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Kerr spacetime in 3+1 Kerr coordinates

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

These computations are based on SageManifolds (v0.9)

Click here to download the worksheet file (ipynb format). To run it, you must start SageMath with the Jupyter notebook, with the command sage -n jupyter

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

In [1]:
%display latex

Spacetime

We declare the spacetime manifold MM:

In [2]:
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:

In [3]:
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))\left(M,(t, r, {\theta}, {\phi})\right)
In [4]:
X.coord_range()
t: (,+);r: (,+);θ: (0,π);ϕ: (0,2π)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:

In [5]:
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:

In [6]:
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=(a2cos(θ)22mr+r2a2cos(θ)2+r2)dtdt+(2mra2cos(θ)2+r2)dtdr+(2amrsin(θ)2a2cos(θ)2+r2)dtdϕ+(2mra2cos(θ)2+r2)drdt+(a2cos(θ)2+2mr+r2a2cos(θ)2+r2)drdr+((a3cos(θ)2+2amr+ar2)sin(θ)2a2cos(θ)2+r2)drdϕ+(a2cos(θ)2+r2)dθdθ+(2amrsin(θ)2a2cos(θ)2+r2)dϕdt+((a3cos(θ)2+2amr+ar2)sin(θ)2a2cos(θ)2+r2)dϕdr+(2a2mrsin(θ)4+(a2r2+r4+(a4+a2r2)cos(θ)2)sin(θ)2a2cos(θ)2+r2)dϕdϕg = \left( -\frac{a^{2} \cos\left({\theta}\right)^{2} - 2 \, m r + r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \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{a^{2} \cos\left({\theta}\right)^{2} + 2 \, m r + r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( -\frac{{\left(a^{3} \cos\left({\theta}\right)^{2} + 2 \, a m r + a r^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \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 + \left( -\frac{{\left(a^{3} \cos\left({\theta}\right)^{2} + 2 \, a m r + a r^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} r + \left( \frac{2 \, a^{2} m r \sin\left({\theta}\right)^{4} + {\left(a^{2} r^{2} + r^{4} + {\left(a^{4} + a^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \right) \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}
In [7]:
g.display_comp()
gtttt=a2cos(θ)22mr+r2a2cos(θ)2+r2gtrtr=2mra2cos(θ)2+r2gtϕtϕ=2amrsin(θ)2a2cos(θ)2+r2grtrt=2mra2cos(θ)2+r2grrrr=a2cos(θ)2+2mr+r2a2cos(θ)2+r2grϕrϕ=(a3cos(θ)2+2amr+ar2)sin(θ)2a2cos(θ)2+r2gθθθθ=a2cos(θ)2+r2gϕtϕt=2amrsin(θ)2a2cos(θ)2+r2gϕrϕr=(a3cos(θ)2+2amr+ar2)sin(θ)2a2cos(θ)2+r2gϕϕϕϕ=2a2mrsin(θ)4+(a2r2+r4+(a4+a2r2)cos(θ)2)sin(θ)2a2cos(θ)2+r2\begin{array}{lcl} g_{ \, t \, t }^{ \phantom{\, t } \phantom{\, t } } & = & -\frac{a^{2} \cos\left({\theta}\right)^{2} - 2 \, m r + r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ 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{a^{2} \cos\left({\theta}\right)^{2} + 2 \, m r + r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ g_{ \, r \, {\phi} }^{ \phantom{\, r } \phantom{\, {\phi} } } & = & -\frac{{\left(a^{3} \cos\left({\theta}\right)^{2} + 2 \, a m r + a r^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{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 } } & = & -\frac{{\left(a^{3} \cos\left({\theta}\right)^{2} + 2 \, a m r + a r^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ g_{ \, {\phi} \, {\phi} }^{ \phantom{\, {\phi} } \phantom{\, {\phi} } } & = & \frac{2 \, a^{2} m r \sin\left({\theta}\right)^{4} + {\left(a^{2} r^{2} + r^{4} + {\left(a^{4} + a^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \end{array}

