When the black hole is electrically charged, the Schwarzschild solution is no longer valid.
The photon sphere is always present, but it's not shown on this diagram.
The most important consequence of this fact is the following : the exchange between space and time, which arises when you cross the horizon, now appears twice : in the sphere included in the inner horizon (sometimes called the Cauchy horizon), space and time have gone back to their usual roles. So, it's possible to avoid the singularity, which is said to be a temporal one.
If the charge of the black hole is high enough, the two horizons
disappear : the singularity is naked. Many physicists think that such a situation
can't arise- the universe applies a self-censorship.
We'll talk further about this fact, in the case of a rotating black hole.
A charged black hole is studied like a model, it may not actually exist. The star, which has generated it, is unlikely to be electrically charged.
This is the most realistic model, when you remember that the initial
star was rotating.
Its name comes from the New-Zealand mathematician Roy Kerr, who was the first, in 1963, to succeed in solving the full equations of General Relativity around a rotating massive body.
An oblic term appears in
which is responsible for the frame dragging effect (Lense-Thirring effect).
If a = 0 (no rotation), we find the Schwarzschild metric.
So, a strange effect arises from the resolution of these equations
: in the neighborhood of such a body, the space-time itself, curved
by the mass, is dragged into a rotating movement.
Of course, near the Earth, or even near the Sun, this effect is negligible, but near a black hole, it's not the same.
The outer sphere is counterrotating : photons are moving against
the rotation of the BH at its equator.
The faster the black hole spins, the further apart are the two spheres.
Between the two spheres, there is a sea of photons, orbiting around the BH.
The area between the static limit and the outer horizon is called the ergosphere.
At the poles of the BH, the static limit joins the outer horizon.
The static limit is a boundary. Inside this border, nothing can stay stationary,
even if it moves at the speed of light.
This limit is the result of the dragging of space-time by the rotation of the BH itself.
In a Schwarzschild black hole, the horizon represents the static limit. Once you are inside it, you can only move towards the singularity.
You can notice the dependence of the size of the static limit on the
angle θ with the equatorial plane.
(these formulae are written with G=c=1 according to the usual notation in order to lighten the équations).
The path, in dark blue, of an observer coming from the past in universe #1 can cross the outer horizon, then the inner one. By avoiding the singularity, he can cross horizons again, and reapper in another universe.
Possibly, he can (light blue path) cross the singularity and travel to the "negative universe".
By continuing on this journey, it seems possible to go from one universe
to another. So, the black hole has created an infinite number of universes.
Some physicists think that these other universes are just another area of our one universe (of space and time), which can happen on the assumption that the universe is turned in on itself.
In the area between the two horizons, light will be endlessly blue-shifted,
in the same way that outside the outer horizon, it was endlessly red-shifted
for the observer : you can consider the gravitational
redshift near the horizon to be a transfer of the energy of the photons
into a gravitational energy. On the other side of horizon, the opposite
transfer takes place.
The observer is now "bathed" in a sea of gamma rays
All this is very theoretical. In fact, at the slightest disturbance on the Cauchy horizon it becomes singular (in the mathematical meaning). New physics will have to be invented in order to investigate this situation.
We've seen that the faster the BH is rotating, the nearer are the
two horizons. If the rotation rate is high enough, the two horizons don't exist,
and the singularity is "naked". The cosmic censorship conjecture,
expressed by the physicist Roger Penrose, could apply in such a case.
Beyond the event horizon, the singularity is isolated of our universe.
If it's naked, this area, which breaks the usual physical rules, is free to
interact with the entire universe.
Specifically, an observer, orbiting around it, could travel back in time, and transgress the causality principle.
All this is very theoretical, because the Kerr solution is very unstable : it corresponds to a black hole alone in an absolute emptyness. Every addition of matter, even the simple approach af an observer, is enough to destabilize the black hole, and this travel simply becomes unrealistic.
If we want to examine further the inside of a black hole, we must
use quantum mechanics. It's the
only way to investigate the behaviour of the singularity.
Quantum mechanics, with the uncertainty principle, prevents the singularity having a null size, and therefore producing an infinite curvature of space-time.
The ideal scenario would be, of course, to couple together quantum
mechanics and general relativity, i.e. to obtain a quantum theory of gravity.
That's the aim of superstrings theories, for instance, but they are far away from any practical result...
We could wonder why all these studies are undertaken about black
holes, which are only marginal phenomena ?
The only answer is simply because they may be a key to understand the actual nature of our universe, and beyond it of space and time.