Can black holes bounce to white holes?

Fast track to wisdom: Sure, but who cares if they can? We want to know if they do.

Black holes are defined by the presence of an event horizon which is the boundary of a region from which nothing can escape, ever. The word black hole is also often used to mean something that looks for a long time very similar to a black hole and that traps light, not eternally but only temporarily. Such space-times are said to have an “apparent horizon.” That they are not strictly speaking black holes was origin of the recent Stephen Hawking quote according to which black holes may not exist, by which he meant they might have only an apparent horizon instead of an eternal event horizon.

A white hole is an upside-down version of a black hole; it has an event horizon that is a boundary to a region in which nothing can ever enter. Static black hole solutions, describing unrealistic black holes that have existed forever and continue to exist forever, are actually a combination of a black hole and a white hole.

The horizon itself is a global construct, it is locally entirely unremarkable and regular. You would not note crossing the horizon, but the classical black hole solution contains a singularity in the center. This singularity is usually interpreted as the breakdown of classical general relativity and is expected to be removed by the yet-to-be-found theory of quantum gravity. 

You do however not need quantum gravity to construct singularity-free black hole space-times. Hawking and Ellis’ singularity theorems prove that singularities must form from certain matter configurations, provided the matter is normal matter and cannot develop negative pressure and/or density. All you have to do to get rid of the singularity is invent some funny type of matter that refuses to be squeezed arbitrarily. This is not possible with any type of matter we know, and so just pushes around the bump under the carpet: Now rather than having to explain quantum effects of gravity you have to explain where the funny matter comes from. It is normally interpreted not as matter but as a quantum gravitational contribution to the stress-energy tensor, but either way it’s basically the physicist’s way of using a kitten photo to cover the hole in wall.

Singularity-free black hole solutions have been constructed almost for as long as the black hole solution has been known – people have always been disturbed by the singularity. Using matter other than normal ones allowed constructing both wormhole solutions as well as black holes that turn into white holes and allow an exit into a second space-time region. Now if a black hole is really a black hole with an event horizon, then the second space-time region is causally disconnected from the first. If the black hole has only an apparent horizon, then this does not have to be so, and also the white hole then is not really a white hole, it just looks like one.

The latter solution is quite popular in quantum gravity. It basically describes matter collapsing, forming an apparent horizon and a strong quantum gravity region inside but no singularity, then evaporating and returning to an almost flat space-time. There are various ways to construct these space-times. The details differ, but the corresponding causal diagrams all look basically the same.

This recent paper for example used a collapsing shell turning into an expanding shell. The title “Singularity free gravitational collapse in an effective dynamical quantum spacetime” basically says it all. Note how the resulting causal diagram (left in figure below) looks pretty much the same as the one Lee and I constructed based on general considerations in our 2009 paper (middle in figure below), which again looks pretty much the same as the one that Ashtekar and Bojowald discussed in 2005 (right in figure below), and I could go on and add a dozen more papers discussing similar causal diagrams. (Note that the shaded regions do not mean the same in each figure.)

One needs a concrete ansatz for the matter of course to be able to calculate anything. The general structure of the causal diagram is good for classification purposes, but not useful for quantitative reasoning, for example about the evaporation.

Haggard and Rovelli and recently added to this discussion with a new paper about black holes bouncing to white holes.

Black hole fireworks: quantum-gravity effects outside the horizon spark black to white hole tunneling
Hal M. Haggard, Carlo Rovelli
arXiv: 1407.0989

Ron Cowen at Nature News announced this as a new idea, and while the paper does contain new ideas, that black holes may turn into white holes is in and by itself not new. And so it follows some clarification.

Haggard and Rovelli’s paper contains two ideas that are connected by an argument, but not by a calculation, so I want to discuss them separately. Before we start it is important to note that their argument does not take into account Hawking radiation. The whole process is supposed to happen already without outgoing radiation. For this reason the situation is completely time-reversal invariant, which makes it significantly easier to construct a metric. It is also easier to arrive at a result that has nothing to do with reality.

