Carter constant

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:

${\displaystyle C=p_{\theta }^{2}+\cos ^{2}\theta {\Bigg (}a^{2}(m^{2}-E^{2})+\left({\frac {L_{z}}{\sin \theta }}\right)^{2}{\Bigg )}}$,

where ${\displaystyle p_{\theta }}$ is the latitudinal component of the particle's angular momentum, ${\displaystyle E}$ is the energy of the particle, ${\displaystyle L_{z}}$ is the particle's axial angular momentum, ${\displaystyle m}$ is the rest mass of the particle, and ${\displaystyle a}$ is the spin parameter of the black hole.[2] Because functions of conserved quantities are also conserved, any function of ${\displaystyle C}$ and the three other constants of the motion can be used as a fourth constant in place of ${\displaystyle C}$. This results in some confusion as to the form of Carter's constant. For example it is sometimes more convenient to use:

${\displaystyle K=C+(L_{z}-aE)^{2}}$

in place of ${\displaystyle C}$. The quantity ${\displaystyle K}$ is useful because it is always non-negative. In general any fourth conserved quantity for motion in the Kerr family of spacetimes may be referred to as "Carter's constant".

Noether's theorem states that each conserved quantity of a system generates a continuous symmetry of that system. Carter's constant is related to a higher order symmetry of the Kerr metric generated by a second order Killing tensor field ${\displaystyle K}$ (different ${\displaystyle K}$ than used above). In component form:

${\displaystyle C=K^{\mu \nu }u_{\mu }u_{\nu }}$,

where ${\displaystyle u}$ is the four-velocity of the particle in motion. The components of the Killing tensor in Boyer–Lindquist coordinates are:

${\displaystyle K^{\mu \nu }=2\Sigma \ l^{(\mu }n^{\nu )}+r^{2}g^{\mu \nu }}$,

where ${\displaystyle g^{\mu \nu }}$ are the components of the metric tensor and ${\displaystyle l^{\mu }}$ and ${\displaystyle n^{\nu }}$ are the components of the principal null vectors:

${\displaystyle l^{\mu }=\left({\frac {r^{2}+a^{2}}{\Delta }},1,0,{\frac {a}{\Delta }}\right)}$

${\displaystyle n^{\nu }=\left({\frac {r^{2}+a^{2}}{2\Sigma }},-{\frac {\Delta }{2\Sigma }},0,{\frac {a}{2\Sigma }}\right)}$

with

${\displaystyle \Sigma =r^{2}+a^{2}\cos ^{2}\theta \ ,\ \ \Delta =r^{2}-r_{s}\ r+a^{2}}$.

The parentheses in ${\displaystyle l^{(\mu }n^{\nu )}}$ are notation for symmetrization:

${\displaystyle l^{(\mu }n^{\nu )}={\frac {1}{2}}(l^{\mu }n^{\nu }+l^{\nu }n^{\mu })}$

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 ${\displaystyle E}$, ${\displaystyle L_{z}}$, and ${\displaystyle m}$ to determine the motion; however, the symmetry leading to Carter's constant still exists. Carter's constant for Schwarzschild space is:

${\displaystyle C=p_{\theta }^{2}+\left(L_{z}\sin \theta \right)^{2}}$.

By a rotation of coordinates we can put any orbit in the ${\displaystyle \theta =\pi /2}$ plane so ${\displaystyle p_{\theta }=0}$. In this case ${\displaystyle C=L_{z}^{2}}$, the square of the orbital angular momentum.

• Kerr metric
• Kerr–Newman metric
• Boyer–Lindquist coordinates
• Hamilton–Jacobi equation
• Euler's three-body problem