**Of course, the black hole information puzzle has been discussed in dozens of older TRF blog entries**

A Western European physics blog has posted a "Black Hole Information Paradox Flow Diagram" that can't hide its similarity with the scheme of a female brain. Unfortunately, among lots of wrong answers to questions about black holes, it doesn't include the right answer to the question what happens with the information stored in an evaporating black hole.

So here are the questions and answers.

- When a massive object of mass \(M\gg m_\text{Planck}\) collapses (let's assume \(Q=0\) and \(\vec J=0\) although we don't have to), does a horizon form?

However, no such forces can exist for a large enough black hole. If you have a really huge conglomerate of matter, the density may be as low as the density of water and the gravitational force becomes large enough so that the collapse is inevitable. To derive this collapse, we only need to rely on the validity of Einstein's equations in each region – and they may be, locally speaking, totally unspectacular reasons whose density only matches that of water and the curvature is correspondingly low.

The inevitability of the emergence of the event horizon was proved by the Hawking-Penrose singularity theorems of the 1970s. Why did they care about the singularity if it is the event horizon that defines the black hole? Well, the event horizon is the boundary of the black hole interior. And the black hole interior is composed of all the spacetime points whose future time-like geodesics can't get to the spatial infinity. They must get elsewhere; and the "elsewhere" means the singularity because there are not too many other options. (Let's assume that one can't form or connect to new infinite spacetimes by a collapse of a star.)

So they needed to prove that one actually forms a singularity, the alternative future fate of an observer. And indeed, such singularities arise rather generically when a massive enough object collapses. So black holes have to exist. Their existence has also been independently derived from string theory although this theory is based on an independent starting point from that of general relativity. And the evidence for astrophysical black holes – those in the real world – has become overwhelming, too.

- Do black holes radiate?

The thermal radiation – whose black body temperature is equal to the gravitational acceleration at the event horizon (surface gravity) in certain natural units – was originally derived by Stephen Hawking around the mid 1970s. A simplified calculation for the Rindler space – a wedge of the flat Minkowski spacetime as seen by a uniformly accelerating observer – was later derived by William Unruh.

Hawking's and Unruh's calculations only rely on the validity of the semiclassical approximation; for large black holes, one may consistently ignore all effects and corrections that are of higher order in Planck's constant than those that are considered. The existence of the Hawking radiation may also be partly independently derived from string theory.

In the operator formalism, one needs to discuss the Bogoliubov transformation mixing creation and annihilation operators which is needed if we redefine the Hamiltonian from an inertial observer's one to an accelerated observer's one (and if we have to redefine the ground state in a corresponding way). In the path integral approach, the black hole temperature may be derived using the Gibbons-Hawking method.

- Does the radiation at \(M\gg T\) i.e. for black hole masses much heavier than the typical mass/energy of the Hawking particles (dictated by the temperature) carry information?

(Alternatively, the external observer always has the right to imagine that the infalling matter got stuck at [or right above] the event horizon and there is no interior at all.)

This is in no contradiction with the validity of the semiclassical approximation for operationally meaningful questions: the code in which the Hawking radiation stores the information is incredibly subtle and scrambled and using the fat and awkward probes that are compatible with the semiclassical approximation, we can't decode the information. The semiclassical approximation gives the right approximate answers to arbitrary quantitative questions which have the form of continuous numbers (with small errors); however, it may fail and it does fail to give the right Yes/No answers to some qualitative questions.

The main theoretical weapons that allowed us to learn that the information is preserved were new approaches to string theory, especially the AdS/CFT correspondence and Matrix theory (that I plan to write about soon). These descriptions of string theory are manifestly unitary – they evolve quantum states in a one-to-one way from the past to the future just like your textbook models of quantum mechanics – but they may also be shown to incorporate (evaporating) black holes.

Stephen Hawking has admitted that he was wrong and he surrendered a famous bet against John Preskill. If quantum gravity is treated properly, the information comes out.

- Do black holes leave remnants with a lot of information after they evaporate?

In fact, remnants – essentially point-like objects that may carry an arbitrarily high amount of information – would yield the theory (or Nature, if she suffered from the remnant illness) inconsistent. Remnants would violate the holographic entropy bounds: one can't really squeeze too much information to too small a volume. Also, there would be infinitely many types of remnants and their pair production would be infinitely frequent and it would correct many ordinary physical quantities by infinite amounts, and so on.

So yes, the information is getting out of the black hole as it Hawking-radiates. When the black hole gets very small, the higher-derivative terms in quantum gravity and various stringy/M-corrections become very important. In the very final stages of the evaporation, the evaporating black hole behaves just like an unstable elementary particle. The transition from a large black hole microstate to a particular elementary particle species is gradual; there is no qualitative difference between them.

These "unified objects" may be described as black holes (whose corrections to Einstein's equations are small) if they're much heavier than the Planck mass; and they may be described as elementary particles of different species (whose gravitational force may be neglected) if they're much lighter than the Planck mass.

