Kerr-Newman metric

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The Kerr-Newman metric is a solution of Einstein's general relativity field equation that describes the spacetime geometry in the exterior region around a charged ( $Q\neq0$ ), rotating ( $J\neq0$ ) black hole of mass m. Like the Kerr metric, the interior solution exists mathematically and satisfies Einstein's field equations, but is probably not representative of the actual metric of a physical black hole due to stability issues. The Kerr-Newman metric is:

$ds^{2}=-\frac{\Delta}{\rho^{2}}\left(dt-a\sin^{2}\theta d\phi\right)^{2}+\frac{\sin^{2}\theta}{\rho^{2}}\left[\left(r^{2}+a^{2}\right)d\phi-{a}dt\right]^{2} +\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}$

$\Delta\ \stackrel{\mathrm{def}}{=}\ r^{2}-2Mr+a^{2}+Q^{2}$

$\rho^{2}\ \stackrel{\mathrm{def}}{=}\ r^{2}+a^{2}\cos^{2}\theta$

$a\ \stackrel{\mathrm{def}}{=}\ \frac{J}{M}$

where

M is the mass of the black hole

J is the angular momentum of the black hole

Q is the charge of the black hole

and all quantities are in geometrized units.

The Kerr-Newman metric reduces to the Schwarzschild metric in the uncharged and non-rotating case $Q = a = 0$ , the Reissner-Nordstrom metric in the non-rotating case $a = 0$ and the Kerr metric in the uncharged case $Q = 0$ . The case $M = Q = 0$ reduces to empty Minkowski space but in a usual spheroidal coordinate system.

As for the Kerr metric, the Kerr-Newman metric defines a black hole only when

$a^2 + Q^2 \leq M^2.$

Newman's result represents the most general stationary, axisymmetric asymptotically flat solution of Einstein's equations in the presence of an electromagnetic field in four dimensions. Since the matter content of the solution reduces to an electromagnetic field, it is referred as an electrovac solution of Einstein's equations. Although it represents a generalization of the Kerr metric, it is not considered as very important for astrophysical purposes since one does not expect that realistic black holes have an important electric charge.

The Kerr-Newman solution is named after Roy Kerr, discoverer of the uncharged rotating solution named after him (see Kerr metric) and Ezra T. Newman, co-discoverer of the charged solution in 1965.