5)sin(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} \Gamma_{ \phantom{\, v} \, v \, v }^{ \, v \phantom{\, v} \phantom{\, v} } & = & -\frac{a^{4} m - m r^{4} - {\left(a^{4} m + a^{2} m r^{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{\, v} \, v \, {\theta} }^{ \, v \phantom{\, v} \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{\, v} \, v \, {\phi} }^{ \, v \phantom{\, v} \phantom{\, {\phi}} } & = & -\frac{{\left(a^{5} m + a^{3} m r^{2}\right)} \sin\left({\theta}\right)^{4} - {\left(a^{5} m - a m r^{4}\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{\, v} \, r \, {\theta} }^{ \, v \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{\, v} \, r \, {\phi} }^{ \, v \phantom{\, r} \phantom{\, {\phi}} } & = & \frac{a r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, v} \, {\theta} \, {\theta} }^{ \, v \phantom{\, {\theta}} \phantom{\, {\theta}} } & = & -\frac{a^{2} r + r^{3}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, v} \, {\theta} \, {\phi} }^{ \, v \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{\, v} \, {\phi} \, {\phi} }^{ \, v \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & \frac{{\left(a^{4} m r^{2} + a^{2} m r^{4} - {\left(a^{6} m + a^{4} m r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{4} - {\left(a^{2} r^{5} + r^{7} + {\left(a^{6} r + a^{4} r^{3}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{3} + 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{\, r} \, v \, v }^{ \, r \phantom{\, v} \phantom{\, v} } & = & \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} \, v \, r }^{ \, r \phantom{\, v} \phantom{\, r} } & = & \frac{a^{2} m \cos\left({\theta}\right)^{2} - m r^{2}}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \\ \Gamma_{ \phantom{\, r} \, v \, {\phi} }^{ \, r \phantom{\, v} \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 \, {\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} \, r \, {\phi} }^{ \, r \phantom{\, r} \phantom{\, {\phi}} } & = & \frac{{\left(a^{5} m \cos\left({\theta}\right)^{2} + a^{5} m\right)} \sin\left({\theta}\right)^{4} + {\left(a^{5} r \cos\left({\theta}\right)^{4} + 2 \, a^{3} r^{3} \cos\left({\theta}\right)^{2} - a^{5} m + a m r^{4} + a r^{5}\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} \, {\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}} \, v \, v }^{ \, {\theta} \phantom{\, v} \phantom{\, v} } & = & -\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}} \, v \, {\phi} }^{ \, {\theta} \phantom{\, v} \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 \, {\theta} }^{ \, {\theta} \phantom{\, r} \phantom{\, {\theta}} } & = & \frac{r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, {\theta}} \, r \, {\phi} }^{ \, {\theta} \phantom{\, r} \phantom{\, {\phi}} } & = & \frac{a \cos\left({\theta}\right) \sin\left({\theta}\right)}{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}} \, v \, v }^{ \, {\phi} \phantom{\, v} \phantom{\, v} } & = & -\frac{a^{3} m \cos\left({\theta}\right)^{2} - a m r^{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{\, {\phi}} \, v \, {\theta} }^{ \, {\phi} \phantom{\, v} \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}} \, v \, {\phi} }^{ \, {\phi} \phantom{\, v} \phantom{\, {\phi}} } & = & \frac{{\left(a^{4} m \cos\left({\theta}\right)^{2} - a^{2} m r^{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{\, {\phi}} \, r \, {\theta} }^{ \, {\phi} \phantom{\, r} \phantom{\, {\theta}} } & = & -\frac{a \cos\left({\theta}\right)}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} \sin\left({\theta}\right)} \\ \Gamma_{ \phantom{\, {\phi}} \, r \, {\phi} }^{ \, {\phi} \phantom{\, r} \phantom{\, {\phi}} } & = & \frac{r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, {\phi}} \, {\theta} \, {\theta} }^{ \, {\phi} \phantom{\, {\theta}} \phantom{\, {\theta}} } & = & -\frac{a r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, {\phi}} \, {\theta} \, {\phi} }^{ \, {\phi} \phantom{\, {\theta}} \phantom{\, {\phi}} } & = & \frac{a^{4} \cos\left({\theta}\right)^{5} - 2 \, {\left(a^{2} m r - a^{2} r^{2}\right)} \cos\left({\theta}\right)^{3} + {\left(2 \, a^{2} m r + 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)} \\ \Gamma_{ \phantom{\, {\phi}} \, {\phi} \, {\phi} }^{ \, {\phi} \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & -\frac{{\left(a^{5} m \cos\left({\theta}\right)^{2} - a^{3} m r^{2}\right)} \sin\left({\theta}\right)^{4} + {\left(a^{5} r \cos\left({\theta}\right)^{4} + 2 \, a^{3} r^{3} \cos\left({\theta}\right)^{2} + a r^{5}\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}} \end{array}

