Tuesday, August 03, 2010 ... /////

Stephen Hsu's confusion about white holes

The physics arXiv blog promotes a preprint by Stephen Hsu,

White holes and eternal black holes.
Much like John Wiley Price, Hsu tries to claim that the white holes are being discriminated against and that they do exist in isolation where they do differ from black holes. In his opinion, their main special property is that they like to spontaneously explode. ;-)

All these views are completely nonsensical and reveal Hsu's misunderstanding of elementary thermodynamics. I am surprised how difficult it is for the people - many of whom even have physics PhDs, e.g. Stephen Hsu and Sean Carroll - to understand some of the most rudimentary principles of statistical physics, e.g. the fact that the entropy never decreases.

What is a white hole?

Well, whatever it is, we want this object to behave as the time-reversal of a black hole. We want the life of a white hole to be the same thing as the life of a black hole but the future should be exchanged with the past. We must ask two key questions:
1. Does such a reversal of the black hole processes predict the existence of new objects?
2. Do these processes exist at all?
To make things simple, Hsu essentially answers Yes, Yes. However, the correct answers are No, No (with some additions that will be clarified below). It's really not difficult to see why the answers are what they are. First, let us ask whether there can exist new objects that would describe the white holes.
Update: After you finish reading this criticism of mine, you may try a reply from Stephen Hsu
White hole states in the Hilbert space

In physics, the state of a physical system - including all the information about the "objects" that exist in it at a given moment and their motion - is described by a point in the phase space (in a classical, i.e. non-quantum theory) or a state in the Hilbert space (in quantum physics).

Whenever classical physics emerges, it emerges as a limit of quantum mechanics. In this limit, the Hilbert space is replaced by the phase space (parameterized by coordinates and momenta), too. Without a loss of generality, I will only discuss the full - quantum - theory in which the states are described by states in the Hilbert space.

There must exist states that represent the black holes. A few years after the great visions of Jacob Bekenstein, it was realized in the 1970s that black holes carry a huge entropy. Consequently, there exists an exponentially gigantic number of microstates that are associated with a single black hole.

String theory has also made it very clear that these microstates are the "generic" states of localized matter with high enough mass - the final states of an entropy-increasing evolution. Also, these states may be described by the quark-gluon plasma, excessively excited strings, or other pictures produced by string theory. These perspectives make sense and they're among the strongest pieces of evidence that string theory is the right theory of quantum gravity.

Now, what about a white hole? Well, because we have "defined" white holes as the time reversal of the black holes, we can simply act with the "T" anti-linear operator on a black hole microstate to obtain a white hole microstate. What will we get? Well, we obviously get the same thing. At least qualitatively. There's no secret label that would distinguish black holes from white holes.

The "T" operator doesn't really change the type of objects in your state; it only reverses its velocities (which is sometimes not a possible symmetry - if the "CP" i.e. "T" is violated). So when you take a microstate representing a black hole, e.g. a bunch of some quark-gluon plasma in AdS/CFT, you will obtain a very similar quark-gluon plasma again.

It will be just another state - or even the same state - of the quark gluon plasma. It will still be a black hole. You won't get any "new class" of objects that you could identify as "new" white holes. White hole microstates are the same thing as black hole microstates.

The arguments above are completely obvious and should have been understood from the very first moment when people tried to discuss black holes and/or revert them in time. But if you need some paper that makes the point about the equivalence of black hole and white hole microstates explicit, check the 1976 paper Black holes and thermodynamics by Stephen Hawking.

The paper contains all the qualitative points I am making in this text and some additional quantitative ones.

Hsu refers to this paper and does realize that the paper has explained that the white hole microstates are identical to the black hole microstates. But he completely misunderstands this basic point and tries to obscure it and deny it in various ways. And this misunderstanding is the first necessary pre-requisite for Hsu to write his fundamentally wrong paper. Another pre-requisite is related to the second question:

How do white holes evolve in time?

Let me assume that you have understood that the white holes correspond to the very same objects as the black holes. There are simply no "qualitatively new" excitations of a string or a quark-gluon plasma to produce "qualitatively new" objects that could be sold as white holes.

The white holes are obtained from the black holes by keeping all the building blocks and reverting the relative velocities of all of them, so to say. For example, if you imagine that your black hole microstate had vanishing velocities (or other first time derivatives), the state won't be changed by the "T" operation at all.

However, we should ask how these "white hole" states are going to evolve in time. If we flip the signs of all the relative velocities, will the time evolution be the time reversal of the black hole evolution?

Much like in all of statistical physics, the answer would be Yes in principle: if we managed to exactly flip all the velocities, the evolution would be the time reversal of the original one. Except that it is insanely unlikely that we could succeed in this hyper-precise time reversal. In reality, we will never do it exactly and the macroscopic processes will never look like the time reversal of the original processes.

Technically speaking, states in the real world can never be fine-tuned accurately enough for them to evolve into lower-entropy states in the future.

The second law of thermodynamics is the principle that summarizes all these basic facts...

The second law of thermodynamics says that the entropy - the amount of information that can be carried by the detailed microscopic degrees of freedom of an object but that has been lost because the object looks chaotic - never decreases by a macroscopic amount. In most situations involving macroscopic objects, the entropy is strictly increasing. The time reversal of an increase would be a decrease which is prohibited. So generic processes can never be observed in their time-reversed form.

There are no exceptions to this rule because the rule only depends on basic statistics and probability theory.

