Before getting to his comments, let me say a few things about what I did in the paper.

A. The "white hole" I analyzed is just a classical background which is the time reversal of (part of) a black hole spacetime. The initial data for this spacetime can be obtained from a spacelike slice across the usual black hole spacetime ("after" the horizon has formed), and need not include the singularity.

B. I imposed the condition of isolation (vacuum) outside the white hole at early times. This is equivalent to requiring no radiation in the future of the original black hole spacetime. An unusual boundary condition, but corresponds to the "isolated" white hole I was interested in investigating.

C. I used Hawking's method (i.e., Bogoliubov transformation of in- and out- modes) to study the future behavior of the white hole, or, equivalently, the initial preparation of the black hole required to prevent it from radiating.

It should be clear from the summary that my paper can be read as merely analyzing the

**black hole**spacetime with a nonstandard future boundary condition and then

**interpreting**the results in time reversed language. ("Look ma, no white hole"!) That is, the methods I use have exact time reversal invariance built in. In the paper I note that an isolated white hole and an isolated black hole behave differently. That is

**not**a violation of time reversal invariance

*applied to an entire spacelike slice*because the time reversal of an isolated black hole (which radiates into its future) is

**not**an isolated white hole -- rather, it is a white hole bathed in incoming radiation (from its past) at the Hawking temperature (see figures in the paper). However it does contradict the idea that the time reversed evolution of the hole can be understood independently of its environment (i.e., what is outside its horizon).

Lubos makes a number of remarks in his blog post. I try to summarize them below, together with my comments. He may make other claims as well that I haven't addressed.

1.We know from string theory that black holes and white holes are the same thing.Well, let me first point out that not everyone believes in string theory as the correct theory of quantum gravity (i.e., describing our universe) at 99.9 percent confidence level. Secondly, if a semi-classical calculation like mine suggests differences between the behavior of an isolated white hole and an isolated black hole, isn't it interesting to reconcile that with what AdS/CFT predicts? Although I am not an expert on AdS/CFT I suspect that the time reversal invariance of the CFT boundary state only implies time reversal of the entire bulk state (i.e., on an entire spacelike slice) and not of the black hole alone. If so, there is no contradiction with my results -- see above. Perhaps someone can clarify this for me?

2.Hawking clearly said the same in his 1976 paper.That was my impression on first reading, but since all of his arguments center on the case of a black hole in equilibrium with a bath at equal temperature, it is unclear (at least to me) how this can be generalized to an isolated white hole. That was one motivation for my investigation.

3.Entropic arguments imply that white holes (as obtained via (A) above) are extremely unlikely: specifically, a highly entropic white hole should not explode into lower entropy ejecta. I understand the argument but don't place as much confidence in it as Lubos does. The uncertainty is not about the 2nd Law but about the interpretation of black or white hole entropy.

Lubos does not want me to consider classical spacetimes generated by the initial data obtained in (A) above. Even if one accepts that such spacetimes are highly improbable, that does not mean that they shouldn't be studied. (For example, if you are a many worlder there are some branches on which exploding white holes are observed!) Apparently it is in bad taste to think about (exploding) white holes, but perhaps Lubos should tell this to, e.g., Frolov and Novikov.

Now, a little analysis of the cognitive dissonance (conflicting priors ;-) between Lubos and me. When I say "white hole" I mean the time reversal of some classical black hole spacetime. I consider this (time reversed) spacetime of theoretical interest, even if it results from strange initial conditions. I use what I know (general relativity + quantum fields in slightly curved space) to probe the more mysterious issues (black hole entropy, quantum gravity, ... ). This follows Wheeler's approach of "radical conservatism" -- take the physics you trust with high confidence, and extrapolate to extreme conditions until something interesting happens! Lubos is a true believer in string theory, so to him a black hole is this stringy thing about which we already know almost everything, including that its entropy is due to countable internal microstates, it is dual to some YM configuration through holography, etc. This will likely elicit a shriek of anger from Lubos (or he will just call me dumb), but I consider all of those claims

**plausible but perhaps not true**in our universe: string theory may turn out not to describe Nature.

Finally, there is also some discussion of my paper here and here, but it seems that both authors are slightly confused about the results (perhaps this is my fault for not being clear :-). For example, the requirement that white holes "explode" is not a consequence of my analysis, but just follows from time reversal of the black hole formation event (see, e.g., Frolov and Novikov or figures in the paper). I am only studying the quantum effects (i.e., equivalent of Hawking radiation), which are a

*correction*to the classical evolution.

