SharedKerr Spacetime.ipynbOpen in CoCalc
Author: Chan Park
Views : 108
Description: Kerr Spacetime
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%display latex
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M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='Lorentzian'); M
M\mathcal{M}
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var('a', domain='real') assume(a > 0) assume(a < 1)
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BL.<t,r,th,ph> = M.chart(r't r:(1+sqrt(1-a^2),+oo) th:(0,pi):\theta ph:(0,2*pi):\phi'); BL
(M,(t,r,θ,ϕ))\left(\mathcal{M},(t, r, {\theta}, {\phi})\right)
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g = M.metric()
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Sigma = r^2 + (a*cos(th))^2 Delta = r^2 - 2*r + a^2 g[0,0] = -(1-2*r/Sigma) g[0,3] = -2*a*r*sin(th)^2/Sigma g[1,1] = Sigma/Delta g[2,2] = Sigma g[3,3] = (r^2 + a^2 + 2*r*(a*sin(th))^2/Sigma)*sin(th)^2 g.display()
g=(2ra2cos(θ)2+r21)dtdt+(2arsin(θ)2a2cos(θ)2+r2)dtdϕ+(a2cos(θ)2+r2a2+r22r)drdr+(a2cos(θ)2+r2)dθdθ+(2arsin(θ)2a2cos(θ)2+r2)dϕdt+(2a2rsin(θ)2a2cos(θ)2+r2+a2+r2)sin(θ)2dϕdϕg = \left( \frac{2 \, r}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} - 1 \right) \mathrm{d} t\otimes \mathrm{d} t + \left( -\frac{2 \, a 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} + r^{2} - 2 \, r} \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 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} 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}
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g.christoffel_symbols_display()
Γttrttr=a4r4(a4+a2r2)sin(θ)2a2r4+r62r5+(a6+a4r22a4r)cos(θ)4+2(a4r2+a2r42a2r3)cos(θ)2Γttθttθ=2a2rcos(θ)sin(θ)a4cos(θ)4+2a2r2cos(θ)2+r4Γtrϕtrϕ=(a3r2+3ar4(a5a3r2)cos(θ)2)sin(θ)2a2r4+r62r5+(a6+a4r22a4r)cos(θ)4+2(a4r2+a2r42a2r3)cos(θ)2Γtθϕtθϕ=2(a5rcos(θ)sin(θ)5(a5r+a3r3)cos(θ)sin(θ)3)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γrttrtt=a2r2+r42r3(a4+a2r22a2r)cos(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γrtϕrtϕ=(a3r2+ar42ar3(a5+a3r22a3r)cos(θ)2)sin(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γrrrrrr=(a2ra2)sin(θ)2+a2r2a2r2+r42r3+(a4+a2r22a2r)cos(θ)2Γrrθrrθ=a2cos(θ)sin(θ)a2cos(θ)2+r2Γrθθrθθ=a2r+r32r2a2cos(θ)2+r2Γrϕϕrϕϕ=(a4r2+a2r42a2r3(a6+a4r22a4r)cos(θ)2)sin(θ)4(a2r5+r72r6+(a6r+a4r32a4r2)cos(θ)4+2(a4r3+a2r52a2r4)cos(θ)2)sin(θ)2a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γθttθtt=2a2rcos(θ)sin(θ)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γθtϕθtϕ=2(a3r+ar3)cos(θ)sin(θ)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γθrrθrr=a2cos(θ)sin(θ)a2r2+r42r3+(a4+a2r22a2r)cos(θ)2Γθrθθrθ=ra2cos(θ)2+r2Γθθθθθθ=a2cos(θ)sin(θ)a2cos(θ)2+r2Γθϕϕθϕϕ=((a6+a4r22a4r)cos(θ)5+2(a4r2+a2r42a2r3)cos(θ)3+(a2r4+r6+2a4r+4a2r3)cos(θ))sin(θ)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6Γϕtrϕtr=a3cos(θ)2ar2a2r4+r62r5+(a6+a4r22a4r)cos(θ)4+2(a4r2+a2r42a2r3)cos(θ)2Γϕtθϕtθ=2arcos(θ)(a4cos(θ)4+2a2r2cos(θ)2+r4)sin(θ)Γϕrϕϕrϕ=r5+(a4ra4)cos(θ)4a2r22r4+(2a2r3+a4a2r2)cos(θ)2a2r4+r62r5+(a6+a4r22a4r)cos(θ)4+2(a4r2+a2r42a2r3)cos(θ)2Γϕθϕϕθϕ=a4cos(θ)5+2(a2r2a2r)cos(θ)3+(r4+2a2r)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^{4} - r^{4} - {\left(a^{4} + a^{2} r^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} r^{4} + r^{6} - 2 \, r^{5} + {\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{2} + a^{2} r^{4} - 2 \, a^{2} r^{3}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, t} \, t \, {\theta} }^{ \, t \phantom{\, t} \phantom{\, {\theta}} } & = & -\frac{2 \, a^{2} 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} r^{2} + 3 \, a r^{4} - {\left(a^{5} - a^{3} r^{2}\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{2}}{a^{2} r^{4} + r^{6} - 2 \, r^{5} + {\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{2} + a^{2} r^{4} - 2 \, a^{2} r^{3}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, t} \, {\theta} \, {\phi} }^{ \, t \phantom{\, {\theta}} \phantom{\, {\phi}} } & = & -\frac{2 \, {\left(a^{5} r \cos\left({\theta}\right) \sin\left({\theta}\right)^{5} - {\left(a^{5} r + a^{3} 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} r^{2} + r^{4} - 2 \, r^{3} - {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\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} r^{2} + a r^{4} - 2 \, a r^{3} - {\left(a^{5} + a^{3} r^{2} - 2 \, a^{3} r\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{{\left(a^{2} r - a^{2}\right)} \sin\left({\theta}\right)^{2} + a^{2} - r^{2}}{a^{2} r^{2} + r^{4} - 2 \, r^{3} + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\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 + r^{3} - 2 \, r^{2}}{a^{2} \cos\left({\theta}\right)^{2} + r^{2}} \\ \Gamma_{ \phantom{\, r} \, {\phi} \, {\phi} }^{ \, r \phantom{\, {\phi}} \phantom{\, {\phi}} } & = & \frac{{\left(a^{4} r^{2} + a^{2} r^{4} - 2 \, a^{2} r^{3} - {\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} \cos\left({\theta}\right)^{2}\right)} \sin\left({\theta}\right)^{4} - {\left(a^{2} r^{5} + r^{7} - 2 \, r^{6} + {\left(a^{6} r + a^{4} r^{3} - 2 \, a^{4} r^{2}\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{3} + a^{2} r^{5} - 2 \, a^{2} r^{4}\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} 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} r + a 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} + r^{4} - 2 \, r^{3} + {\left(a^{4} + a^{2} r^{2} - 2 \, a^{2} r\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} + a^{4} r^{2} - 2 \, a^{4} r\right)} \cos\left({\theta}\right)^{5} + 2 \, {\left(a^{4} r^{2} + a^{2} r^{4} - 2 \, a^{2} r^{3}\right)} \cos\left({\theta}\right)^{3} + {\left(a^{2} r^{4} + r^{6} + 2 \, a^{4} r + 4 \, a^{2} r^{3}\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} \cos\left({\theta}\right)^{2} - a r^{2}}{a^{2} r^{4} + r^{6} - 2 \, r^{5} + {\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{2} + a^{2} r^{4} - 2 \, a^{2} r^{3}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, {\phi}} \, t \, {\theta} }^{ \, {\phi} \phantom{\, t} \phantom{\, {\theta}} } & = & -\frac{2 \, a 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{r^{5} + {\left(a^{4} r - a^{4}\right)} \cos\left({\theta}\right)^{4} - a^{2} r^{2} - 2 \, r^{4} + {\left(2 \, a^{2} r^{3} + a^{4} - a^{2} r^{2}\right)} \cos\left({\theta}\right)^{2}}{a^{2} r^{4} + r^{6} - 2 \, r^{5} + {\left(a^{6} + a^{4} r^{2} - 2 \, a^{4} r\right)} \cos\left({\theta}\right)^{4} + 2 \, {\left(a^{4} r^{2} + a^{2} r^{4} - 2 \, a^{2} r^{3}\right)} \cos\left({\theta}\right)^{2}} \\ \Gamma_{ \phantom{\, {\phi}} \, {\theta} \, {\phi} }^{ \, {\phi} \phantom{\, {\theta}} \phantom{\, {\phi}} } & = & \frac{a^{4} \cos\left({\theta}\right)^{5} + 2 \, {\left(a^{2} r^{2} - a^{2} r\right)} \cos\left({\theta}\right)^{3} + {\left(r^{4} + 2 \, a^{2} r\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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