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 $r, \theta$, $\phi$ to cartesian coordinates x, y, z is given by

${x} = \sqrt {r^2 + a^2} \sin\theta\cos\phi$

${y} = \sqrt {r^2 + a^2} \sin\theta\sin\phi$

${z} = r \cos\theta \quad$

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

$ds^2 = -\frac{\Delta}{\Sigma}\left(dt - a \sin^2\theta d\phi \right)^2 +\frac{\sin^2\theta}{\Sigma}\Big((r^2+a^2)d\phi - a dt\Big)^2 + \frac{\Sigma}{\Delta}dr^2 + \Sigma d\theta^2$

where

$\Delta = r^2 - 2Mr + a^2 + Q^2$

$\Sigma = r^2 + a^2 \cos^2\theta$

$a = J/M$, the angular momentum per unit mass of the black hole

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 was separable in Boyer–Lindquist coordinates. Using Hamilton-Jacobi theory one can derive a fourth constant of the motion known as Carter's constant.^[1]

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.

• Boyer, R. H. and Lindquist, R. W. Maximal Analytic Extension of the Kerr Metric. J. Math. Phys. 8, 265-281, 1967.
• Shapiro, S. L. and Teukolsky, S. A. Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects. New York: Wiley, p. 357, 1983.