Kernel: SageMath (stable)
Equatorial geodesics in Kerr spacetime
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%display latex
The spacetime manifold and Boyer-Lindquist coordinates:
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M = Manifold(4, 'M') X.<t,r,th,ph> = M.chart(r"t r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi") X
(M,(t,r,θ,ϕ))
The spacetime metric:
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g = M.lorentzian_metric('g') var('m, a', domain='real') rho2 = r^2 + (a*cos(th))^2 Delta = r^2 -2*m*r + a^2 g[0,0] = -(1-2*m*r/rho2) g[0,3] = -2*a*m*r*sin(th)^2/rho2 g[1,1], g[2,2] = rho2/Delta, rho2 g[3,3] = (r^2+a^2+2*m*r*(a*sin(th))^2/rho2)*sin(th)^2 g.display()
g=(a2cos(θ)2+r22mr−1)dt⊗dt+(−a2cos(θ)2+r22amrsin(θ)2)dt⊗dϕ+(a2−2mr+r2a2cos(θ)2+r2)dr⊗dr+(a2cos(θ)2+r2)dθ⊗dθ+(−a2cos(θ)2+r22amrsin(θ)2)dϕ⊗dt+(a2cos(θ)2+r22a2mrsin(θ)2+a2+r2)sin(θ)2dϕ⊗dϕ
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u = M.vector_field('u') var('eps', latex_name=r'\varepsilon') var('ell', latex_name=r'\ell') u[0] = ((r^2 + a^2*(1+2*m/r))*eps - 2*a*m/r*ell)/Delta u[1] = sqrt(eps^2 - 1 + 2*m/r - (ell^2-a^2*(eps^2-1))/r^2 + 2*m/r^3*(ell-a*eps)^2) u[3] = (2*a*m/r*eps + (1-2*m/r)*ell)/Delta u.display_comp()
utturruϕϕ===a2−2mr+r2(a2(r2m+1)+r2)ε−r2aℓmε2+r32(aε−ℓ)2m+r2m+r2(ε2−1)a2−ℓ2−1a2−2mr+r2r2aεm−ℓ(r2m−1)
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norm = g(u,u) norm.coord_function()
a2r3cos(θ)2+r5(a2ε2−2a2−ℓ2)r3cos(θ)2−r5+(2(a4ε2−2a3ℓε+a2ℓ2)m+(a4ε2−a4−a2ℓ2)r)cos(θ)4
Value of g(u,u) in the equatorial plane (θ=2π):
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norm.coord_function()(t,r,pi/2,ph)
−1
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nabla = g.connection() print(nabla)
Levi-Civita connection nabla_g associated with the Lorentzian metric g on the 4-dimensional differentiable manifold M
The 4-acceleration vector a=∇uu:
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Du = nabla(u) a = u.contract(0, Du, 1) a.set_name('a') a.display_comp()
attarraθθaϕϕ====−(a2r7−2mr8+r9+(a6r3−2a4mr4+a4r5)cos(θ)4+2(a4r5−2a2mr6+a2r7)cos(θ)2)r2(((a4ε−a3ℓ)mr2+(a6ε−a5ℓ)m)cos(θ)4+3((a2ε−aℓ)mr4+(a4ε−a3ℓ)mr2)cos(θ)2)(ε2−1)r3+2mr2+2(a2ε2−2aℓε+ℓ2)m+(a2ε2−a2−ℓ2)ra2r10−2mr11+r12+(a8r4−2a6mr5+a6r6)cos(θ)6+3(a6r6−2a4mr7+a4r8)cos(θ)4+3(a4r8−2a2mr9+a2r10)cos(θ)2((a6ε2+4a5ℓε−4a6−5a4ℓ2)mr4−(a6ε2−a6−a4ℓ2)r5−2(a8ε2−a8−a6ℓ2−2(a6ε2−3a5ℓε+a6+2a4ℓ2)m2)r3−2(2(a6ε2−2a5ℓε+a4ℓ2)m3+(a8ε2−5a7ℓε+2a8+4a6ℓ2)m)r2−3(a10ε2−2a9ℓε+a8ℓ2)m−(a10ε2−a10−a8ℓ2−8(a8ε2−2a7ℓε+a6ℓ2)m2)r)cos(θ)6+((a4ε2+6a3ℓε−8a4−7a2ℓ2)mr6−2(a4ε2−a4−a2ℓ2)r7−4(a6ε2−a6−a4ℓ2−(a4ε2−2a3ℓε+2a4+a2ℓ2)m2)r5+2(2(a4ε2−2a3ℓε+a2ℓ2)m3−(3a6ε2−10a5ℓε+4a6+7a4ℓ2)m)r4−7(a8ε2−2a7ℓε+a6ℓ2)mr2−2(a8ε2−a8−a6ℓ2−6(a6ε2−2a5ℓε+a4ℓ2)m2)r3)cos(θ)4−(2(a2ε2−3aℓε+2a2+2ℓ2)mr8+(a2ε2−a2−ℓ2)r9+2(4a4ε2−9a3ℓε+2a4+5a2ℓ2)mr6+2(a4ε2−a4−a2ℓ2−2(2a2ε2−3aℓε+a2+ℓ2)m2)r7+6(a6ε2−2a5ℓε+a4ℓ2)mr4