The inverse metric is pretty simple:

In [8]:
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)\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:

In [9]:
g.determinant().display()
MR(t,r,θ,ϕ)a4cos(θ)6(a42a2r2)cos(θ)4r4(2a2r2r4)cos(θ)2\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}
In [10]:
g.determinant() == - (rho2*sin(th))^2
True\mathrm{True}

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

In [11]:
g.ricci().display() # long: takes 1 or 2 minutes
Ric(g)=0\mathrm{Ric}\left(g\right) = 0

The Christoffel symbols w.r.t. the 3+1 Kerr coordinates:

In [12]:
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\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}

In [13]:
dr = X.coframe()[1] print(dr) dr.display()
1-form dr on the 4-dimensional differentiable manifold M
dr=dr\mathrm{d} r = \mathrm{d} r
In [14]:
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\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} }
In [15]:
assume(a^2<m^2) rp = m + sqrt(m^2-a^2) rp
m+a2+m2m + \sqrt{-a^{2} + m^{2}}
In [16]:
p = M.point(coords=(t,rp,th,ph), name='p') print(p)
Point p on the 4-dimensional differentiable manifold M
In [17]:
X(p)
(t,m+a2+m2,θ,ϕ)\left(t, m + \sqrt{-a^{2} + m^{2}}, {\theta}, {\phi}\right)
In [18]:
nrH = nr.at(p) print(nrH)
Tangent vector at Point p on the 4-dimensional differentiable manifold M
In [19]:
Tp = M.tangent_space(p) print(Tp)
Tangent space at Point p on the 4-dimensional differentiable manifold M
In [20]:
Tp.default_basis()
(t,r,θ,ϕ)\left(\frac{\partial}{\partial t },\frac{\partial}{\partial r },\frac{\partial}{\partial {\theta} },\frac{\partial}{\partial {\phi} }\right)
In [21]:
nrH[:]
[2(a+ma+mm+m2)a2sin(θ)22a+ma+mm2m2,0,0,aa2sin(θ)22a+ma+mm2m2]\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]
In [22]:
OmegaH = a/(2*m*rp) OmegaH
a2(m+a2+m2)m\frac{a}{2 \, {\left(m + \sqrt{-a^{2} + m^{2}}\right)} m}
In [23]:
xi = X.frame()[0] xi
t\frac{\partial}{\partial t }
In [24]:
eta = X.frame()[3] eta
ϕ\frac{\partial}{\partial {\phi} }
In [25]:
chi = xi + OmegaH*eta chi.display()
t+a2(a+ma+mm+m2)ϕ\frac{\partial}{\partial t } + \frac{a}{2 \, {\left(\sqrt{a + m} \sqrt{-a + m} m + m^{2}\right)}} \frac{\partial}{\partial {\phi} }

Ingoing principal null geodesics

In [26]:
k = M.vector_field(name='k') k[0] = 1 k[1] = -1 k.display()
k=trk = \frac{\partial}{\partial t }-\frac{\partial}{\partial r }

Let us check that kk is a null vector:

In [27]:
g(k,k).display()
g(k,k):MR(t,r,θ,ϕ)0\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:

In [28]:
nab = g.connection() acc = nab(k).contract(k) acc.display()
00
In [29]:
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\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

In [30]:
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()
=(a2+2mr+r22(a2+r2))t+(a22mr+r22(a2+r2))r+(aa2+r2)ϕ\ell = \left( \frac{a^{2} + 2 \, m r + r^{2}}{2 \, {\left(a^{2} + r^{2}\right)}} \right) \frac{\partial}{\partial t } + \left( \frac{a^{2} - 2 \, m r + r^{2}}{2 \, {\left(a^{2} + r^{2}\right)}} \right) \frac{\partial}{\partial r } + \left( \frac{a}{a^{2} + r^{2}} \right) \frac{\partial}{\partial {\phi} }