# Kerr metric

The solution of the Einstein equation describing the external gravity field of a rotating source with mass $m$ and angular momentum $L$. It is of type $D$ according to the classification of A.Z. Petrov. The simplest description is as the Kerr–Schild metric:

$$g _ {\mu \nu } = \eta _ {\mu \nu } + 2 h K _ \mu K _ \nu ,$$

where $K _ \mu$ is the null vector $( K _ \mu K _ \nu g ^ {\mu \nu } = 0 )$, tangent to the special principal null congruence with rotation (of non-gradient type), and $\eta _ {\mu \nu }$ is the metric tensor of Minkowski space. The characteristic parameter of the Kerr metric is $a = L / m$. In the general case in the presence of a charge $e$( a Kerr–Newman metric) the scalar function $h$ has the form

$$h = \frac{m}{2} ( \rho ^ {-} 1 + \overline{ {\rho ^ {-} 1 }}\; ) - \frac{e ^ {2} }{2 \rho \overline \rho \; } ,$$

where

$$\rho ^ {2} = x ^ {2} + y ^ {2} + ( z + i a ) ^ {2} .$$

The field is singular on the annular thread of radius $a$( when $\rho = 0$). For $a = 0$ the singularity contracts to a point; when $a = e = 0$ the Kerr metric becomes the Schwarzschild metric.

The Kerr metric was obtained by R.P. Kerr [1].

#### References

 [1] R.P. Kerr, "Gravitational field of a spinning mass as an example of algebraically special matrices" Phys. Rev. Letters , 11 (1963) pp. 237–238 [2] C.W. Misner, K.S. Thorne, J.A. Wheeler, "Gravitation" , Freeman (1973) [3] M. Rees, R. Ruffini, J. Wheeler, "Black holes, gravitational waves and cosmology" , Gordon & Breach (1974)
How to Cite This Entry:
Kerr metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kerr_metric&oldid=47493
This article was adapted from an original article by A.Ya. Burinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article