+(a6ε2−a6−a4ℓ2−12(a4ε2−2a3ℓε+a2ℓ2)m2)r5)cos(θ)2−a2r9−2mr10+r11+(a8r3−2a6mr4+a6r5)cos(θ)6+3(a6r5−2a4mr6+a4r7)cos(θ)4+3(a4r7−2a2mr8+a2r9)cos(θ)2((2(2a5ℓε−a6−2a4ℓ2)mr2−(a6ε2−a6−a4ℓ2)r3−2(a8ε2−2a7ℓε+a6ℓ2)m−(a8ε2−a8−a6ℓ2−4(a6ε2−2a5ℓε+a4ℓ2)m2)r)cos(θ)5+2(2(2a3ℓε−a4−2a2ℓ2)mr4−(a4ε2−a4−a2ℓ2)r5−2(a6ε2−2a5ℓε+a4ℓ2)mr2−(a6ε2−a6−a4ℓ2−4(a4ε2−2a3ℓε+a2ℓ2)m2)r3)cos(θ)3+(2(a2ε2−a2−ℓ2)mr6−(a2ε2−a2−ℓ2)r7−(a4ε2−a4−a2ℓ2)r5)cos(θ))sin(θ)−(a2r7−2mr8+r9+(a6r3−2a4mr4+a4r5)cos(θ)4+2(a4r5−2a2mr6+a2r7)cos(θ)2)r2(3(a3ε−a2ℓ)mr2cos(θ)2+(a5ε−a4ℓ)mcos(θ)4)(ε2−1)r3+2mr2+2(a2ε2−2aℓε+ℓ2)m+(a2ε2−a2−ℓ2)r
Values of the 4-acceleration in the equatorial plane (θ=2π):
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a[0](t,r,pi/2,ph)
0
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a[1](t,r,pi/2,ph)
0
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a[2](t,r,pi/2,ph)
0
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a[3](t,r,pi/2,ph)
0
The (non-zero and non-redundant) Christoffel symbols in Boyer-Lindquist coordinates:
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g.christoffel_symbols_display()
ΓttrttrΓttθttθΓtrϕtrϕΓtθϕtθϕΓrttrttΓrtϕrtϕΓrrrrrrΓrrθrrθΓrθθrθθΓrϕϕrϕϕΓθttθttΓθtϕθtϕΓθrrθrrΓθrθθrθΓθθθθθθΓθϕϕθϕϕΓϕtrϕtrΓϕtθϕtθΓϕrϕϕrϕΓϕθϕϕθϕ====================a2r4−2mr5+r6+(a6−2a4mr+a4r2)cos(θ)4+2(a4r2−2a2mr3+a2r4)cos(θ)2a2mr2+mr4−(a4m+a2mr2)cos(θ)2−a4cos(θ)4+2a2r2cos(θ)2+r42a2mrcos(θ)sin(θ)−a2r4−2mr5+r6+(a6−2a4mr+a4r2)cos(θ)4+2(a4r2−2a2mr3+a2r4)cos(θ)2(a3mr2+3amr4−(a5m−a3mr2)cos(θ)2)sin(θ)2−a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r62(a5mrcos(θ)sin(θ)5−(a5mr+a3mr3)cos(θ)sin(θ)3)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6a2mr2−2m2r3+mr4−(a4m−2a2m2r+a2mr2)cos(θ)2−a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6(a3mr2−2am2r3+amr4−(a5m−2a3m2r+a3mr2)cos(θ)2)sin(θ)2a2r2−2mr3+r4+(a4−2a2mr+a2r2)cos(θ)2a2r−mr2+(a2m−a2r)cos(θ)2−a2cos(θ)2+r2a2cos(θ)sin(θ)−a2cos(θ)2+r2a2r−2mr2+r3a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6(a4mr2−2a2m2r3+a2mr4−(a6m−2a4m2r+a4mr2)cos(θ)2)sin(θ)4−(a2r5−2mr6+r7+(a6r−2a4mr2+a4r3)cos(θ)4+2(a4r3−2a2mr4+a2r5)cos(θ)2)sin(θ)2−a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r62a2mrcos(θ)sin(θ)a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r62(a3mr+amr3)cos(θ)sin(θ)a2r2−2mr3+r4+(a4−2a2mr+a2r2)cos(θ)2a2cos(θ)sin(θ)a2cos(θ)2+r2r−a2cos(θ)2+r2a2cos(θ)sin(θ)−a6cos(θ)6+3a4r2cos(θ)4+3a2r4cos(θ)2+r6((a6−2a4mr+a4r2)cos(θ)5+2(a4r2−2a2mr3+a2r4)cos(θ)3+(2a4mr+4a2mr3+a2r4+r6)cos(θ))sin(θ)−a2r4−2mr5+r6+(a6−2a4mr+a4r2)cos(θ)4+2(a4r2−2a2mr3+a2r4)cos(θ)2a3mcos(θ)2−amr2−(a4cos(θ)4+2a2r2cos(θ)2+r4)sin(θ)2amrcos(θ)−a2r4−2mr5+r6+(a6−2a4mr+a4r2)cos(θ)4+2(a4r2−2a2mr3+a2r4)cos(θ)2a2mr2+2mr4−r5+(a4m−a4r)cos(θ)4−(a4m−a2mr2+2a2r3)cos(θ)2(a4cos(θ)4+2a2r2cos(θ)2+r4)sin(θ)a4cos(θ)sin(θ)4−2(a4−a2mr+a2r2)cos(θ)sin(θ)2+(a4+2a2r2+r4)cos(θ)
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