Do black holes have a temperature?

It seems I've come to the party late, and the number one response is predictably that "Yes, blackholes have a temperature because that is what Hawking predicted in his famous paper." But Hawking predicted no such thing! Before I can answer this question I will have to tell you what Hawking actually predicted, which is slightly different than what is often attributed to him.

A remarkable analogy was discovered in 1973 between blackhole physics and thermodynamics by Larry Smarr. What he realized is that just as an object in thermal equilibrium can be completely characterized by 3 quantities: temperature, pressure, and chemical potential (note, these are the so-called intensive variables), so, too, can a black hole be defined by 3 quantities: mass, charge, and angular momentum (using extensive variables this time) in precisely the same way, meaning that the equations that define these two very different physical phenomenon are surprisingly identical. What Bekenstein did is realize that the analogy is perfect if and only if black holes had entropy, which he calculated was proportional to the area of the event horizon. But if black holes have entropy, then they must have temperature according to the laws of thermodynamics. What Hawking did, and what made him famous, is calculate the numerical value of the temperature (it is T=1/(8 pi M), where M is the mass of the black hole).

So you might think that answers your question is "yes," but wait! Hawking never said blackholes have temperature. If you go back to his paper what he shows is very clear: He states that IF empty space has a temperature, and IF the blackhole is in thermodynamic equilibrium with the ambient space, THEN there is only one temperature the blackhole can have and that is the one quoted above. Now neither of these statements needs to be true. However, we know that the universe does have a finite temperature in empty space (it is 3 kelvin), however, we don't know that a blackhole is in thermodynamics equilibrium with space, and indeed, if it were true then all black holes would have the same temperature as empty space, namely 3 kelvin.

So the question is difficult to answer because it is still in the line of current research. As previous commenters noted, Hawking then claimed that radiation is emitted from black holes in the form of so-called "Hawking radiation" such that heat can escape, but this is an unsubstantiated claim made by Hawking that his own equations never vindicated. Indeed, just after describing the mechanism of "Hawking radiation" by which a particle and anti-particle appear in the vacuum, and then then anti-particle falls into the black hole while the particle escapes to infinity, Hawking then immediately back pedals and states "this is just a heuristic, and not meant to be taken literally" (!) Hawking has to say this, because his equations show no such thing. There are no anti particles in his equations, and he has to assume the vacuum already has a temperature even before the black hole has formed. However, the story is too compelling to ignore and Hawking rocketed to fame in the following years.

Current research is beginning to unravel this yarn. If Hawking radiation exists, then Susskind has shown there is an information paradox that violates the most important law of physics, that no information (energy) is created or destroyed. This has been brought further into focus by Polchinski who has shown that if Hawking is right, then Einstein is wrong. The greatest concession came from Hawking himself when he declared just after Polchinski's paper came out that black holes couldn't even exist, that they were only approximately black, or grey holes, and this meant no information paradox has to exist. This is all as recent as 2013, so you can see the question is still open, but if you want something to feel good about, it was in trying to answer this very question that Juan Maldecena worked out the Holographic priniciple which has become the top cited physics paper of all time. So keep thinking about it and maybe you'll be the one to write the next chapter in the great history of physics.

/r/askscience Thread