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Kernel: SageMath 9.2.beta14

Kerr spacetime in 3+1 Kerr coordinates

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

The corresponding tools have been developed within the SageManifolds project.

NB: a version of SageMath at least equal to 9.1 is required to run this notebook:

version()
'SageMath version 9.2.beta14, Release Date: 2020-09-30'

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') print(M)
4-dimensional differentiable manifold M

and the 3+1 Kerr coordinates (t,r,θ,ϕ)(t,r,\theta,\phi) as a chart on MM:

X.<t,r,th,ph> = M.chart(r't r th:(0,pi):\theta ph:(0,2*pi):\phi') X
(M,(t,r,θ,ϕ))\renewcommand{\Bold}[1]{\mathbf{#1}}\left(M,(t, r, {\theta}, {\phi})\right)
X.coord_range()
t: (,+);r: (,+);θ: (0,π);ϕ: (0,2π)\renewcommand{\Bold}[1]{\mathbf{#1}}t :\ \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 3+1 Kerr coordinates:

g = M.lorentzian_metric('g') rho2 = r^2 + (a*cos(th))^2 g[0,0] = -(1 - 2*m*r/rho2) g[0,1] = 2*m*r/rho2 g[0,3] = -2*a*m*r*sin(th)^2/rho2 g[1,1] = 1 + 2*m*r/rho2 g[1,3] = -a*(1 + 2*m*r/rho2)*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)dtdt+(2mra2cos(θ)2+r2)dtdr+(2amrsin(θ)2a2cos(θ)2+r2)dtdϕ+(2mra2cos(θ)2+r2)drdt+(2mra2cos(θ)2+r2+1)drdra(2mra2cos(θ)2+r2+1)sin(θ)2drdϕ+(a2cos(θ)2+r2)dθdθ+(2amrsin(θ)2a2cos(θ)2+r2)dϕdta(2mra2cos(θ)2+r2+1)sin(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} t\otimes \mathrm{d} t + \left( \frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} t\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} t\otimes \mathrm{d} {\phi} + \left( \frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} r\otimes \mathrm{d} t + \left( \frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} + 1 \right) \mathrm{d} r\otimes \mathrm{d} r -a {\left(\frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} + 1\right)} \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} t -a {\left(\frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} + 1\right)} \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()
gtttt=2mra2cos(θ)2+r21gtrtr=2mra2cos(θ)2+r2gtϕtϕ=2amrsin(θ)2a2cos(θ)2+r2grtrt=2mra2cos(θ)2+r2grrrr=2mra2cos(θ)2+r2+1grϕrϕ=a(2mra2cos(θ)2+r2+1)sin(θ)2gθθθθ=a2cos(θ)2+r2gϕtϕt=2amrsin(θ)2a2cos(θ)2+r2gϕrϕr=a(2mra2cos(θ)2+r2+1)sin(θ)2gϕϕϕϕ=(2a2mrsin(θ)2a2cos(θ)2+r2+a2+r2)sin(θ)2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{lcl} g_{ \, t \, t }^{ \phantom{\, t}\phantom{\, t} } & = & \frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} - 1 \\ g_{ \, t \, r }^{ \phantom{\, t}\phantom{\, r} } & = & \frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ g_{ \, t \, {\phi} }^{ \phantom{\, t}\phantom{\, {\phi}} } & = & -\frac{2 \, a m r \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ g_{ \, r \, t }^{ \phantom{\, r}\phantom{\, t} } & = & \frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ g_{ \, r \, r }^{ \phantom{\, r}\phantom{\, r} } & = & \frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} + 1 \\ g_{ \, r \, {\phi} }^{ \phantom{\, r}\phantom{\, {\phi}} } & = & -a {\left(\frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} + 1\right)} \sin\left({\theta}\right)^{2} \\ g_{ \, {\theta} \, {\theta} }^{ \phantom{\, {\theta}}\phantom{\, {\theta}} } & = & a^{2} \cos\left({\theta}\right)^{2} + r^{2} \\ g_{ \, {\phi} \, t }^{ \phantom{\, {\phi}}\phantom{\, t} } & = & -\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 {\left(\frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} + 1\right)} \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()[:]
(a2cos(θ)2+2mr+r2a2cos(θ)2+r22mra2cos(θ)2+r2002mra2cos(θ)2+r2a22mr+r2a2cos(θ)2+r20aa2cos(θ)2+r2001a2cos(θ)2+r200aa2cos(θ)2+r201a2sin(θ)4(a2+r2)sin(θ)2)\renewcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} -\frac{a^{2} \cos\left({\theta}\right)^{2} + 2 \, m r + r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} & \frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} & 0 & 0 \\ \frac{2 \, m r}{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 \\ 0 & \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 3+1 Kerr coordinates:

