Inside a black hole (2)



The Reissner-Nordstrøm black hole

When the black hole is electrically charged, the Schwarzschild solution is no longer valid.

Trou noir chargé
Now, we have two separate horizons. When the black hole becomes charged, the event horizon shrinks, and another one appears, near the singularity.
The more charged the black hole is, the closer the two horizons are.

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.


The Kerr black hole

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.

In this case, the metric of the space-time becomes -
with - , - are the polar coordinates, if J is the angular momentum and M the mass, a is the rotation parameter J/M.

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.


Cosmic censorship conjecture

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.

The radii of the horizons of the black hole are -, if a becomes greater than M, this formula has no meaning.

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.

A singularity which is not enclosed in an event horizon becomes the past for some points of the space-time.
Closed timelike curves are possible, this can lead to a violation of causality.

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.


References :
Finkelstein Black Hole Kerr Newman (T. Smith)
Topological Censorship (K. Schleich, D. Witt)
The nature of space and time (R. Penrose, S. Hawking)
General Relativity and Quantum Cosmology (I. Novikov)