The only region where the full quantum theory of gravity i.e. string/M-theory fully exposes its muscles (and the aforementioned approximations are not enough) is the regime in which the mass of the objects is comparable to the Planck mass. String/M-theory is the peacemaker that is needed for the smooth interpolation between the elementary particles propagating on a mildly curved, nearly flat background; and the general relativity that is needed for heavy black holes. Macroscopic physical phenomena in these two opposite extremes are captured by two copies of an effective quantum field theory (and in fact, the classical field theory limit is the most important part of them, especially on the black hole side); however, the interpolation required by consistency is inevitably a slightly more general theory than quantum field theory, namely string/M-theory.

- Does the Hawking radiation carry nonlocal correlations that contain some information?

It shouldn't be shocking that the black hole is able to produce this subtle entanglement. The Hawking radiation itself may be interpreted as quantum tunneling. The information tunnels out of the black hole interior, too. In other words, the causal diagram that seemingly strictly tells us that the black hole interior is separated from the exterior shouldn't be taken too seriously because the metric tensor is a fluctuating observable. Much like the alpha particle can't be guaranteed to remain inside a nucleus – that's why we occasionally observe alpha-decay – Hawking particles can't be "totally confined" within the black hole interior and may appear in random directions outside the black hole. They're still coming from the same origin and may be entangled.

- How to make the semiclassical limit nonlocal?

The semiclassical limit is what Stephen Hawking calculated in the 1970s and he correctly determined that the radiation couldn't carry the information away in the form of (nonlocal) correlations because that would violate the locality. In fact, even if he calculated the exact answer to all orders in the perturbation theory in \(\hbar\), it would still be right that the information can't get away.

The preservation of the information may only be seen if one goes beyond the perturbative expansion – beyond all orders in a Taylor expansion in Planck's constant. Because the question above appeared in the aforementioned chart, one must say that the right answer to the questions about the fate of the information in black holes isn't included in the chart, despite the plethora of wrong answers that are included.

- Do black holes have hair, after all?

Samir Mathur is behind the most convincing proposals that would yield a "Yes" answer to this question. His fuzz balls literally look like very complicated objects that totally change the character of the black hole interior – fill it with complicated fuzz that carries a huge amount of information. It should still be true from locality (validity of general relativity at long distances, for fat enough probes) that an infalling observer generically experiences empty space. This emptiness must result from some averaging over all the complicated types of fuzz.

In some simple enough contexts, Mathur and collaborators have constructed a literal "local" representation of hair – the microstates are genuine solutions of ordinary extended Einstein's equations. In more generic situations, however, the degrees of freedom needed to describe the hair probably need to be more nonlocal themselves.

These are cute constructions that didn't have to exist and many people still believe that they either don't exist or they're wrong. But whether such "visualizations" of the information carried by black holes (and by their Hawking radiation) may be found or not, the answers to the remaining questions listed in this blog entry are almost certainly irreversible at this point. What would many people love to see is some readable answer to the question "where in space and how" the information is stored on the black hole horizon or in the radiation. However, the exact description of these objects doesn't seem to have a form of a local field theory in the spacetime so the questions of the type "where do things live" may be misplaced.

On the other hand, it's conceivable that they're not misplaced. There could exist some "much more spacetime-local" description of the black hole evaporation than what you can get from the AdS/CFT correspondence or Matrix theory. It wouldn't be the first time when an "ordinary description" of a structure that was believed by many experts to inevitably transcend field theory was found: the membrane minirevolution found some explicit field-theoretical Lagrangians for theories that many people had believed to be inequivalent to Lagrangian field theories.

"Alternatively, the external observer always has the right to imagine that the infalling matter got stuck at [or right above] the event horizon and there is no interior at all."

ReplyDeleteThis is something which has been bothering me lately. This picture is fine for, say, a collapsing shell of matter, but I don't see how it applies to a more realistic case like a collapsing star. In that case, there are lots of particles which are "located well inside the horizon" from the moment it forms.

I realise that the above is stated in a very naive way, but I think it accurately conveys my point of confusion!

Dear Rhys, right, there are particles in the black hole interior from the beginning of the formation of the black hole.

ReplyDeleteThey can't use the membrane description that an external observer can use. They're not external observers. They're doomed.

The difference between the particles inside and those outside doesn't look sharp at all at the beginning - in fact, the location of the event horizon is only determined later - but it's still true that the places inside and outside have vastly different futures and only those who have the chance to be saved and escape from the gravitational hole may use the description in which the black hole is a hole where things like on the membrane on the event horizon. The event horizon is slowly growing from 0 size when the black hole starts to form and it looks artificial to consider those particles that are doomed "non-existent" from the very beginning but the later evolution will reaffirm that it is a consistent picture for the observer who stays outside.

Cheers

LM