So, the one thing that is new in the Haggard and Rovelli paper is that they construct a space-time diagram, describing a black hole turning into a white hole, both with apparent horizons, and do so by a cutting-procedure rather than altering the equation of state of the matter. As source they use a collapsing shell that is supposed to bounce. This cutting procedure is fine in principle, even though it is not often used. The problem is that you end up with a metric that exists as solution to some source, but you then have to calculate what the source has to do in order to give you the metric. This however is not done in the paper. I want to offer you a guess though as to what source would be necessary to create their metric.

The cutting that is done in the paper takes a part of the black hole metric (describing the inside of the shell) with an arm extending into the horizon region, then squeezes this arm together so that it shrinks in radial extension no longer extends into the regime below the Schwarzschild radius, which is normally behind the horizon. This squeezed part of the black hole metric is then matched to empty space, describing the inside of the shell. See image below

Figure 4 from arXiv: 1407.0989

They do not specify what happens to the shell after it has reached the end of the region that was cut, explaining one would need quantum gravity for this. The result is glued together with the time-reversed case, and so they get a metric that forms an apparent horizon and bounces at a radius where one normally would not expect quantum gravitational effects. (Working towards making more concrete the so far quite vague idea of Planck stars that we discussed here.)

The cutting and squeezing basically means that the high curvature region from inside the horizon was moved to a larger radius, and the only way this makes sense is if it happens together with the shell. So I think effectively they take the shell from a small radius and match the small radius to a large radius while keeping the density fixed (they keep the curvature). This looks to me like they blow up the total mass of the shell, but keep in mind this is my interpretation, not theirs. If that was so however, then makes sense that the horizon forms at a larger radius if the shell collapses while its mass increases. This raises the question though why the heck the mass of the shell should increase and where that energy is supposed to come from.

This brings me to the second argument in the paper, which is supposed to explain why it is plausible to expect this kind of behavior. Let me first point out that it is a bold claim that quantum gravity effects kick in outside the horizon of a (large) black hole. Standard lore has it that quantum gravity only leads to large corrections to the classical metric if the curvature is large (in the Planckian regime). This happens always after horizon crossing (as long as the mass of the black hole is larger than the Planck mass). But once the horizon is formed, the only way to make matter bounce so that it can come out of the horizon necessitates violations of causality and/or locality (keep in mind their black hole is not evaporating!) that extend into small curvature regions. This is inherently troublesome because now one has to explain why we don’t see quantum gravity effects all over the place.

The way they argue this could happen is that small, Planck size, higher-order correction to the metric can build up over time. In this case it is not solely the curvature that is relevant for an estimate of the effect, but also the duration of the buildup. So far, so good. My first problem is that I can’t see what their estimate of the long-term effects of such a small correction has to do with quantum gravity. I could read the whole estimate as being one for black hole solutions in higher-order gravity, quantum not required. If it was a quantum fluctuation I would expect the average solution to remain the classical one and the cases in which the fluctuations build up to be possible but highly improbable. In fact they seem to have something like this in mind, just that they for some reason come to the conclusion that the transition to the solution in which the initially small fluctuation builds up becomes more likely over time rather than less likely.

What one would need to do to estimate the transition probability is to work out some product of wave-functions describing the background metric close by and far away from the classical average, but nothing like this is contained in the paper. (Carlo told me though, it’s in the making.) It remains to be shown that the process of all the matter of the shell suddenly tunneling outside the horizon and expanding again is more likely to happen than the slow evaporation due to Hawking radiation which is essentially also a tunnel process (though not one of the metric, just of the matter moving in the metric background). And all this leaves aside that the state should decohere and not just happily build up quantum fluctuations for the lifetime of the universe or so.

By now I’ve probably lost most readers so let me just sum up. The space-time that Haggard and Rovelli have constructed exists as a mathematical possibility, and I do not actually doubt that the tunnel process is possible in principle, provided that they get rid of the additional energy that has appeared from somewhere (this is taken care of automatically by the time-reversal). But this alone does not tell us whether this space-time can exist as a real possibility in the sense that we do not know if this process can happen with large probability (close to one) in the time before the shell reaches the Schwarzschild radius (of the classical solution).

I have remained skeptical, despite Carlo’s infinitely patience in explaining their argument to me. But if they are right and what they claim is correct, then this would indeed solve both the black hole information loss problem and the firewall conundrum. So stay tuned...