g.determinant().display()
MR(t,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(t, 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 3+1 Kerr coordinates:

g.christoffel_symbols_display()
Γtttttt=2(a2m2rcos(θ)2m2r3)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γttrttr=a4mcos(θ)4+2a2m2rcos(θ)22m2r3mr4a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γttθttθ=2a2mrcos(θ)sin(θ)a4cos(θ)4+2a2r2cos(θ)2+r4Γttϕttϕ=2(a3m2rcos(θ)2am2r3)sin(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γtrrtrr=2(a4mcos(θ)4+a2m2rcos(θ)2m2r3mr4)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γtrθtrθ=2a2mrcos(θ)sin(θ)a4cos(θ)4+2a2r2cos(θ)2+r4Γtrϕtrϕ=(a5mcos(θ)4+2a3m2rcos(θ)22am2r3amr4)sin(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γtθθtθθ=2mr2a2cos(θ)2+r2Γtθϕtθϕ=2a3mrcos(θ)sin(θ)3a4cos(θ)4+2a2r2cos(θ)2+r4Γtϕϕtϕϕ=2((a4m2rcos(θ)2a2m2r3)sin(θ)4+(a4mr2cos(θ)4+2a2mr4cos(θ)2+mr6)sin(θ)2)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+r6Γrtrrtr=2a2m2rcos(θ)22m2r3(a4mcos(θ)2a2mr2)sin(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=2a4mcos(θ)4+a2mr22m2r3mr4(a4m2a2m2r+a2mr2)cos(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γrrθrrθ=a2cos(θ)sin(θ)a2cos(θ)2+r2Γrrϕrrϕ=(a5mcos(θ)2a3mr2)sin(θ)4+(a5rcos(θ)4+2am2r3+ar52(a3m2ra3r3)cos(θ)2)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+r6Γθttθtt=2a2mrcos(θ)sin(θ)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γθtrθtr=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=2a2mrcos(θ)sin(θ)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γθrθθrθ=ra2cos(θ)2+r2Γθrϕθrϕ=(a5cos(θ)5+2a3r2cos(θ)3+(2a3mr+2amr3+ar4)cos(θ))sin(θ)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γθθθθθθ=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+r6Γϕttϕtt=a3mcos(θ)2amr2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γϕtrϕtr=a3mcos(θ)2amr2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γϕtθϕtθ=2amrcos(θ)(a4cos(θ)4+2a2r2cos(θ)2+r4)sin(θ)Γϕtϕϕtϕ=(a4mcos(θ)2a2mr2)sin(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γϕrrϕrr=a3mcos(θ)2amr2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γϕrθϕrθ=a3cos(θ)3+(2amr+ar2)cos(θ)(a4cos(θ)4+2a2r2cos(θ)2+r4)sin(θ)Γϕrϕϕrϕ=a4rcos(θ)4+2a2r3cos(θ)2+r5+(a4mcos(θ)2a2mr2)sin(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γϕθθϕθθ=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{\, t} \, t \, t }^{ \, t \phantom{\, t} \phantom{\, t} } & = & -\frac{2 \, {\left(a^{2} m^{2} r \cos\left({\theta}\right)^{2} - m^{2} r^{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{\, t} \, t \, r }^{ \, t \phantom{\, t} \phantom{\, r} } & = & -\frac{a^{4} m \cos\left({\theta}\right)^{4} + 2 \, a^{2} m^{2} r \cos\left({\theta}\right)^{2} - 2 \, m^{2} r^{3} - m r^{4}}{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{\, 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} \, t \, {\phi} }^{ \, t \phantom{\, t} \phantom{\, {\phi}} } & = & \frac{2 \, {\left(a^{3} m^{2} r \cos\left({\theta}\right)^{2} - a m^{2} r^{3}\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{\, t} \, r \, r }^{ \, t \phantom{\, r} \phantom{\, r} } & = & -\frac{2 \, {\left(a^{4} m \cos\left({\theta}\right)^{4} + a^{2} m^{2} r \cos\left({\theta}\right)^{2} - m^{2} r^{3} - m r^{4}\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{\, t} \, r \, {\theta} }^{ \, t \phantom{\, r} \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^{5} m \cos\left({\theta}\right)^{4} + 2 \, a^{3} m^{2} r \cos\left({\theta}\right)^{2} - 2 \, a m^{2} r^{3} - 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{\, t} \, {\theta} \, {\theta} }^{ \, t \phantom{\, {\theta}} \phantom{\, {\theta}} } & = & -\frac{2 \, m r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, t} \, {\theta} \, {\phi} }^{ \, t \phantom{\, {\theta}} \phantom{\, {\phi}} } & = & \frac{2 \, a^{3} m r \cos\left({\theta}\right) \sin\left({\theta}\right)^{3}}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \\ \Gamma_{ \phantom{\, t} \, {\phi} \, {\phi} }^{ \, t \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & -\frac{2 \, {\left({\left(a^{4} m^{2} r \cos\left({\theta}\right)^{2} - a^{2} m^{2} r^{3}\right)} \sin\left({\theta}\right)^{4} + {\left(a^{4} m r^{2} \cos\left({\theta}\right)^{4} + 2 \, a^{2} m r^{4} \cos\left({\theta}\right)^{2} + m r^{6}\right)} \sin\left({\theta}\right)^{2}\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 \, r }^{ \, r \phantom{\, t} \phantom{\, r} } & = & \frac{2 \, a^{2} m^{2} r \cos\left({\theta}\right)^{2} - 2 \, m^{2} r^{3} - {\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{\, 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{2 \, a^{4} m \cos\left({\theta}\right)^{4} + 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} \, 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^{3} m r^{2}\right)} \sin\left({\theta}\right)^{4} + {\left(a^{5} r \cos\left({\theta}\right)^{4} + 2 \, a m^{2} r^{3} + a r^{5} - 2 \, {\left(a^{3} m^{2} r - a^{3} r^{3}\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} \, {\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 \, r }^{ \, {\theta} \phantom{\, t} \phantom{\, r} } & = & -\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{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}} \, 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{{\left(a^{5} \cos\left({\theta}\right)^{5} + 2 \, a^{3} r^{2} \cos\left({\theta}\right)^{3} + {\left(2 \, a^{3} m r + 2 \, a m r^{3} + a r^{4}\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{\, {\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 \, t }^{ \, {\phi} \phantom{\, t} \phantom{\, t} } & = & -\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}} \, t \, r }^{ \, {\phi} \phantom{\, t} \phantom{\, r} } & = & -\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}} \, 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}} \, t \, {\phi} }^{ \, {\phi} \phantom{\, t} \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 \, r }^{ \, {\phi} \phantom{\, r} \phantom{\, r} } & = & -\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}} \, r \, {\theta} }^{ \, {\phi} \phantom{\, r} \phantom{\, {\theta}} } & = & -\frac{a^{3} \cos\left({\theta}\right)^{3} + {\left(2 \, a m r + a r^{2}\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}} \, r \, {\phi} }^{ \, {\phi} \phantom{\, r} \phantom{\, {\phi}} } & = & \frac{a^{4} r \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{3} \cos\left({\theta}\right)^{2} + r^{5} + {\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}} \, {\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 differentiable 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 differentiable manifold M
(2mra2cos(θ)2+r2)t+(a22mr+r2a2cos(θ)2+r2)r+(aa2cos(θ)2+r2)ϕ\renewcommand{\Bold}[1]{\mathbf{#1}}\left( \frac{2 \, m r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \frac{\partial}{\partial t } + \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=(t,rp,th,ph), name='p') print(p)
Point p on the 4-dimensional differentiable manifold M
X(p)
(t,m+a2+m2,θ,ϕ)\renewcommand{\Bold}[1]{\mathbf{#1}}\left(t, m + \sqrt{-a^{2} + m^{2}}, {\theta}, {\phi}\right)
nrH = nr.at(p) print(nrH)
Tangent vector at Point p on the 4-dimensional differentiable manifold M
Tp = M.tangent_space(p) print(Tp)
Tangent space at Point p on the 4-dimensional differentiable manifold M
Tp.default_basis()
(t,r,θ,ϕ)\renewcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{\partial}{\partial t },\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
t\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{\partial}{\partial t }
eta = X.frame()[3] eta
ϕ\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{\partial}{\partial {\phi} }
chi = xi + OmegaH*eta chi.display()
t+a2(m+a2+m2)mϕ\renewcommand{\Bold}[1]{\mathbf{#1}}\frac{\partial}{\partial t } + \frac{a}{2 \, {\left(m + \sqrt{-a^{2} + m^{2}}\right)} m} \frac{\partial}{\partial {\phi} }

