- Conservative solutions to the black hole information problem
By Sabine Hossenfelder and Lee Smolin
which provides a classification of solution attempts to the black hole information loss problem. As a warm-up, I recommend you read my post on the Black Hole Information Loss Paradox. You will notice that in this earlier post the basic argument of the paper is already outlined. The paper just makes the definitions more precise, and discusses the options one has to solve the problem based on how radical departures from semi-classical gravity they require. Not to mention that the paper has a lot of nice figures. We have made some effort to make the paper understandable for a broad audience, so don't be shy and download the full thing.
The Core of the Problem: The Singularity
The essence of the argument is the following: A singularity is something you don't want to cross your path. Why? Because infinities are dangerous. They crunch and destroy things, they literally set an end to existence, and in doing so they are indifferent as to what exactly crossed their way. A singularity is always singular. Infinity is always infinity. As such, crossing a singularity is an irreversible process. The problem is that once an initial state ended up being singular, you can't figure out what it looked like originally.
The problem with black hole information is that evolution is not unitary if you believe that the initial state of the black hole gets converted mostly into thermal radiation to excellent precision. Non-unitarity is generally considered an unappealing property because it is in conflict with quantum mechanics and can cause all kinds of nasty side-effects you don't want. But an evolution that is not reversible cannot be unitary. Reversibility is not a sufficient, but a necessary condition for unitarity. However, since irreversibility is a characteristic of the presence of singularities, first thing you want to allow for a unitary evolution is to remove the singularity. Classically, this singularity is unavoidable. But we know that close to the singularity the curvature gets very strong (into the Planckian regime) and classical General Relativity (GR) is no longer valid. It should be replaced by a more fundamental theory that can be expected to remove the singularity, though the details are not well understood today.
The paper offers a generalization of the classical singularities in GR that can be used for spacetimes that might have quantum gravitational regions. Throughout the paper we have tried not to make any specific assumptions about the unknown fundamental theory. The problem with the classical definition of geodesic completeness is that the notion of a geodesic, which relies on the presence of a metric, might not make sense any longer in the presence of strong quantum gravitational effects. The definition we are suggesting is motivated by the classical definition, and is then what I outlined above: a space-time is non-singular if evolution is time-reversible. It then follows trivially that a singular space-time generically suffers from information loss. Thus, a black hole space-time without information loss can not be singular in the so defined sense. If you want to understand what happens to the black hole information, first thing you should do is thus to get rid of the singularity.
It is honestly a mystery to me why some people are so obsessed with the black hole horizon, believing that the horizon is the problem. The horizon is not where information gets destroyed. It is merely some surface where the information becomes irretrievable from the outside. Not to mention that the horizon can be at arbitrarily small background curvature. One thus shouldn't expect any quantum gravitational effects to be relevant at the horizon, and no reason to seek a solution there.
Radical and Conservative Solutions
Removing the singularity removes an obstacle to unitary evolution, but it doesn't explain how information survives. In the paper we discuss the possibilities one has if one just accepts that quantum gravitational effects are negligible until the very endstate of the evaporation. These solutions we have dubbed “conservative” . Everything else that requires non-locality on horizon scales or quantum gravitational effects in the weak curvature regime and so on, we have called “radical”.
The conservative solutions can be classified into three cases. In all of them it is assumed the singularity is removed by quantum gravitational effects:
- The information is released in the final Planck phase, in which case there never is a real event horizon (in the paper, that's option 3).
- The information survives in a baby universe that disconnects from our universe ( option 4A).
- The information survives in a permanent, massive remnant (option 4B).
Most importantly, conservative solutions imply that the endstate of black hole evaporation - when the black hole has about Planck mass and Planck size - carries a (potentially arbitrarily large) amount of information. The reason is simply that, if one accepts that the semi-classical approximation holds, Hawking radiation does not carry any information (except its temperature). Thus, the information has to remain inside. We thus have an endstate that must be able to store a large amount of information, even though it has a small surface area. This speaks in particular for the surface-interpretation of the black hole entropy. Objects with these properties are known to be possible in General Relativity, we have discussed such “bags of gold” and “monsters” in a recent post.
The three above mentioned possible cases have been discussed for some while in the literature until some time in the mid 90s. There are some objections to all of them that we address in the paper. All in all, though valid objections, they are not terribly convincing. It is thus puzzling to some extend why there hasn't been more effort invested in what seem to be the most straightforward outcomes of black hole evaporation. Unfortunately, I have had many times the impression these conservative solutions were abandoned prematurely for the sake of creating more fanciful radical solutions, for not say, absurd speculations.
A note on the definition of singularities we are using: If one had a fundamental theory to describe spacetime in the regions with strong quantum graviational effects, one could consider other notions of singular spacetimes, for example by using divergence of operators describing the background curvature or likewise. Then there arises the question how this definition would coincide with the one we have been using. One could imagine cases where they do not. Eg, the information of fields propagating in the background might not be sensitive to a curvature singularity, or the singularity itself could encode information.
The sane thing to do is to stick with conservative options until we are sure it's a no-go. That requires in particular understanding the properties of Planck-sized quantum graviational objects with high entropy.