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). In particular, the
often shown Kruskal-Szekeres diagram like this one he
http://i.stack.imgur.com/U2EFS.gif
is certainly *not* the correct description of formation,
temporary existence and final complete evaporation!

What it may describe correctly is an eternally existing
black hole evaporating at the same rate as accreting new
mass (in homogeneous form, GW150914 is not a good example!)
Any mass in Region II (positive V axis) would never come out
and would not contribute to any paradox in this case. The
mass falling in would linger very long (in Schwarzschild
time) above the horizon and would obviously have ample time
to imprint its information on the Hawking radiation being
emitted there. In fact the in-falling information is just
bouncing of the horizon for this "eternal equilibrium".

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.

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"?

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. In
any case, the event horizon cannot have the simple light-
like infinite wedge shape as it has for the eternal case.

...
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!)

...
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). 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..

[...]
The problem only
appears quantum mechanically, when you allow black holes
to evaporate thermally via Hawking radiation.

And claiming that Hawking radiation is a pure state, statement
i) in the above Polchinsky paper, does not avoid the paradox
either. I would really like to see a diagram/coordinate system
describing the non-eternal black hole for a better judgement!

--
Jos