Ingoing principal null geodesics

k = M.vector_field(name='k') k[0] = 1 k[1] = -1 k.display()
k=tr\renewcommand{\Bold}[1]{\mathbf{#1}}k = \frac{\partial}{\partial t }-\frac{\partial}{\partial r }

Let us check that kk is a null vector:

g(k,k).display()
g(k,k):MR(t,r,θ,ϕ)0\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} g\left(k,k\right):& M & \longrightarrow & \mathbb{R} \\ & \left(t, 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
nab(k).display()
gk=(a2mcos(θ)2mr2a4cos(θ)4+2a2r2cos(θ)2+r4)tdt+(a2mcos(θ)2mr2a4cos(θ)4+2a2r2cos(θ)2+r4)tdr+((a3mcos(θ)2amr2)sin(θ)2a4cos(θ)4+2a2r2cos(θ)2+r4)tdϕ+(a2mcos(θ)2mr2a4cos(θ)4+2a2r2cos(θ)2+r4)rdt+(a2mcos(θ)2mr2a4cos(θ)4+2a2r2cos(θ)2+r4)rdr+(a2cos(θ)sin(θ)a2cos(θ)2+r2)rdθ+((a3ma3r)sin(θ)4(a3ma3ramr2ar3)sin(θ)2a4cos(θ)4+2a2r2cos(θ)2+r4)rdϕ+(ra2cos(θ)2+r2)θdθ+(acos(θ)sin(θ)a2cos(θ)2+r2)θdϕ+acos(θ)(a2cos(θ)2+r2)sin(θ)ϕdθ+(ra2cos(θ)2+r2)ϕdϕ\renewcommand{\Bold}[1]{\mathbf{#1}}\nabla_{g} k = \left( \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}} \right) \frac{\partial}{\partial t }\otimes \mathrm{d} t + \left( \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}} \right) \frac{\partial}{\partial t }\otimes \mathrm{d} r + \left( -\frac{{\left(a^{3} m \cos\left({\theta}\right)^{2} - a m r^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \frac{\partial}{\partial t }\otimes \mathrm{d} {\phi} + \left( -\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}} \right) \frac{\partial}{\partial r }\otimes \mathrm{d} t + \left( -\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}} \right) \frac{\partial}{\partial r }\otimes \mathrm{d} r + \left( \frac{a^{2} \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \frac{\partial}{\partial r }\otimes \mathrm{d} {\theta} + \left( -\frac{{\left(a^{3} m - a^{3} r\right)} \sin\left({\theta}\right)^{4} - {\left(a^{3} m - a^{3} r - a m r^{2} - a r^{3}\right)} \sin\left({\theta}\right)^{2}}{a^{4} \cos\left({\theta}\right)^{4} + 2 \, a^{2} r^{2} \cos\left({\theta}\right)^{2} + r^{4}} \right) \frac{\partial}{\partial r }\otimes \mathrm{d} {\phi} + \left( -\frac{r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \frac{\partial}{\partial {\theta} }\otimes \mathrm{d} {\theta} + \left( -\frac{a \cos\left({\theta}\right) \sin\left({\theta}\right)}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \frac{\partial}{\partial {\theta} }\otimes \mathrm{d} {\phi} + \frac{a \cos\left({\theta}\right)}{{\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} \sin\left({\theta}\right)} \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} {\theta} + \left( -\frac{r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \frac{\partial}{\partial {\phi} }\otimes \mathrm{d} {\phi}

