Boyer–Lindquist coordinates

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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 , to cartesian coordinates x, y, z is given by

The line element for a black hole with mass , angular momentum , and charge in Boyer–Lindquist coordinates and natural units () is


, the angular momentum per unit mass of the black hole

Note that in natural units , , and 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]


  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.