|Soft hair. Redshifted.|
Last year August, Stephen Hawking announced he had been working with Malcom Perry and Andrew Strominger on a solution to the black hole information loss problem, and they were closing in on a solution. But little was explained other than that this solution rests on a symmetry group by name of supertranslations.Yesterday then, Hawking, Perry, and Strominger, had a new paper on the arxiv that fills in a little more detail
- Soft Hair on Black Holes
Stephen W. Hawking, Malcolm J. Perry, Andrew Strominger
First of all, the paper seems only a first step in a longer argument. Several relevant questions are not addressed and I assume further work will follow. As the authors write: “Details will appear elsewhere.”
The present paper does not study information retrieval in general. It instead focuses on a particular type of information, the one contained in electrically charged particles. The benefit in doing this is that the quantum theory of electric fields is well understood.
Importantly, they are looking at black holes in asymptotically flat (Minkowski) space, not in asymptotic Anti-de-Sitter (AdS) space. This is relevant because string theorists believe that the black hole information loss problem doesn’t exist in asymptotic AdS space. They don’t know however how to extend this argument to asymptotically flat space or space with a positive cosmological constant. To best present knowledge we don’t live in AdS space, so understanding the case with a positive cosmological constant is necessary to describe what happens in the universe we actually inhabit.
In the usual treatment, a black hole counts only the net electric charge of particles as they fall in. The total charge is one of the three classical black hole “hairs,” next to mass and angular momentum. But all other details about the charges (eg in which chunks they came in) is lost: there is no way to store anything in or on an object that has no features, has no “hairs”.
In the new paper the authors argue that the entire information about the infalling charges is stored on the horizon in form of 'soft photons', that are photons of zero energy. These photons are the “hair” which previously was believed to be absent.
Since these photons can carry information but have zero energy, the authors conclude that the vacuum is degenerate. A 'degenerate' state is one on which several distinct quantum states share the same energy. This means there are different vacuum states which can surround the black hole and so the vacuum can hold and release information.
It is normally assumed that the vacuum state is unique. If it is not, this allows one to have information in the outgoing radiation (which is the ingoing vacuum). A vacuum degeneracy is thus a loophole in the argument originally lead by Hawking according to which information must get lost.
What the ‘soft photons’ are isn't further explained in the paper; they are simply identified with the action of certain operators and supposedly Goldstone bosons of a spontaneously broken symmetry. Or rather of an infinite amount of symmetries that, basically, belong to the conserved charges of something akin multipole moments. It sounds plausible, but the interpretation eludes me. I haven’t yet read the relevant references.
I think the argument goes basically like this: We can expand the electric field in form of all these (infinitely many) higher moments and show that each of them is associated with a conserved charge. Since the charge is conserved, the black hole can’t destroy it. Consequently, it must be maintained somehow. In the presence of a horizon, future infinity is not a Cauchy surface, so we add the horizon as boundary. And on this additional boundary we put the information that we know can’t get lost, which is what the soft photons are good for.
The new paper adds to Hawking’s previous short note by providing an argument for why the amount of information that can be stored this way by the black hole is not infinite, but instead bounded by the Bekenstein-Hawking entropy (ie proportional to the surface area). This is an important step to assure this idea is compatible with everything else we know about black holes. Their argument however is operational and not conceptual. It is based on saying, not that the excess degrees of freedom don't exist, but that they cannot be used by infalling matter to store information. Note that, if this argument is correct, the Bekenstein-Hawking entropy does not count the microstates of the black hole, it instead sets an upper limit to the possible number of microstates.
The authors don’t explain just how the information becomes physically encoded in the outgoing radiation, aside from writing down an operator. Neither, for that matter, do they demonstrate that by this method actually all of the information of the initial can be stored and released. Focusing on photons of course they can't do this anyway. But they don’t have an argument how it can be extended to all degrees of freedom. So, needless to say, I have to remain skeptical that they can live up to the promise.
In particular, I still don’t see that the conserved charges they are referring to actually encode all the information that’s in the field configuration. For all I can tell they only encode the information in the angular directions, not the information in the radial direction. If I were to throw in two concentric shells of matter, I don’t see how the asymptotic expansion could possibly capture the difference between two shells and one shell, as long as the total charge (or mass) is identical. The only way I see to get around this issue is to just postulate that the boundary at infinity does indeed contain all the information. And that in return we only know to work in AdS space. (At least it’s believed to work in this case.)
Also, the argument for why the charges on the horizon are bounded and the limit reproduces the Bekenstein-Hawking entropy irks me. I would have expected the argument for the bound to rely on taking into account that not all configurations that one can encode in the infinite distance will actually go on to form black holes.
Having said that, I think it’s correct that a degeneracy of the vacuum state would solve the black hole information loss problem. It’s such an obvious solution that you have to wonder why nobody thought of this before, except that I thought of it before. In a note from 2012, I showed that a vacuum degeneracy is the conclusion one is forced to draw from the firewall problem. And in a follow-up paper I demonstrated explicitly how this solves the problem. I didn’t have a mechanism though to transfer the information into the outgoing radiation. So now I’m tempted to look at this, despite my best intentions to not touch the topic again...
In summary, I am not at all convinced that the new idea proposed by Hawking, Perry, and Strominger solves the information loss problem. But it seems an interesting avenue that is worth further exploration. And I am sure we will see further exploration...