Outgoing principal null geodesics

el = M.vector_field(name='el', latex_name=r'\ell') el[0] = 1/2 + m*r/(r^2+a^2) el[1] = 1/2 - m*r/(r^2+a^2) el[3] = a/(r^2+a^2) el.display()
=(mra2+r2+12)t+(mra2+r2+12)r+(aa2+r2)ϕ\renewcommand{\Bold}[1]{\mathbf{#1}}\ell = \left( \frac{m r}{a^{2} + r^{2}} + \frac{1}{2} \right) \frac{\partial}{\partial t } + \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(t,r,θ,ϕ)0\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} g\left(\ell,\ell\right):& M & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & 0 \end{array}

Computation of \nabla_\ell \ell:

acc = nab(el).contract(el) acc.display()
(a4m+2a2m2r2m2r3mr42(a6+3a4r2+3a2r4+r6))t+(a4m2a2m2r+2m2r3mr42(a6+3a4r2+3a2r4+r6))r+(a3mamr2a6+3a4r2+3a2r4+r6)ϕ\renewcommand{\Bold}[1]{\mathbf{#1}}\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 t } + \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}

The gg-dual 1-form \underline{\ell} (i.e. the 1-form of components α=gαμμ\ell_\alpha = g_{\alpha\mu} \ell^\mu) is obtained via the method down():

el_form = el.down(g) el_form.display()
(a22mr+r22(a2+r2))dt+(2a2cos(θ)2a2+2mr+r22(a2+r2))dr+((a52a3mr+a3r2)sin(θ)4(a52a3mr+2a3r22amr3+ar4)sin(θ)22(a2r2+r4+(a4+a2r2)cos(θ)2))dϕ\renewcommand{\Bold}[1]{\mathbf{#1}}\left( -\frac{a^{2} - 2 \, m r + r^{2}}{2 \, {\left(a^{2} + r^{2}\right)}} \right) \mathrm{d} t + \left( \frac{2 \, a^{2} \cos\left({\theta}\right)^{2} - a^{2} + 2 \, m r + r^{2}}{2 \, {\left(a^{2} + r^{2}\right)}} \right) \mathrm{d} r + \left( -\frac{{\left(a^{5} - 2 \, a^{3} m r + a^{3} r^{2}\right)} \sin\left({\theta}\right)^{4} - {\left(a^{5} - 2 \, a^{3} m r + 2 \, a^{3} r^{2} - 2 \, a m r^{3} + a r^{4}\right)} \sin\left({\theta}\right)^{2}}{2 \, {\left(a^{2} r^{2} + r^{4} + {\left(a^{4} + a^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}\right)}} \right) \mathrm{d} {\phi}

