Kerr Metric - Mathematical Form

Mathematical Form

The Kerr metric describes the geometry of spacetime in the vicinity of a mass M rotating with angular momentum J

$c^{2} d\tau^{2} = \left( 1 - \frac{r_{s} r}{\rho^{2}} \right) c^{2} dt^{2} - \frac{\rho^{2}}{\Delta} dr^{2} - \rho^{2} d\theta^{2} -$

$\left( r^{2} + \alpha^{2} + \frac{r_{s} r \alpha^{2}}{\rho^{2}} \sin^{2} \theta \right) \sin^{2} \theta \ d\phi^{2} + \frac{2r_{s} r\alpha \sin^{2} \theta }{\rho^{2}} \, c \, dt \, d\phi$

where the coordinates are standard spherical coordinate system, and r_s is the Schwarzschild radius

$r_{s} = \frac{2GM}{c^{2}}$

and where the length-scales α, ρ and Δ have been introduced for brevity

$\alpha = \frac{J}{Mc}$

$\ \rho^{2} = r^{2} + \alpha^{2} \cos^{2} \theta$

$\ \Delta = r^{2} - r_{s} r + \alpha^{2}$

In the non-relativistic limit where M (or, equivalently, r_s) goes to zero, the Kerr metric becomes the orthogonal metric for the oblate spheroidal coordinates

$c^{2} d\tau^{2} = c^{2} dt^{2} - \frac{\rho^{2}}{r^{2} + \alpha^{2}} dr^{2} - \rho^{2} d\theta^{2} - \left( r^{2} + \alpha^{2} \right) \sin^{2}\theta d\phi^{2}$