Further discussion in a follow up post about white/black hole entropy.

## 3 comments:

Steve, I have no idea what you're talking about but I still find it fascinating :)

One thing I'm interested in is how you store mental models of physics in your head..? It seems to go completely against the folk physics that we've evolved to intuitively understand.

As you write correctly in your paper, whether you have radiation in the initial or the final state depends on the choice of vacuum at asymptotic infinity. It's not something that follows from the space-time geometry. You can already do that for black holes. Ie you can assume there's constantly radiation infalling from scri -, then the black hole will never evaporate. I've actually been wondering some years back why none of the black-hole-will-eat-the-earth doomsday proclaimers pointed that out ;-) It's mathematically a perfectly correct possibility. It just doesn't make a lot of sense because the initial state that you need for that is extremely improbable, much like it is not impossible but very improbable to be able to arrange initial conditions for an inverse supernovae. The white hole's dynamics is only a time-reversal of the black hole's dynamics if you also exchange the vacuum states.

In any case, it is funny that last year I was looking for a paper that examined white hole's evaporation since I needed it for one of our papers. In the end, I added an argument about the choice of vacuum that has some resemblance to yours. Wish I had had your paper then!

White holes may turn out to play a role in the universe. These emerge from the Schwarzschild metric for black holes. There is time symmetry between two spacelike regions with a singularity that have event horizons bounding a timelike region (actually two of them) that is our universe. Geodesics of matter and radiation from one spacelike region are outgoing and the other has ingoing geodesics. These define the white hole and black hole respectively. Now this is an exact solution result of the Einstein field equations, an eternal black hole, so we don’t expect every black hole to have a corresponding white hole. Generally the conformal theory or Penrose diagram is truncated off so the black hole is not an idealized eternal system and is instead the result of mass-energy imploding into a small region of space. In this physical sense the white hole is removed. However, white holes return in a way in inflationary cosmology.

Inflationary cosmology is an early phase of the universe where the scale factor for space exponentially expanded. The FLRW dynamical equation is

(a’/a)^2 = 8πGρ/3

a’ = time derivative of the scale factor and ρ the density of mass-energy. During inflationary period this density is about constant in time (not exactly though) and the differential equation gives the scale factor with time as a = exp(sqrt{Λ/3}t), where Λ is the cosmological constant. During the inflationary phase of the universe this Lambda was ~ 10^{120} times larger than it is today, where the first 10^{-20} seconds saw exponentially expansion. If the universe had any inhomogeneous distribution of mass-energy and spacetime curvature prior to inflation these were stretched out enormously as spacetime was flattened out. Then in reverse time space would be exponentially shrinking until the wrinkles begin to appear. These clumps could focus inwards to form black holes. In ordinary forward directed time these are white holes!

Is a white hole symmetrical to a black hole? Symmetry between the two happens if the ingoing and outgoing particles are in some equilibrium on both cases. A Hawking black hole quantum radiating out mass energy in photons or bosons is similar to a white hole gushing out mass-energy at the same rate particles quantum tunnel into the white hole. This is a bit akin to turning an Eddington-Finkelstein diagram upside down. A large black hole stands in stark contrast to a putative large white hole. The black hole is stable, the horizon remains constant in area and thus entropy S = kA/4L_p. Outgoing geodesics of the white hole would disgorge its mass energy at a huge rate. This may have existed in the pre-inflationary or at the start of cosmological inflation. So the entropy of a white hole can’t be equal in form to that of a black hole, for otherwise this would clearly violate the second law of thermodynamics. Departures between black and white holes may be a feature time asymmetry with respect to thermodynamics.

Lawrence B. Crowell

Post a Comment