Vector normal to the hypersurfaces r=constr=\mathrm{const}

dr = X.coframe()[1] print(dr) dr.display()
1-form dr on the 4-dimensional Lorentzian manifold M
dr=dr\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{d} r = \mathrm{d} r
nr = dr.up(g) print(nr) nr.display()
Vector field on the 4-dimensional Lorentzian manifold M
(a2+r2a2cos(θ)2+r2)v+(a22mr+r2a2cos(θ)2+r2)r+(aa2cos(θ)2+r2)ϕ\renewcommand{\Bold}[1]{\mathbf{#1}}\left( \frac{a^{2} + r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \frac{\partial}{\partial v } + \left( \frac{a^{2} - 2 \, m r + r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \frac{\partial}{\partial r } + \left( \frac{a}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \frac{\partial}{\partial {\phi} }
assume(a^2<m^2) rp = m + sqrt(m^2-a^2) rp
m+a2+m2\renewcommand{\Bold}[1]{\mathbf{#1}}m + \sqrt{-a^{2} + m^{2}}
p = M.point(coords=(v,rp,th,ph), name='p') print(p)
Point p on the 4-dimensional Lorentzian manifold M
X(p)
(v,m+a2+m2,θ,ϕ)\renewcommand{\Bold}[1]{\mathbf{#1}}\left(v, m + \sqrt{-a^{2} + m^{2}}, {\theta}, {\phi}\right)
nrH = nr.at(p) print(nrH)
Tangent vector at Point p on the 4-dimensional Lorentzian manifold M
Tp = M.tangent_space(p) print(Tp)
Tangent space at Point p on the 4-dimensional Lorentzian manifold M
Tp.default_basis()
(v,r,θ,ϕ)\renewcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{\partial}{\partial v },\frac{\partial}{\partial r },\frac{\partial}{\partial {\theta} },\frac{\partial}{\partial {\phi} }\right)
nrH[:]
[2(a+ma+mm+m2)a2sin(θ)22a+ma+mm2m2,0,0,aa2sin(θ)22a+ma+mm2m2]\renewcommand{\Bold}[1]{\mathbf{#1}}\left[-\frac{2 \, {\left(\sqrt{a + m} \sqrt{-a + m} m + m^{2}\right)}}{a^{2} \sin\left({\theta}\right)^{2} - 2 \, \sqrt{a + m} \sqrt{-a + m} m - 2 \, m^{2}}, 0, 0, -\frac{a}{a^{2} \sin\left({\theta}\right)^{2} - 2 \, \sqrt{a + m} \sqrt{-a + m} m - 2 \, m^{2}}\right]
OmegaH = a/(2*m*rp) OmegaH
a2(m+a2+m2)m\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{a}{2 \, {\left(m + \sqrt{-a^{2} + m^{2}}\right)} m}
xi = X.frame()[0] xi
v\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{\partial}{\partial v }
eta = X.frame()[3] eta
ϕ\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{\partial}{\partial {\phi} }
chi = xi + OmegaH*eta chi.display()
v+a2(a+ma+mm+m2)ϕ\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{\partial}{\partial v } + \frac{a}{2 \, {\left(\sqrt{a + m} \sqrt{-a + m} m + m^{2}\right)}} \frac{\partial}{\partial {\phi} }

Ingoing principal null geodesics

k = M.vector_field(name='k') k[1] = -1 k.display()
k=r\renewcommand{\Bold}[1]{\mathbf{#1}}k = -\frac{\partial}{\partial r }

