pt
p\phi
\sqrt{|p\mup\mu|}
Carter noticed that the Hamiltonian for motion in Kerr spacetime was separable in Boyer–Lindquist coordinates, allowing the constants of such motion to be easily identified using Hamilton–Jacobi theory.[1] The Carter constant can be written as follows:
C=
2 | |
p | |
\theta |
+\cos2\theta(a2(m2-E2)+\left(
Lz | |
\sin\theta |
\right)2)
where
p\theta
E=pt
Lz=p\phi
m=\sqrt{|p\mup\mu|}
a
p\mu
X\mu=(t,r,\theta,\phi)
\tau
U\mu=dX\mu/d\tau
p\mu=g\mu\nup\nu
p\mu=mU\mu
g\mu\nu
\mu | |
U | |
\rmobs |
p\mu
L=\boldsymbol{x}\wedge\boldsymbol{p}=rp\theta\boldsymbol{dr}\wedge\boldsymbol{d\theta}+rp\phi\boldsymbol{dr}\wedge\boldsymbol{d\phi} =mr3
\theta |
\boldsymbol{dr}\wedge\boldsymbol{d\theta}+mr3\sin2\theta
\phi\boldsymbol{dr}\wedge\boldsymbol{d\phi} |
z
Lxy
p\phi
Because functions of conserved quantities are also conserved, any function of
C
C
K=C+(Lz-aE)2
in place of
C
K
a=0
2 | |
C=L | |
z |
K=L2
L
K
K
C=K\mu\nuu\muu\nu
where
u
K\mu\nu=2\Sigma l(\mun\nu)+r2g\mu\nu
where
g\mu\nu
l\mu
n\nu
l\mu=\left(
r2+a2 | ,1,0, | |
\Delta |
a | |
\Delta |
\right)
n\nu=\left(
r2+a2 | ,- | |
2\Sigma |
\Delta | ,0, | |
2\Sigma |
a | |
2\Sigma |
\right)
with
\Sigma=r2+a2\cos2\theta , \Delta=r2-rs r+a2
The parentheses in
l(\mun\nu)
l(\mun\nu)=
1 | |
2 |
(l\mun\nu+l\nun\mu)
The spherical symmetry of the Schwarzschild metric for non-spinning black holes allows one to reduce the problem of finding the trajectories of particles to three dimensions. In this case one only needs
E
Lz
m
C=
2 | |
p | |
\theta |
+
2 | |
L | |
z |
\cot2\theta
Lij=xi\wedgepj
\boldsymbol{x}=r\boldsymbol{dr}
\boldsymbol{p}=pr\boldsymbol{dr}+p\theta\boldsymbol{d\theta}+p\phi\boldsymbol{d\phi}
p\theta=g\theta\thetap\theta=r2m
\theta |
p\phi=g\phi\phip\phi=r2\sin2\thetam
\phi |
\phi |
=d\phi/d\tau
\theta |
L=\boldsymbol{x}\wedge\boldsymbol{p}=rp\theta\boldsymbol{dr}\wedge\boldsymbol{d\theta}+rp\phi\boldsymbol{dr}\wedge\boldsymbol{d\phi} =mr3
\theta |
\boldsymbol{dr}\wedge\boldsymbol{d\theta}+mr3\sin2\theta
\phi\boldsymbol{dr}\wedge\boldsymbol{d\phi} |
Since
\boldsymbol{\hat{\theta}}=r\boldsymbol{d\theta}
\boldsymbol{\hat{\phi}}=r\sin\theta\boldsymbol{d\phi}
L
\boldsymbol{L*}=mr2
\theta |
\boldsymbol{\hat{\theta}}+mr2\sin\theta
\phi |
\boldsymbol{\hat{\phi}}
\vec{\boldsymbol{r}} x m\vec{\boldsymbol{v}}
\theta |
\phi |
L2=g\theta\thetar2
2 | |
p | |
\theta |
+g\phi\phir2
2 | |
p | |
\phi |
=g\theta\thetar2(p\theta)2+g\phi\phir2(p\phi)2=m2r4
\theta |
2+m2r4\sin2\theta
\phi |
2
Further since
p\theta=g\theta\thetap\theta=mr2
\theta |
Lz=p\phi=g\phi\phip\phi=mr2\sin2\theta
\phi |
C=m2r4
\theta |
2+m2r4\sin2\theta\cos2\theta
\phi |
2=m2r4
\theta |
2+m2r4
| |||
\sin |
2-m2r4
| |||
\sin |
2=L2-
2 | |
L | |
z |
In the Schwarzschild case, all components of the angular momentum vector are conserved, so both
L2
2 | |
L | |
z |
C
Lz=p\phi
p\theta
L2
C
The other form of Carter's constant is
K=C+(Lz-aE)2=(L2-
2) | |
L | |
z |
+(Lz-aE)2=L2
a=0
C\geq0
K\geq0
K=0
C=0
K>0
\theta=\pi/2