There are a couple of things to keep in mind when discussing
this problem. First, in classical general relativity the
horizon is empty space -- there's nothing material there.
Second, for a large black hole, the curvature at the horizon
is very small. That means that, by the equivalence principle,
a freely falling (classical) observer will notice nothing
special happening as she crosses the horizon. The consistent
problem with many proposed "solutions" to the information
loss paradox is that they require drastic changes in the
physics in these empty, low curvature regions that are, at
least classically, locally indistinguishable from anywhere
else.

On 4/15/16 8:24 AM, Jos Bergervoet wrote:
On 4/3/2016 11:32 PM, Steven Carlip wrote:
On 4/1/16 4:47 PM, Jos Bergervoet wrote:

For the black hole information paradox I would expect
two seemingly conflicting results (as is necessary for
a paradox by definition,) e.g. like:

1) We know that information is lost because [...]

We believe that matter in a pure quantum state can collapse
to form a black hole.

This is not quite clear if we cannot provide the solution
of the GR equations that show the formation (so indeed we
now can only say that we "believe" it).

Well, we can certainly write a classical solution in which
matter collapses to form a black hole. We can also make
this "semiclassical" -- that is, we can write down a quantum
state describing, say, a thin shell of radiation and use
the expectation value of its stress-energy tensor as a source.

Now, it *could* be that if we could somehow carry out this
same analysis in a completely quantum mechanical setting, the
results might be completely different. There are proposals
along this line, such as the "fuzzball" proposal of Mathur
et al. But if that's the answer, you have to explain why
the semiclassical approximation breaks down so drastically
in a region in which space is empty and curvatures can be
arbitrarily small -- that is, why the correspondence principle
of quantum mechanics fails in a place where there's no obvious
reason it should fail.

[...]

Such a black hole will then evaporate
by Hawking radiation, which is thermal.

This again is not so clear, as the Polchinsky-paper
http://arxiv.org/abs/1207.3123
states immediately in the abstract: we want it to be a pure
state! You are now claiming something that is not granted.

You can *calculate* Hawking radiation, using what are now
standard methods of quantum field theory in a curved background.
The result is definitely not a pure state.

Again, it could be that this calculation is badly wrong. But,
again, the problem is to explain *why* it's wrong.

Note that it's not enough to simply have "a pure state" --
for the process to be unitary, one must have a *different* pure
state for each possible initial state of collapsing matter.
This almost certainly means that "late" Hawking radiation,
emitted near the end of evaporation, must be correlated with
much "earlier" Hawking radiation. But late Hawking quanta
are never in causal contact with early quanta. So this would
seem to require some highly nonlocal interactions. This would
mean a breakdown of the effective field theory description,
again in a place where curvatures are small and there's no
evident reason for the description to break down.

When the black hole
has completely evaporated, the net result will be the conversion
of a pure state to a mixed state.

For that, we clearly cannot use the static Schwarzschild
solution, and just using Kruskal-Szekeres coordinates won't
help since they still describe the same solution. Is there
any closed form solution for the "transient black hole"?

There are many proposals. We have no idea which, if any, of them
is correct.

In any case I cannot quickly find it (so I can't even rule
out it might fit in the margin of this post..) But we can
look at Rindler coordinates for a uniformly accelerating
frame. The Unruh effect corresponds to Hawking radiation
and the Rindler horizon is the correct event horizon for
the accelerating observer. Now, for the non-eternal case,
we only have to make the observer reduce its acceleration,
corresponding to the black hole losing its mass. For the
observer, the Rindler horizon fades away as acceleration
is reduced (of course we get perturbed Rindler coordinates
by the "variation of constant" method).

So whatever matter was forever behind the horizon in the
eternal case, will simply come into view again with non-
eternal acceleration. This is not yet the correct solution
for the non-eternal black hole, but it suggests that the
paradox will be resolved just by allowing the black hole
mass to decrease, which the correct solution must do.

Many people have tried this. It's very hard (so far, not
possible) to get it to work. In simple models one can
calculate quantitatively exactly what sorts of interactions
in the late stages of evaporation are needed to "purify"
the earlier Hawking radiation. (Eugenio Bianchi has some
nice work on this.) The answer seems to be that either
normal Hawking evaporation has to change drastically while
the black hole is still very large -- again, when there is
no apparent reason to expect the semiclassical description
to fail -- or else one needs to end with a very long-lived,
extremely high entropy "remnant."

[...]
2) But also that it is not lost because [...]

Quantum mechanics is unitary. This means that the "fine-grained"
information in a quantum state is never lost; pure states evolve
to pure states.

Yes, this part we all want to keep. (It might even be true!)

Or might not -- this is as much of a possible loophole as
anything else.

There are some obvious places to look for loopholes. I suspect
that at least the obvious ones have all been analyzed to death,
and all have unpleasant consequences. For instance, Hawking
radiation might not really be thermal; but to get back enough
correlations seems to require some very nonlocal interactions.

That's not clear. if in-falling information just bounces back
at the horizon then that's local (as could happen in the
eternal equilibrium hole).

That makes no sense. There's nothing *at* the horizon. It's
just empty space. It's true that "new" matter falling in can
interact with Hawking radiation coming out -- this is something
't Hooft has been working on lately -- but I don't see how it's
relevant to the setting in which the paradox is posed, which is
a black hole forming from collapse of a pure state and then
evaporating.

And if information in region II
comes back into view by some curved shape of the non-eternal
horizon, then also nothing non-local is needed..

Again, the devil is in the details. No one has managed to make
a model in which this works without drastically changing the
physics in low curvature, nearly classical regions where there
seems to be no reason for it to change. That is, either the
information in region II has to reappear very early on, when
the black hole is still very large and quantum effects should
be negligible; or else it has to stay "in view" for an extremely
long time after the black hole has gotten very small, basically
stretching out the black hole lifetime to nearly infinity and
creating all sorts of problems associated with ultra-high
entropy "remnants."

It's easy to identify assumptions and say, "Maybe this one is
wrong" or "maybe that one is wrong." It's a whole lot harder
to identify a plausible reason that any of the assumptions is
wrong, and harder yet to actually show that that solves the
problem.

Steve Carlip