Thermodynamically reversed black holes, firewalls, Casimir effect, null energy condition violations

Scott Aaronson asked a very deep question at Hawking radiation and reversibility about what happens if black hole evolution is reversed thermodynamically. Most of the commenters missed his point entirely. Of course it's overwhelmingly improbable statistically to set up the initial conditions needed to get a thermodynamically reversed black hole evolution. But that's not the point of the question at all. It is theoretically possible to have a thermodynamically reversed black hole evolution as a valid evolving state by CPT symmetry, even if it's improbable for all practical purposes. As it exists as a possible solution, we can ask questions about the properties of such a solution.

In a thermodynamically probable evolution, we can have a massive object with a significant fraction of the black hole's mass fall into a black hole, and the blak hole automatically increases in mass by a huge fraction over a timescale of order of the Schwarzschild radius R. Then, it slowly evaporates away Hawking radiation with a remaining lifetime of order $R^3$. In the thermodynamically reversed version of this event, over a time period of order $R^3$, thermal radiation is fed into a black hole which increases in size very gradually as a result. There is an incredible fine-tuning of the ingoing radiation fed in with fine-tuned higher order multipartite entanglement (this is a theoretical state, not a practical engineering one) so that as a result, over the time period of order $R^3$, no Hawking radiation is emitted from the black hole because of fine-tuned quantum cancellations between the contributions from the black hole, and the contribution from frequency mixing of the radiation fed into the hole. Then, after a time period of order $R^3$, suddenly over a time period of order $R$, the black hole emits as "Hawking radiation" a massive body a huge fraction of its mass, and the black hole's mass decreases accordingly. The area of its event horizon goes down by a huge fraction over a time period of order R. According to Raychaudhuri's optical equation, this requires a significant negative null energy. This is too large to come from Casimir effects, as that would require a time period of order $R^3$ to lose that much black hole mass. So, what is the origin of this huge negative null energy?

Let's look at what happens just outside the black hole before the massive body is ejected. In the semiclassical approximation, the massive body must have come from a firewall ( What are cosmological "firewalls"?) at an exponentially small distance above the event horizon. This exponential factor comes from the very high boost factor and Lorentz contraction. Semiclassically, this firewall couldn't have originated from entangled Hawking pair production just outside the horizon. So, it must have originated from the fine-tuned radiation fed into the hole.

What if a probe is sent during an earlier time to measure this firewall? Unfortunately, this probably can't be done. See, we are conditioning a significant part of the future state by stipulating that a huge massive body is ejected as Hawking radiation. To further specify a probe hovering just above the horizon requires in addition a significant condition on the past. Ordinarily, with no thermodynamic conspiracies, we only have past conditions and no future conditions, and so, there's no problem. However, such fine-tuned conditioning of both the past and future might actually be impossible?

-

The question is only getting one point wrong, which is that the Raychoudhuri equation requires the reversed black hole whose horizon shrinks by emitting matter to have negative stress energy.

The reason this last point fails in the reversed case is becuase the horizon character is reversed--- the white hole horizon is a past horizon, it's another extension of the exterior solution. Stuff can come out of the white hole horizon, leading the black hole to shrink.

The past horizon in a white hole can be understood by considering the four quadrants of Minkowski space in Rindler coordinates: in the metric

$$ds^2 = dx^2 - dt^2$$

Has four regions picked out by the origin: I --- (spacelike and to the right of the origin), II-- (spacelike and to the left of the origin) III-(interior of future light cone of origin) IV--- (interior of past light cone of origin).

When you make a horizon by shifting to Rindler coodinates, which are relativistic polar coordinates for the origin, or the coordinates appropriate to a family of accelerated observers that consider each other mutually stationary, you find that the null rays from the horizon, both past and future, become event horizons for the accelerated observers.

In a Schwarzschild metric, near the horizon, the metric becomes Rindler, and therefore the maximal extension interprets each point on the horizon as both a future horizon, into which things can fall, and a past horizon from which things can come out. The past-horizon doesn't make sense near the point of formation of the black hole, and the future horizon doesn't make sense near the point of evaporation. Because collapse in classical GR doesn't have evaporation, people generally said "the past horizon intepretation is unphysical, and the future horizon is physical". But this is not 100% accurate, as with evaporation, both horizons are equally physical for a black hole in equilibrium. The future region, the analog of III, is where infalling perturbations go, and the past region, the analog of IV, is where the past of fluctuations in the Hawking radiation comes from.

This means that in your example, the time-reversed process emits the collapsing star from a past horizon, and this emission shrinks the black hole without any violation of positive energy. Past horizons can only shrink for the same reason that future horizons can only grow. The horizon in thermal equilibrium can only grow only by entropic thermodynamic considerations--- the entropy is unlikely to go down.

So once you accept Susskind's holographic duality, and the complementarity, a static black hole formed long ago has both extensions by different transformation on the physical exterior-measurable variables, it has a past region equally as a future region. It is just a conspiracy for such a black hole to emit stuff non-thermally from it's past region.

This argument is only valid, the nonthermal emissions are only conspiratorial thermodynamically, when the horizon has rung-down to thermal equilibrium. This doesn't happen for near-extremal black holes, whose normal modes are only slightly damped, with a damping time which diverges at extremality. There is no theoretical barrier for non-thermal emissions from such near-extremal black holes, which is because they have a hard time thermalizing.

