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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|Schwarzschild metric]].

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.P. Kerr,   "Gravitational field of a spinning mass as an example of algebraically special matrices"  ''Phys. Rev. Letters'' , '''11'''  (1963)  pp. 237–238</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.W. Misner,   K.S. Thorne,   J.A. Wheeler,   "Gravitation" , Freeman  (1973)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Rees,   R. Ruffini,   J. Wheeler,   "Black holes, gravitational waves and cosmology" , Gordon &amp; Breach  (1974)</TD></TR></table>