# Boyer–Lindquist coordinates

In the mathematical description of general relativity, the Boyer–Lindquist coordinates[1] 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 ${\displaystyle r,\theta }$, ${\displaystyle \phi }$ to Cartesian coordinates x, y, z is given by

${\displaystyle {x}={\sqrt {r^{2}+a^{2}}}\sin \theta \cos \phi }$
${\displaystyle {y}={\sqrt {r^{2}+a^{2}}}\sin \theta \sin \phi }$
${\displaystyle {z}=r\cos \theta \quad }$.

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

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

where

${\displaystyle \Delta =r^{2}-2Mr+a^{2}+Q^{2}}$
${\displaystyle \Sigma =r^{2}+a^{2}\cos ^{2}\theta }$
${\displaystyle a=J/M}$, the angular momentum per unit mass of the black hole.

Note that in natural units ${\displaystyle M}$, ${\displaystyle a}$, and ${\displaystyle Q}$ 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.[2]

## References

1. ^ Boyer, Robert H.; Lindquist, Richard W. (1967). "Maximal Analytic Extension of the Kerr Metric". J. Math. Phys. 8 (2): 265–281. Bibcode:1967JMP.....8..265B. doi:10.1063/1.1705193.
2. ^ 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.
• 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.