We apply extra trigonometric simplifications on the components and factor them, via the method apply_map:

el_form.apply_map(lambda x: x.trig_reduce()) el_form.apply_map(lambda x: x.simplify_trig()) el_form.apply_map(factor) el_form.display()
(a22mr+r22(a2+r2))dt+(2a2cos(θ)2a2+2mr+r22(a2+r2))dr+(a22mr+r2)asin(θ)22(a2+r2)dϕ\renewcommand{\Bold}[1]{\mathbf{#1}}\left( -\frac{a^{2} - 2 \, m r + r^{2}}{2 \, {\left(a^{2} + r^{2}\right)}} \right) \mathrm{d} t + \left( \frac{2 \, a^{2} \cos\left({\theta}\right)^{2} - a^{2} + 2 \, m r + r^{2}}{2 \, {\left(a^{2} + r^{2}\right)}} \right) \mathrm{d} r + \frac{{\left(a^{2} - 2 \, m r + r^{2}\right)} a \sin\left({\theta}\right)^{2}}{2 \, {\left(a^{2} + r^{2}\right)}} \mathrm{d} {\phi}

The scalar product k=,k\ell\cdot k = \langle \underline\ell, k \rangle:

el_form(k).display()
MR(t,r,θ,ϕ)a2cos(θ)2+r2a2+r2\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & -\frac{a^{2} \cos\left({\theta}\right)^{2} + r^{2}}{a^{2} + r^{2}} \end{array}
el_form(k) == g(el, k)
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}

A variant for \ell:

el = M.vector_field(name='el', latex_name=r'\ell') el[0] = r^2+a^2 + 2*m*r el[1] = r^2+a^2 - 2*m*r el[3] = 2*a el.display()
=(a2+2mr+r2)t+(a22mr+r2)r+2aϕ\renewcommand{\Bold}[1]{\mathbf{#1}}\ell = \left( a^{2} + 2 \, m r + r^{2} \right) \frac{\partial}{\partial t } + \left( a^{2} - 2 \, m r + r^{2} \right) \frac{\partial}{\partial r } + 2 \, a \frac{\partial}{\partial {\phi} }
g(el,el).display()
g(,):MR(t,r,θ,ϕ)0\renewcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} g\left(\ell,\ell\right):& M & \longrightarrow & \mathbb{R} \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & 0 \end{array}
acc = nab(el).contract(el) acc.display()
(2a2m+2mr2+2r3+2(a22m2)r)t+(2a2m6mr2+2r3+2(a2+2m2)r)r+(4am+4ar)ϕ\renewcommand{\Bold}[1]{\mathbf{#1}}\left( -2 \, a^{2} m + 2 \, m r^{2} + 2 \, r^{3} + 2 \, {\left(a^{2} - 2 \, m^{2}\right)} r \right) \frac{\partial}{\partial t } + \left( -2 \, a^{2} m - 6 \, m r^{2} + 2 \, r^{3} + 2 \, {\left(a^{2} + 2 \, m^{2}\right)} r \right) \frac{\partial}{\partial r } + \left( -4 \, a m + 4 \, a r \right) \frac{\partial}{\partial {\phi} }
for i in [0,1,3]: pretty_print(acc[i] / el[i])
2m+2r\renewcommand{\Bold}[1]{\mathbf{#1}}-2 \, m + 2 \, r
2m+2r\renewcommand{\Bold}[1]{\mathbf{#1}}-2 \, m + 2 \, r
2m+2r\renewcommand{\Bold}[1]{\mathbf{#1}}-2 \, m + 2 \, r
kappa = (acc[0] / el[0]).expr() kappa
2m+2r\renewcommand{\Bold}[1]{\mathbf{#1}}-2 \, m + 2 \, r
acc == kappa * el
True\renewcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}