The spacetime metric is, in Boyer-Lindquist coordinates,
![{\displaystyle ds^{2}={\frac {\Delta ^{2}}{\rho ^{2}}}(dct-a\,\sin ^{2}\theta \,d\phi )^{2}-{\frac {\sin ^{2}\theta }{\rho ^{2}}}[(r^{2}+a^{2})d\phi -a\,dct]^{2}-{\frac {\rho ^{2}}{\Delta ^{2}}}dr^{2}-\rho ^{2}d\theta ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cd4658c98cc3c2affed8aab677243d09ccfa1b5)
where





This represents the exact solution to General relativity/Einstein equations for the stress-energy tensor for an electromagnetic field from a charged rotating black hole.
Defining three more functions of the coordinates



The solution can now be written

is an invariant line element, a measure of spacetime displacement between neighboring events. The displacement four vector between those events is
, and being a four-vector, would yield an invariant scalar for the inner product of it with itself using the metric tensor
as a spacetime inner product operator as
. We call that invariant scalar
.

So though technically it is the set of elements
that is the metric tensor, since its elements can be directly read off of this line element as the coefficients of the coordinate differentials, in jargon
is often referred to as just "the metric".
In the case that the charge
is zero it becomes an exact vacuum solution to Einstein's field equations and is called just "the Kerr solution".
The solution

may also be written as

where

Lets say something neutral is equatorially orbiting in this spacetime with an angular velocity of
, then in using the solution in describing its path through spacetime, or world line, the
term vanishes and it is said to be "locally nonrotating". If it emits according to its local free fall frame a frequency
, then the frequency received by a remote observer
will be red shifted by

There are three important mathematical surfaces for this line element, the static limit and the inner and outer event horizons.
The static limit is the outermost place something can be outside the outer horizon with a zero angular velocity. It is

The event horizons are coordinate singularities in the metric where
.
The outer event horizon is at

and the inner horizon is at

An external observer can never see an event at which something crosses into the outer horizon. A remote observer reckoning with these coordinates will reckon that it takes an infinite time for something infalling to reach the outer horizon even though it takes a finite proper time till the event according to what fell in.
The exact equations of equatorial geodesic motion for a neutral test mass in a charged and rotating black hole's spacetime are




where
is the conserved energy parameter, the energy per
of the test mass and
is the conserved angular momentum per mass
for the test mass.
The exact equations of polar geodesic motion for a neutral test mass in a charged and rotating black hole's spacetime are


where
is the conserved energy parameter, the energy per
of the test mass.
Above we see a Penrose diagram representing a coordinate extension (1) for a charged or rotating black hole.
The same way as mapping Schwarzschild coordinates onto Kruskal-Szekeres coordinate reveals two separate external regions for the Schwarzschild black hole, such a mapping done for a charged or rotating hole reveals an even more multiply connected region for charged and rotating black holes.
Lets say region I represents our external region outside a charged black hole. In the same way that the other external region is inaccesible as the wormhole connection is not transversible, external region II is also not accessible from region I. The difference is that there are other external regions VII and VIII which are ideed accesible from region I by transversible paths at least one way. One should expect this as the radial movement case of geodesic motion for a neutral test particle written above leads back out of the hole without intersecting the physical singularity at
.
(1)Black Holes-Parts 4&5 pp 26-42