Not having studied General Relativity, I have sometimes been puzzled by references to the behaviour for "classic" black holes — as they are popularly portrayed — as being true for black holes which are not rotating and which have no charge. I don't understand the role rotation does play, but at least because it has to do with the motion of massive bodies, I can understand why it could play an important role. But I have no such intuition for electrical charge.
How does charge change the behaviour of a black hole, aside from the obvious role of electrostatic force? (It will preferentially attract particles of the opposite charge, obviously.) But it would seem that charge plays a more intriguing role than just this.
The current status of the Wikipedia page on black holes claims that you can theoretically avoid the singularity of a charged black hole. It also describes there that there is a theoretical upper bound on the charge/mass ratio of a black hole: that any would-be black hole exceeding it (which is generally thought to be impossible — see this related question on trying to force saturation of the charge/mass ratio or black holes) would lack an event horizon (and therefore presumably not be a black hole). Why should that be? Furthermore: from this other related question on repulsion of pairs of charged black holes (and from Willie Wong's comment, below), ir seems that the size of the event horizon may change depending on how close it is to being extremal! Why would the event horizon of a highly charged black hole be different than the event horizon of a neutral black hole, of similar mass?
Is there a clear reason for such an interplay is there between electrodynamics (aside from local Lorentz invariance) and general relativity that bring these things about?
Best Answer
This is an area where there is a lot of confusion, and a lot of opinion cannot be supported by sound mathematical theory.
Not-analytic continuation
The theory of black holes requires continuation beyond regions which are accessible. These continuations are often called "analytic continuations", but this is ridiculous, because the fields do not have to be analytic in order to do it. Hyperbolic equations don't have general analytic solutions, and generally covariant equations can't have solutions which are analytic in non-analytic coordinate systems, which are in no way bad for GR.
The term I would use is "Minkowski horizon equivalence principle continuation", because it is continuation past a horizon by assuming that the horizon looks like Minkowski space locally, that is, by assuming the equivalence principle includes horizons too. When you reach a horizon which is locally like a Rindler horizon, like special relativity in accelerating coordinates, so that the curvature does not blow up, then you can continue a little ways past the horizon by extending the solution to be locally flat. In this way, you can continue into the interior of a black hole knowing the outside solution.
A Rindler horizon is vertical sheet in time for the accelerated observer. There is another kind of horizon which is locally a horizontal sheet. The Rindler horizon is formed by the spatial light cone going to the right in a 2d space-time diagram where time is vertical. If you tip this cone by 90 degrees clockwise so that it goes into the past, this is called a Cauchy horizon. A Cauchy horizon is a limit to predictability, because it is where new light rays come from infinity. This Cauchy horizon in question is the limit of the future development of a past pointing half-hyperbola which is just the 90 degree rotated trajectory of an accelearated observer.
Cauchy horizons are also locally flat, so one may continue past them by assuming the equivalence principle, just as for Rindler horizons. The issue is that when one does this, there is always new information coming in from someplace that you haven't seen before, so it is difficult to predict what happens, at least in classical GR.
But the full continuation is not in principle any more difficult than the continuation past the event horizon. The equivalence principle should hold at all horizons. But the results are strange, and require interpretation
Charged black hole solutions
I will restrict myself to 4 dimensions here, and to classical General Relativity. In string theory, there is a dilaton and the charges are d-form fields, so the "charged" black holes are extended objects in higher dimensions, sometimes with very different interior structure.
The charged black hole is nice because it is still spherically symmetric, and yet looks like the generic case of an arbitrary rotating black hole. The reason charge and rotation look alike is Kaluza Klein theory--- a charged black hole can be thought of as moving in a small circular fifth dimension. This is similar to rotation.
But the solution in GR is just as simple as the Schwartschild solution: for general mass M and charge Q:
$ ds^2 = - f(r)dt^2 + {dr^2\over f(r)} + r^2 dS_2^2$
Where
$f(r)= 1-{2m\over R} + {Q^2\over R^2}$
The important thing is that there are now two radii where f is zero, and these are the two horizons. The outer horizon is continuously deformable to the Schwartschild horizon, and is an Event horizon. The inner horizon is a Cauchy horizon. As M=Q, the black hole is extremal, because the two zeros of f(r) have the same value of r, so you get a double root. Although the two horizons are collided in the r coordinates, they are not at the same point! The radius r breaks down, because the distance between the horizons approaches a finite limit at extremality.
