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)