Boyer-Lindquist coordinates
Encyclopedia
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,
, 
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
where
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
.
The coordinate transformation from Boyer–Lindquist coordinates r,
, 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 unitsNatural units
In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...
(
) is
where
-
, 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 metricKerr-Newman metric
The Kerr–Newman metric is a solution of the Einstein–Maxwell equations in general relativity, describing the spacetime geometry in the region surrounding a charged, rotating mass. It is assumed that the cosmological constant equals zero...
.
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
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...
.