Here's the proof: the probability of a transition from "A" to "B", where "A" and "B" represent classes of macroscopically indistinguishable microstates, must be averaged over the states in the "A" group (we don't know which microstate occurred, so we must divide our priors among them) and summed over the states in the "B" group (all of the final microstates are OK so their contributions are simply added).

The averaging over "A" adds the factor of "1/N_{initial,A}" or "exp(-S_{initial,A}" which is a tiny factor (for macroscopic "S") multiplying a "dynamical" number that is much closer to "1" in any reasonable units. If this tiny factor is not beaten by a comparable "exp(+S_{final,B})" from the summing over an equally large or larger group of the final "B" states, the probability will be de facto zero. That's why the entropy can never decrease by large amounts: "S_B" can never be smaller than "S_A".

The rule is independent of all dynamical details of a physical system you want to consider. The law holds for all physical systems with a large number of degrees of freedom. Any article, preprint, or a book that tries to convince you that the entropy could macroscopically decrease - or that it is even a normal thing - is a product of crackpot pseudoscience.

The probability for a transition of an initial state to a final state whose entropy is smaller by "-S" than the initial state is suppressed by "exp(-S/k)" where "k" is Boltzmann's constant - usually set to one in any system of adult units. In everyday life units, "k" is a tiny number, so whenever "S" is macroscopic, the probability is the exponential of the negative googol (almost), so to say, or the inverse googolplex (almost).

This probability is zero not only for all practical purposes but also for impractical purposes that could become relevant in the visible Universe during the future 50 billion years.

Again, the time reversal of macroscopic processes can only exist if we "saturate" the second law of thermodynamics: if the entropy stays constant. The time reversal of such processes keeps the entropy constant, too. These processes are not real processes because nothing much is changing. Instead, they describe a physical system at equilibrium.

Time-asymmetric processes

But in all the processes where something is actually taking place, the entropy is strictly increasing. And when black holes are involved, the entropy is increasing by amounts that are comparable to the black hole entropy itself. The black hole entropy is the highest entropy that a localized or bound system of the same mass or volume can actually have. It is gigantic.

It follows that when this entropy is increasing by relatively significant percentages, it is a huge increase, too. That's why the time reversal becomes even more impossible than for other macroscopic systems. A black hole of a solar mass can indeed have entropy comparable to 10^{60} (times "k") which is indeed closer to a googol than e.g. its square root. The probability that the entropy will decrease by a similar amount is (almost) one over a googolplex. Recall that a googol is 10^{100} and a googolplex is 10^{googol}.

What are the processes involving black holes that cannot be reversed because of statistical physics?

Well, all of them. Mass accretion is the first example. Black holes are swallowing the mass around them and their own mass keeps on increasing. The entropy of a black hole - and even the entropy difference associated with a mass increment - is much bigger than the entropy of the conventional massive object that the black hole has just swallowed.

So if the black hole at our galactic center eats another star, the entropy increases by a huge amount.

A star collapses into a black hole which eventually evaporates. The reversed process can't occur in Nature. The Hawking radiation never spontaneously produces a black hole and a black hole never spontaneously changes to a star.

Even if two black holes merge, the entropy grows a big time. Recall that in 3+1 dimensions, the radius of a black hole is proportional to its mass, "R=2M", in "c=hbar=G=1" Planck units. So the entropy i.e. area goes like "R^2" or "M^2".

If you have two black holes of mass "M" in the initial state and they merge, the final state is a black hole of mass "2M", by mass conservation. However, the initial area was two (from two objects) times "4.pi.(2M)^2". However, the final area is two squared (because the radius doubled, and it is squared in the formula for the area) times "4.pi.(2M)^2". And "two squared" is equal to four which is greater than "two".

If two equally heavy neutral black holes merge, the total entropy gets doubled!

In fact, Hawking has proved a theorem that the total area of event horizons never decreases. This finding has materialized before the black hole thermodynamics was understood - and it was actually one of the big reasons to think that the black hole areas could be "analogous" to the entropy which is increasing, too. These days, we know that it's not just an analogy; black holes actually do carry the entropy that is proportional to the area.

Hawking would later realize that black holes emit his thermal radiation, too. Needless to say, these processes don't violate the second law of thermodynamics, either. The resulting radiation carries an even bigger entropy than the black hole itself. Because the radiation is not bound or localized, it is no longer true that the black hole entropy is the maximum entropy that objects of a fixed total mass can carry.

I could continue with additional processes. But the lesson should be clear. None of the processes - accretion, mergers, Hawking radiation, and many others - can ever be found in the time-reversed form because that would violate the second law of thermodynamics by the most gigantic amounts that are possible in physics.

In particular, a (non-extremal) black hole or a white hole (which is given by the same states) can never "explode" and split into a few smaller objects because that would blatantly violate the second law of thermodynamics. Hsu's paper is wrong at every conceivable level. The only differences between black holes and white holes come from statistical considerations - but these limitations imposed by the second law of thermodynamics are damn real.

In the real world, the black/white hole microstates simply have to behave in ways that agrees with the second law. It implies that they can easily swallow chunks of mass but they can't spontaneously "spit" these chunks. Consequently, they look "black": the (visible frequency) light doesn't come out of large black holes. Consequently, it's more natural to talk about black holes and not white holes because the latter are not allowed. You may call this preference "racist" but it is definitely Mother Nature and Queen of Sciences mathematics (note that both of them are female) who are the authors of this racism! ;-)

And that's the memo.