# Hawking radiation and reversibility

As you said, the case of black holes is conceptually totally analogous to the burning books. In principle, the process is reversible, but the probability of the CPT-conjugated process (more accurate a symmetry than just time reversal) is different from the original one because $$ \frac{Prob(A\to B)}{Prob(B^{CPT}\to A^{CPT})} \approx \exp(S_B-S_A ).$$ This is true because the probabilities of evolution between ensembles are obtained by summing over final states but averaging over initial states. The averaging differs from summing by the extra factor of $1/N = \exp(-S)$, and that's why the exponential of the entropy difference quantifies the past-future asymmetry of the evolution.

At the qualitative level, a white hole is exactly as impossible in practice as a burning coal suddenly rearranging into a particular book. Quantitatively speaking, it's more impossible because the drop of entropy would be much greater: black holes have the greatest entropy among all localized or bound objects of the same total mass.

However, the Hawking radiation isn't localized or bound and it actually has an even greater entropy – by a significant factor – than the black hole from which it evaporated. That's needed and that's true because even the Hawking evaporation process agrees with the second law of thermodynamics.

At the level of classical general relativity, nothing prevents us from drawing a white hole spacetime. In fact, the spacetime for an eternal black hole is already perfectly time-reversal-symmetric. We still mostly call it a black hole but it's a "white hole" at the same moment. Such solutions don't correspond to the reality in which black holes always come from a lower-entropy initial state – because the initial state of the Universe couldn't have any black holes.

So the real issue are the realistic diagrams for a star collapsing into a black hole which later evaporates. Such a diagram is clearly time-reversal-asymmetric. The entropy increases during the star collapse as well as during the Hawking radiation. You may flip the diagram upside down and you will get a picture that solves the equations of general relativity. However, it will heavily violate the second law of thermodynamics.

Any consistent classical or quantum theory explains and guarantees the thermodynamic phenomena and laws microscopically, i.e. by statistical physics applied to its phase space or Hilbert space. That's true for burning books but that's true for theories containing black holes, too. So if one has a consistent microscopic quantum theory for this process – but the same comment would hold for a classical theory as well: your question has really nothing to do with quantum mechanics per se – then this theory must predict that the inverted processes that decrease entropy are exponentially unlikely. Whenever there is a specific model with well-defined microstates and a microscopic T or CPT symmetry, it's easy to prove the equation I started with.

A genuine microscopic theory really establishes that the inverted processes (those that lower the total entropy) are possible but very unlikely. A classical theory of macroscopic matter however "averages over many atoms". For solids, liquids, and gases, this is manifested by time-reversal-asymmetric terms in the effective equations - diffusion, heat diffusion, friction, viscosity, all these things that slow things down, heat them up, and transfer heat from warmer bodies to cooler ones.

The transfer of heat from warmer bodies to cooler ones may either occur by "direct contact" which really looks classical but it may also proceed via the black body radiation – which is a quantum process and may be found in the first semiclassical corrections to classical physics. The Hawking radiation is an example of the "transfer of heat from warmer to cooler bodies", too. The black hole has a nonzero temperature so it radiates energy away to the empty space whose temperature is zero. Again, it doesn't "realistically" occur in the opposite chronological order because the entropy would decrease and a cooler object would spontaneously transfer its heat to a warmer one.

In an approximate macroscopic effective theory that incorporates the microscopic statistical phenomena collectively, much like friction terms in mechanics, those time-reversal-violating terms appear explicitly: they are replacements/results of some statistical physics calculations. In the exact microscopic theory, however, there are no explicit time-reversal-breaking terms. And indeed, according to the full microscopic theory – e.g. a consistent theory of quantum gravity – the entropy-lowering processes aren't strictly forbidden, they may just be calculated to be exponentially unlikely.

The probability that we arrange the initial state of the black hole so that it will evolve into a star with some particular shape and composition is extremely tiny. It is hard to describe the state of the black hole microstates explicitly, but even in setups where we know them in principle, it's practically impossible to locate black hole microstates that have evolved from a recent star (or will evolve into a star soon, which is the same mathematical problem). Your $U^{-1}$ transformation undoubtedly exists in a consistent theory of quantum gravity – e.g. in AdS/CFT – but if you want the final state $U^{-1}|initial\rangle$ to have a lower entropy than the initial one, you must carefully cherry-pick the initial one and it's exponentially unlikely that you will be able to prepare such an initial state, whether it is experimental preparation or a theoretical one. For "realistically preparable" initial states, the final states will have a higher entropy. This is true everywhere in physics and has nothing specific in the context of quantum gravity with black holes.

