Boyer–Lindquist coordinates

In the mathematical description of general relativity, the Boyer–Lindquist coordinates are a generalization of the coordinates used for the metric of a Schwarzschild black hole that can be used to express the metric of a Kerr black hole.

The coordinate transformation from Boyer–Lindquist coordinates r , θ , ϕ to cartesian coordinates x, y, z is given by

The line element for a black hole with mass M , angular momentum J , and charge Q in Boyer–Lindquist coordinates and natural units ( G = c = 1 ) is

where

Note that in natural units M , a , and Q all have units of length. This line element describes the Kerr–Newman metric.

The Hamiltonian for test particle motion in Kerr spacetime is separable in Boyer–Lindquist coordinates. Using Hamilton-Jacobi theory one can derive a fourth constant of the motion known as Carter's constant.