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).

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 ${\displaystyle p_{\theta }}$ 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. 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:

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