Let me also say that the "white hole" microstates exist but they're the same thing as the "black hole microstates". The reason why these microstates almost always behave as black holes and not white holes is the second law of thermodynamics once again: it's just very unlikely for them to evolve to a lower-entropy state (at least if we expect this entropy drop to be imminent: within a long enough, Poincaré recurrence time, such thing may occur at some point). That's true for burned books, too. A "white hole" is analogous to a "burned book that will conspire its atomic vibrations and rearrange itself into a nice and healthy book again". But macroscopically, such "books waiting to be revived" don't differ from other piles of ashes; that's the analogous claim to the claim that there is no visible difference between black hole and white hole microstates, and due to their "very likely" future evolution, the whole class should better be called "black hole microstates" and not "white hole microstates" even the microstates that will drop entropy soon represent a tiny fraction of this set.

My main punch line is that at the level of general reversibility, there has never been any qualitative difference between black holes and other objects that are subject to thermodynamics and, which is related, there has never been (and there is not) any general incompatibility between the general principles of quantum mechanics, microscopic reversibility, and macroscopic irreversibility, whether black holes are present or not. The only "new" feature of black holes that sparked the decades of efforts and debates was the causality. While a burning book may still transfer the information in both ways, the material inside the black hole should no longer be able to transfer the information about itself to infinity because it's equivalent to superluminal signals forbidden in relativity. However, we know today that the laws of causality aren't this strict in the presence of black holes and the information is leaked, so the qualitative features of a collapsing star and evaporating black hole are *literally* the same as in a book that is printed by diffusing ink and then burned.

There are two equivalent descriptions for the same process in terms of the time-forward version, and the time-reversed version. Externally, both look the same; some matter in a pure state collapses together into a dense state — a gravitational hole — and slowly, over time, it evaporates Hawking radiation until nothing is left of it. The totality of all the Hawking radiation remains pure.

In the time-forward version, matter collapses into a black hole with a future singularity, and ends there. Entangled Hawking pairs are produced just outside the horizon. One particle of each pair falls into the hole and hits the singularity. Postselection to an entangled state of the infalling Hawking radiation and infalling matter is imposed at the future singularity. The outgoing Hawking particles carry information about the infalling matter *after* postselection. Before postselection, it remains entangled with the infalling Hawking radiation.

In the time-reversed version, a white hole with a past singularity forms with a white hole horizon. Matter can only emerge from the white hole, not enter it, and all matter emerging from inside it was created at the past singularity. Matter just outside the white hole is still attracted gravitationally, but it only accumulates just outside the horizon, unable to penetrate it. Matter emerging from the past white hole singularity and crossing the horizon, but without enough escape velocity to leave the white hole's pull also accumulates just outside the horizon. The shell of accumulating matter just outside the horizon quickly forms a black hole shell with a future singularity. This is the *time-reversed* firewall, which is very real in the time-reversed version. Postselection occurs at this shell. All the matter emerging from the past white hole singularity at the same location are entangled with each other. After postselection, matter emerging from the white hole singularity which has enough escape velocity carries information about the infalling matter which accumulates at the shell just outside the white hole horizon because it was initially entangled with other matter which collides with the infalling matter at the shell, and they are postselected together in an entangled state.

Far away from the horizon outside the hole, both the time-forward and time-reversed versions look identical after their respective postselections. However, around the horizon and inside the hole, they look very different. This is "time reversal complementarity"! The time reversed version of a time-forward version is a time-reversed version, and vice versa. However, *operationally*, the only information one can have about these regions are those carried by concrete physical information emerging from the hole and recorded far outside it. *Operationally*, one can never tell the difference.

Is there a firewall just outside the horizon? In the time-forward picture, no. In the time-reversed picture, yes. Far outside the hole, we can't tell the difference. Sure, we can send a probe to measure the presence/absence of a firewall, and beam the result outside. Then, external observers will see a signal telling them "I'm from the probe, and I don't see any firewall". However, in the time-reversed picture, there is a firewall shell, and the probe is thermalized there. After postselection, the beamed signal that external observers pick up originated from the past white hole singularity itself, which heads straight to the external observers. Prior to postselection, the radiation from the past singularity carries no such information, but after postselection, it does. This process looks conspiratorial, but then, the time-reversed picture works best when entropy is decreasing, i.e. reversed thermodynamic arrow of time, which isn't the case here. With a reversed time arrow, such conspiracies are the norm.

See also the related question Why are white holes the same thing as black holes in quantum gravity? and What are cosmological "firewalls"?.