Carter constant

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. The Carter constant can be written as follows:

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

in place of C . The quantity 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 K (different K than used above). In component form:

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

where g μ ν are the components of the metric tensor and l μ and n ν are the components of the principal null vectors:

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 E , L , and m to determine the motion; however, the symmetry leading to Carter's constant still exists. Carter's constant for Schwarzschild space is:

By a rotation of coordinates we can put any orbit in the θ = π / 2 plane so p θ = 0 . In this case C = L 2 , the square of the orbital angular momentum.