It is my opinion, not shared by anyone else, that in the extremal case, the matter that falls generally does a full circuit of the interior regions, and comes out of the white hole past region after a finite time. Ignoring the past extension of the black hole is a legacy of pre-evaporation pictures of black holes, and doesn't make sense in the modern AdS/CFT context, where extremal black holes are fully reversible.

-
But surely a description of Aaronson's scenario fully in terms of a blavk hole without invoking white holes ought to be possible? –  Bewildered Oct 29 '12 at 13:34
At any rate, if the only way to explain this scenario is by treating the system as a white hole, what if after ejecting the massive body, we now dump a lot of matter into the hole and forget about any further fine-tuned feeding of matter. If it's a white hole, it now has to convert into a black hole, right? –  Bewildered Oct 29 '12 at 13:37
@Bewildered: Black holes and white holes are the same thing--- they are both described in the stationary limit by maximally extended Schwarzschild. They are both black and white away from the initial point of formation or the end point of evaporation, and both extensions (white and black) are from rearranging exterior variables in the quantum description, but in incompatible ways. –  Ron Maimon Oct 29 '12 at 14:14
That's only the case for an eternal Schwarzschild solution. For a hole which formed a finite time in the past, and will end in a finite time in the future, they have different Penrose diagrams. –  Bewildered Oct 29 '12 at 14:34
@Bewildered: They have mutually inconsistent Penrose diagrams. You can't have a past horizon for a black hole which formed, and you can't have a future horizon for a black hole which decays. Nevertheless, the correct classical description includes both extensions away from the two endpoints, and this is something which was only sorted out with Susskind's stuff. –  Ron Maimon Oct 29 '12 at 14:49

The two-state formalism of Aharanov can answer your question. True, in ordinary QM, there's a limit as to how low the Casimir negative null energy can go.

In What is the physical meaning of weak expectation values?, it was pointed out that the weak expectation value $\langle \chi |A|\psi \rangle/\langle \chi |\psi\rangle$ can be much larger in magnitude than the eigenvalues of A. Similarly, $\langle \chi |\widehat{T}_{vv}(x)|\psi \rangle/\langle \chi |\psi\rangle$ where $\widehat{T}_{vv}(x)$ is the null energy operator can have a much more negative value than permitted by the Casimir effect in ordinary QM where $|\psi\rangle=|\chi\rangle$. What is needed is the case where $|\psi\rangle$ and $|\chi\rangle$ are nearly orthogonal, but not quite.

However, with such huge null energy differences between $\langle \chi |\widehat{T}_{vv}(x)|\chi \rangle$ and $\langle \psi |\widehat{T}_{vv}(x)|\psi \rangle$, backreaction effects will mean both states will have significantly different metrics, and hence, significantly different causal structures, which means the locations of their horizons will differ significantly.

At any rate, don't expect your ejected massive body to come from frequency mixing. If you do your math, you'd find frequency mixing only becomes significant when the wavelength is around the Schwarzschild radius R. The de broglie wavelength of your massive body will be much smaller. This means frequency mixing will be highly suppressed. Your massive body has to come straight from the firewall, and not from Hawking radiation. As another person so aptly put it, it has to be "born in the firewall".

Although come to think about it, if the massive body came from the firewall outside the horizon, why do we need negative null energies at the horizon?

-

The comments about the case with conspiracies until a massive body is emitted, followed by dumping in a massive body without conspiracies is interesting. What if it's the same massive body?

Wild wild crazy thought experiment coming:

A conspiracy to feed the hole until a macroscopic sentient observer is pushed out of the hole. This sentient observer tells his story about his life, his conception at the past singularity, and his birth pushed out violently out of the white hole horizon. Then, after telling his fantastic story, his duty is done, and he jumps back into the hole to die. This same observer.

Black hole Penrose diagram version of events: His story is a LIE!!! We conspired to feed the hole until it pushes out this sentient observer, but as fellow conspirators, we planned out all the fine details of what comes out, including the brain memories of this observer, and all the records in his spaceship. They're all implanted false memories. The observer may swear all he wants about the truth of his experiences and memories but we know better, because we implanted them conspiratorially. We'll just nod and wink among ourselves, and exchange knowing glances, even while this observer keeps on protesting. We know better. There is no white hole. Only a black hole. A black hole which emitted a sentient observer with false memories as Hawking radiation coming out from the firewall just outside the horizon, where he was born. Then, this observer jumps back into the hole, passes through the horizon, and dies violently at the future singularity.

White hole Penrose diagrams version of events: It's a white hole, not a black hole. Miraculously, a sentient observer is conceived at the past singularity, and is pushed out through the horizon. He lives to tell his true miraculous life story. Then, he jumps back into the hole. As he approaches the white horizon, he is bombarded by radiation coming out from the white hole as he asymptotically approaches the horizon. He meets his death by this bombardment. Mercifully, this bombardment causes hallucinations in his brain so that just before he is finally destroyed, he hallucinates a future life inside the hole which in his hallucinatory state, he thinks it's a black hole. He doesn't realize he's already dying even as he hallucinates. He doesn't live to tell the story about his hallucination either.

-