physics - Kerr metric

In physics, the Kerr metric describes the geometry of spacetime around a rotating black hole. (The Schwarzschild metric is used to describe nonrotating black holes.) Discovered in 1963 by Roy Kerr, it is an exact solution to the Einstein field equations.

The Boyer-Lindquist form of the line element is given by

$ds^2=\rho^2(\frac{dr^2}{\Delta}+d\theta^2)+(r^2+a^2)\sin^2\theta d\phi^2-dt^2+\frac{2mr}{\rho^2}(a\sin^2\theta d\phi-dt)^2$

where

ρ²=r² + a²cos²θ

and

Δ=r² - 2mr + a².

Here m is the mass of the black hole, and a is is a parameter describing the rotation of the black hole, related to the angular momentum J by : $a = J / m$ . Note that r does not agree with the radial coordinate of the Schwarzschild solution, except asymptotically.

The Kerr metric is not the most general cylindrically symmetric metric. It is the case for certain vanishing multipole moments.

References

Ronald Adler, Maurice Bazin, Menahem Schiffer, Introductin to General Relativity (Second Edition), (1975) McGraw-Hill New York, ISBN 0-07-000423-4 See chapter 7.

Categories: General relativity | Black holes