Let us check that kk is a null vector:

g(k,k).display()
g(k,k):MR(v,r,θ,ϕ)0\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} g\left(k,k\right):& M & \longrightarrow & \mathbb{R} \\ & \left(v, r, {\theta}, {\phi}\right) & \longmapsto & 0 \end{array}

Computation of kk\nabla_k k:

nab = g.connection() acc = nab(k).contract(k) acc.display()
0\renewcommand{\Bold}[1]{\mathbf{#1}}0

Outgoing principal null geodesics

el = M.vector_field(name='el', latex_name=r'\ell') el[0] = 1 el[1] = 1/2 - m*r/(r^2+a^2) el[3] = a/(r^2+a^2) el.display()
=v+(mra2+r2+12)r+(aa2+r2)ϕ\renewcommand{\Bold}[1]{\mathbf{#1}}\ell = \frac{\partial}{\partial v } + \left( -\frac{m r}{a^{2} + r^{2}} + \frac{1}{2} \right) \frac{\partial}{\partial r } + \left( \frac{a}{a^{2} + r^{2}} \right) \frac{\partial}{\partial {\phi} }

Let us check that \ell is a null vector:

g(el,el).display()
g(,):MR(v,r,θ,ϕ)0\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} g\left(\ell,\ell\right):& M & \longrightarrow & \mathbb{R} \\ & \left(v, r, {\theta}, {\phi}\right) & \longmapsto & 0 \end{array}

Computation of \nabla_\ell \ell:

acc = nab(el).contract(el) acc.display()
(a2mmr2a4+2a2r2+r4)v+(a4m2a2m2r+2m2r3mr42(a6+3a4r2+3a2r4+r6))r+(a3mamr2a6+3a4r2+3a2r4+r6)ϕ\renewcommand{\Bold}[1]{\mathbf{#1}}\left( -\frac{a^{2} m - m r^{2}}{a^{4} + 2 \, a^{2} r^{2} + r^{4}} \right) \frac{\partial}{\partial v } + \left( -\frac{a^{4} m - 2 \, a^{2} m^{2} r + 2 \, m^{2} r^{3} - m r^{4}}{2 \, {\left(a^{6} + 3 \, a^{4} r^{2} + 3 \, a^{2} r^{4} + r^{6}\right)}} \right) \frac{\partial}{\partial r } + \left( -\frac{a^{3} m - a m r^{2}}{a^{6} + 3 \, a^{4} r^{2} + 3 \, a^{2} r^{4} + r^{6}} \right) \frac{\partial}{\partial {\phi} }

We check that \nabla_\ell \ell \propto \ell:

for i in [0,1,3]: show(acc[i] / el[i])
a2mmr2a4+2a2r2+r4\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{a^{2} m - m r^{2}}{a^{4} + 2 \, a^{2} r^{2} + r^{4}}
a2mmr2a4+2a2r2+r4\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{a^{2} m - m r^{2}}{a^{4} + 2 \, a^{2} r^{2} + r^{4}}
a2mmr2a4+2a2r2+r4\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{a^{2} m - m r^{2}}{a^{4} + 2 \, a^{2} r^{2} + r^{4}}

Hence we may write =κ\nabla_\ell\ell = \kappa \ell:

kappa = (acc[0] / el[0]).expr() kappa
a2mmr2a4+2a2r2+r4\renewcommand{\Bold}[1]{\mathbf{#1}}-\frac{a^{2} m - m r^{2}}{a^{4} + 2 \, a^{2} r^{2} + r^{4}}
acc == kappa * el
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}

Surface gravity

On HH, \ell coincides with the Killing vector χ\chi:

el.at(p) == chi.at(p)
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}

Therefore the surface gravity of the Kerr black hole is nothing but the value of the non-affinity coefficient of \ell on HH:

kappaH = kappa.subs(r=rp).simplify_full() kappaH
a2m2a2+m2m2(a2m2m32a2+m2m2)\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{a^{2} - m^{2} - \sqrt{-a^{2} + m^{2}} m}{2 \, {\left(a^{2} m - 2 \, m^{3} - 2 \, \sqrt{-a^{2} + m^{2}} m^{2}\right)}}
bool(kappaH == sqrt(m^2-a^2)/(2*m*(m+sqrt(m^2-a^2))))
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}