Stephen Hawking’s Final Paper: How to Escape from a Black Hole

That must be the understatement of the week. Love it! I guess observations of failed supernovae and possible direct-collapse black holes could shed some light (hah!) on the matter.

Unfortunately, you are. The information that is lost is what went into the black hole in the first place.

It's not strictly a quantum problem: if black holes have no hair, then we cannot tell by looking at a spherically symmetric non-rotating black hole if it was formed by one spherical shell of infalling matter of mass M, or two concentric spherical shells of infalling matter of mass M/2, or three of M/3, etc. When we add electromagnetism to the picture, we get Hawking radiation inversely proportional to the black hole mass; but that mass does not encode the number of shells or their composition, just their total mass.

When we add in quantum electrodynamics, we find that Hawking radiation has a thermal spectrum (so, cold photons for a stellar-mass black hole, but when the black hole is very small you'll get electrons and positrons too; and potentially the whole zoo of particles if we use the full standard-model as the quantum field theory). But we could start with a black hole formed by squashing together neutral composites (positronium, atoms) and with some probability get out nothing but photons: no massive particles at all. With some smaller probability we get mostly photons but also electrons and positrons. The main problem is that we are stuck talking probabilistically about the spectrum Hawking quanta even if we know every single detail of what we threw into the black hole; there is no unitary evolution from known-in-every-detail state to known-in-every-detail state. The "every detail" part is the information that is lost.

There are a variety of ways one can try to deal with the conversion of "we know every detail" (a pure state, quantum mechanically) to "we can only talk probabilistically" (a mixed state, quantum mechanically), and some are listed in the wikipedia page you link to. Hawking's final paper is yet another approach, and throws away the idea that black holes have no hair; that is, a black hole cannot be described with a small number of parameters (dominated by mass and angular momentum) but rather develop an enormous number of parameters encoded as perturbations of the vacuum. Those perturbations in turn influence the spectrum of the Hawking quanta in such a way that it is fully predictable -- even though it looks like a thermal bath, the vacuum perturbations ("soft hairs") fully determine it. It an idea is worth further investigation, but is not much more compelling than several alternatives.

One problem is that when we take an exact analytical black hole solution to the Einstein Field Equations of general relativity, we have "no hair" as a mathematical theorem. If we perturb around such a solution we generate observables that closely match what we see of candidate astrophysical black holes in the sky. Hawking wants to treat astrophysical black holes as even more different than the theoretical models, and while that's not a crazy idea, it's also not very parsimonious as many many many more perturbations ("hairs") are necessary than the minimum required to match the observed systems, and it's not clear that a "no hair" black hole must be measurably different from a "soft hair" black hole.

(More detail here https://news.ycombinator.com/item?id=18327614 )

You're welcome.

> evaporating black hole must be leaking its information to somewhere

The information about the contents of the BH during its formation and growth is in the region of strong gravity. Classically, it's squashed into the gravitational singularity; fully classically the singularity is always hidden behind an event horizon, so it does no harm to predicting events outside the horizon.

However, we now add a quantum field theory to the picture.

The origin of Hawking radiation is the acceleration between observers before the formation of the strong gravity and the observers after that; the accelerated (later) observers see particles where the non-accelerated (early) observers see none. The particles appear in the dynamical spacetime around (but outside) the horizon. The reason they are there is (rougly) that the creation and annihiliation operators that line up in "unstretched" vacuum separate in "stretched" vacuum, and annihilation operators miss the created particles (that is, the annihilation happens at the right spatial coordinate, but too early or too late: the created particle is elsewhere). The analogy with Unruh radiation, which appears for accelerated observers in flat spacetime but not for unaccelerated observers in the same spacetime is not accidental. In the Unruh case, the acceleration mechanism (say, a rocket engine) is the reason the accelerated observer sees the extra particles. In the Hawking case, the acceleration mechanism for later observers is the dynamically collapsing spacetime.

If nothing exits the horizon of a black hole (at least until final evaporation; and for that we are stuck with not knowing enough about the behaviour of quantum fields in strong gravity) then the only parameters available at any instant in the (QED-filled) dynamical spacetime that is the origin of Hawking radiation are mass (1 component), charge (1 component), angular momentum (3 components), linear momentum (3 components), and spatial position (3 components). The last six components fall away for some families of observers with a suitable choice of spatial coordinates. ("Instant" in this context is a coordinate time defining a spacelike hypersurface, and one has lots of freedom there). You get a handful of extra components (individual "charges") as you go from QED to the standard model.

There have been attempts to break this picture by inter alia having things never enter the horizon in the first place, by implanting extra information in the spacetime around the black hole ("hair"), and by locking up all the infalling matter into a crystal that preserves details of the matter's microscopic states either forever or until evaporation is almost entirely complete. It is extremely hard to do this without introducing unlikely observables.

> Conservation of Energy

... is not a global symmetry of a dynamically collapsing spacetime. You only get conservation of energy locally within a suitably small region of spacetime (which can be quite large far from the collapse, assuming asymptotic flatness).

> time-reverible

Locally. This is most sharply obvious in strong curvature.

> CPT symmetric

This is a problem with unitary time evolution of any quantum system in this setting; CPT doesn't enter into it. There is neither antimatter nor chirality in the model non-interacting scalar field that exposes the information loss problem for a collapsing black hole. ("Negative energy" is only a trick used when one wants to use a static background instead of a dynamical one; it does not interact at all with its pair-partner or other "negative energy" quanta; there is no local symmetry, it is the global symmetries of the Schwarzschild solution that are being preserved through the trick. You entangle the real Hawking quanta with false quanta instead of entangling the real Hawking quanta with the spacetime (which would change the metric, which is exactly what one is trying to avoid in some studies)).

Indeed, the problem is mostly centred on "time" in the first sentence of the previous paragraph. There is no unique slicing of a general curved 4-spacetime into 3-spaces, and if one does it wrong, one gets problems (see ref to Giddings 2006 below). This is in some ways an argument that black hole information loss is mainly about the https://en.wikipedia.org/wiki/Problem_of_time .

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See also

https://arxiv.org/abs/1511.08221 (Giddings 2015)

http://inspirehep.net/record/775859 (Unruh 2009)

http://inspirehep.net/record/775859 (Unruh 2007)

https://arxiv.org/abs/hep-th/0606146 (Giddings 2006)

and refs therein (e.g. Unruh 1977).

or with a concise summary of the work above and related work

http://backreaction.blogspot.com/2015/12/hawking-radiation-i...