The Kerr metric is a stationary axisymmetric metric tensor on a Lorentzian manifold with vanishing
Ricci tensor in four dimensions,
parametrized by a mass 
and rotation parameter 
.
In Boyer-Lindquist coordinates 
, it has the line
element
![ds^2=-Delta/Sigma(dt-asin^2thetadphi)^2+Sigma/Deltadr^2+Sigmadtheta^2+(sin^2theta)/Sigma[(r^2+a^2)dphi-adt]^2,](/images/equations/KerrMetric/NumberedEquation1.svg) |
where 
and 
. The signature is 
and the angular momentum is 
(Visser 2008). For 
and 
, its horizon radii in Boyer-Lindquist coordinates are
 |
The rotating subextremal range is 
, distinct from the nonrotating and extremal limits.
A complete two-ended bridge through the bifurcation sphere has initial manifold diffeomorphic
to 
. The smooth inner horizon of
the exact solution motivates the extension-regularity
question in strong cosmic censorship.
