Carter constant

The Carter constant is a conserved quantity for motion around black holes in the general relativistic formulation of gravity. Carter's constant was derived for a spinning, charged black hole by Australian theoretical physicist Brandon Carter in 1968. Carter's constant along with the energy, axial angular momentum, and particle rest mass provide the four conserved quantities necessary to uniquely determine all orbits in the Kerr–Newman spacetime (even those of charged particles).

Formulation

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".

As generated by a Killing tensor

Noether's theorem states that all conserved quantities are related to spacetime symmetries. 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

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

Schwarzschild limit

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:

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

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.

See also

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

References

1. Carter, Brandon (1968). "Global structure of the Kerr family of gravitational fields". Physical Review. 174 (5): 1559–1571. Bibcode:1968PhRv..174.1559C. doi:10.1103/PhysRev.174.1559.
2. Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. New York: W. H. Freeman and Co. p. 899. ISBN 0-7167-0334-3.