For $M<Q$ there is no solution to GR without a naked singularity. Such solutions are absurd, they cannot be reached by continuous deformation of the uncharged black hole, and are forbidden by the Censorship conjecture.
The black hole singularity is at r=0, past both horizons, and trajectories in the interior of the Cauchy horizons are repelled by it. So it is impossible for a massive object to hit the singularity. The Cauchy horizon itself is an outgrowth of the singularity, it is the first light rays from the singularitly
The structure: outer event horizon- inner Cauchy horizon - timelike singularity is generic. It is the situation for rotating/charged/rotating-charged black holes. But it is considered unstable. I will discuss why in the next section, and why I think these arguments are bogus.
Bad arguments that the Cauchy Horizon is the singularity
There are, in the literature, arguments that the Cauchy horizon becomes the singularity in realistic solutions. These arguments are hand-wavy, and based on the intuition that everything should be mushed somewhere. In my opinion they are completely wrong.
The arguments are based on the following properties:
The second point is very important, because it means that photons which come to you from outside are infinitely blue-shifted at the Cauchy horizon, leading people to expect a wall of hard-radiation to appear at this point under generic perturbations, which will make a singularity.
I don't buy these arguments, because the black hole is not eternal. It is not clear to me that a decaying black hole has the same Cauchy horizon structure as a regular black hole. It is also not clear to me why the singularity can't send out mollifying fields to make the Cauchy horizon non-singular rather than singular.
But most damningly, we have models of near extremal black holes in d-branes, slightly off from extremality, and these are shiny and thermodynamically nearly time-reversible. There are arguments by Gubser that d-branes will absorb other d-branes irreversibly by going off-diagonal in the non-commutative description, that is, by producing strings between the branes, but I don't buy these arguments at all.
I think that one should take the story in rotating/charged black holes at face value.
Traversable Cauchy horizon implies you can leave a black hole
If you do take the story at face value, objects going into a black hole will come out pretty much unchanged, except for possibly a little singeing when traversing the hard-radiation at the Cauchy horizon. These objects go in, turn around past the Cauchy horizon, and go out.
There are many problems with this picture. The continuation of the solution for a particle which does a full circuit goes past an event horizon, as many cauchy horizons as it wants to (by choosing whether to leave going "up in t" or "down in t" (remember that t is a spatial coordinate inside the black hole) and then out the event horizon going the other way, the way only Hawking radiation can be said to come.
This picture is crazy, because the continuation goes on producing more sheets, just like analytic continuation (but it's not analytic continuation). Each traversing path puts you in another universe, with another direction of time. The result is total nonsense.
This is why people are adament that one has to throw away interior charged solutions. These would naively lead to information loss, other universes, all that nonsense. In my opinion, the correct solution is to do some cutting and pasting and identify the outgoing and incoming paths. I tried a little, but I never found the right pasting.
If you know the pasting, you can predict what will come out of a charged/rotating black hole, mostly, up to a little randomization which becomes more random as the black hole becomes neutral/non-rotating. This type of idea means that black holes are not black at all, not just slightly thermally non-black, but shiny-mirror non-black.
This idea has exactly one supporter, which is me. I give a reasonable probability to this possibility because the counterarguments about a singular Cauchy horizon are all weak. Perhaps I should take solace in the observation that that mysterious anonymous author of the Wikipedia article also harbors suspicions, but I somehow don't think his opinion is independent.
Heuristic Justification for Pasting
The reason I think pasting is necessary is because of the following classical paradox, which I haven't found in the literature. If you throw a mass into a maximally extended charged black hole, and let it go past the horizons and come out, it comes out in another patch. But the black hole mass went up in your patch after absorption, and the mass of the black hole in the other patch went down after emission. But this means that they no longer fit together as continuations of each other.
So when the object comes out of the other sheet black hole, by consistency, it seems that something must come out of your sheet black hole. But the problem is that coming out can happen in either time direction, depending on how the infalling stuff steers itself, whether it decides to come out going towards positive or towards negative t, and so it means that objects can come out time-reflected, and the only way to make sense of that is to do a full identification with CPT gluing, so matter thrown in can come out as reflected motion-reversed antimatter. This is doubly crazy. So I am not sure if it is true, but I think it is. This might explain some astronomical puzzles of anomalous antimatter production because some astrophysical blackholes are measured to rotate at 99% of extremality.