Oh these rascally black holes! (Part I)

People are fascinated by black holes. You can't see them directly, they can be supermassive, and they are mysterious. Kind of like dinosaurs, which explains the attraction black holes have to (some) kids. But among physics people, it seems black holes create more heated arguments than any other topic, as opposed to child-like wonder. Black holes appear to violate some of our most sacred laws, and people cannot agree on whether they are truly violated, whether we should just go on with our merry lives in the light of such larceny, or what the universe is really doing to prevent this deplorable malfeasance.

So what evil thing are black holes accused of? One of the laws they are purportedly breaking is the law that all dynamics must be time-reversible (barring, perhaps, CP-violating processes). One way in which time reversal invariance can be broken is by processes that lead to a coalescence of trajectories (in phase space, to be precise). If trajectories coalesce (two or more turn into one) then I cannot run time backward unambiguously ("Which branch should I take?") The coalescence of phase space trajectories implies that knowing the future does not allow us to predict the past. It is truly an abomination, and we have to insist that black holes stop it (if in fact they are guilty). 

Another law that we believe in is that wave functions evolve forward in time in a unitary manner, which implies that the entropy of a known state is and remains zero for all time. The latter implies that the (quantum) state is and remains predictable at all times. There is a direct relationship with our law of time-reversal invariance as you see immediately, because if all quantum trajectories can be reversed uniquely, then this means they never coalesce. Two trajectories that have coalesced cannot be time-reversed unambiguously: hence the relationship between predictability and time reversal.  Such a coalescence of trajectories has many outrageous consequences: for example the vanishing entropy of the initial state, upon evaporation of the black hole (something I will explain below), would turn into the non-vanishing entropy of the radiation field left behind. Unitarity would be lost, and with that our conviction that the universe is and remains pure. It is like the loss of innocence.

I will try to convince you here that it is the accused that is innocent, that black holes are just ordinary participants within cosmology. They are quantum, and they are heavy, they are black bodies, but they are not evil and they certainly do not violate any laws.

First, what is the evidence for this violation? This evidence goes back to a 1975 paper by Stephen Hawking, which introduced the world to his eponymous radiation. The paper is not an easy read, but I still encourage everyone who wants to enter the field to read it, and to replicate the calculation as much as he or she can. In my view, nothing replaces actually doing a calculation and re-deriving results. However, there are now much more succinct ways to derive the same result (I think I can do it on a single page), and I'll sketch those here (without equations, though). My simplification relies on ignoring the red shift (the lengthening of wavelengths when light moves within a gravitational field). This may appear problematic, but we can restore the red shift at the end of the calculation, and consider its effect separately. The red shift does not change any of the arguments I give here. It's something practitioners do frequently, if only the informational aspect is of concern. 

The central result of Hawking comes from understanding what a vacuum is. In ordinary language, a vacuum is the absence of anything, but not in quantum field theory. In quantum field theory, the vacuum teems with fluctuations: particles and their respective anti-particles are constantly created in pairs, only in order to decay again (lest they violate our most sacred of laws: energy conservation). In fact, any time a pair is produced, it must borrow a little bit of energy (from the infinite bank of the universe) which may allow them to travel apart from each other for a little bit. But of course, this attempt at separation between the twin particles must be fleeting, because energy bills must be paid.  The pair annihilates in a flash, returning the borrowed good to the bank. Now suppose a system is accelerated, like, a lot. The pairs are still being produced. Now imagine that one pair borrows a lot of energy, and manages to move apart appreciably. Because the system is so strongly accelerated, it can happen that the pair can never be re-united (unless one travels faster than light). One of the particles has disappeared behind a "causal wall", and if you are part of the accelerated system, you will see only one of the two particles, which is now all alone. Now, this looks like radiation. There are now physical particles in a system where there were none when the system was at rest. This curious fact was discovered independently by Stephen Fulling, Paul Davies, and William Unruh, but the effect is usually just abbreviated as the "Unruh effect". If you think about it, it Unruh radiation makes a lot of sense.

Now let's imagine that the pairs are formed (and de-formed) not in an accelerated system, but instead near the horizon of a large black hole. If you paid attention in whatever class taught you general relativity (or whatever book or blog you read to replace said class), you know that Einstein's path to understanding gravity was precisely from seeing the analogy between accelerated systems and gravitational fields. At the edge of the horizon, the same thing can happen to the excited twin-pair as what happened to the accelerated twin pair: one may venture towards the horizon, and one may move the opposite way. But if the daring one goes beyond the point of no return, there will be no happy reunion: the twin moving away from the horizon looks like a particle: he is Hawking radiation. So: Hawking radiation is just like Unruh radiation, only near black hole horizons. Fine, but so what?

Credit: Science Magazine (2004)

Well, there is more. I told you somebody had to pay the energy bill. In this case it will be the black hole who has to pay: there is nobody else around. (In the case of the accelerated observer, it is this observer/detector who will lose mass.)  If this process happens often enough, the black hole will lose all of its mass: it is said to have evaporated. So what? So big deal, as I demonstrate now.

The stuff  that made the black hole isn't just stuff: it's particles and radiation. So yes, particles and radiation turn into particles and radiation, but the stuff that made the star can be seen as special: quantum mechanically, we can say that it is completely known. But after evaporation, nothing of that knowledge remains. After all, according to this picture, all there is to the black hole is mass: the details of how this mass was formed are completely gone. Trajectories have merged in a most heinous way. Information is lost. Or is it?

Part 2 is here.

Part 3 is here.