Where Does the Entropy Go?

Gravity is a weak force, which makes it extremely difficult to do actual experiments (or perform astronomical observations) that would give us any detailed, up-close-and-personal data about the behavior of quantum gravity. We should be thankful, therefore, that we've been able to learn as much as we have about quantum gravity (and we do know some things) just by sitting in our chairs and doing thought experiments, constrained only by the basic principles of general relativity and quantum mechanics. Undoubtedly the most prolific thought-experiment laboratories have been black holes. In particular, Hawking's discovery that black holes radiate and have entropy has driven an enormous amount of research, and some of it has actually been productive! One of the highlights was certainly the calculation in 1996 by Strominger and Vafa, who used some tricks from string theory to actually count the number of quantum states hidden in a black hole, in a way that would have made Boltzmann proud, and come up with an answer that matched Hawking's formula precisely. There are still puzzles, however, as you might guess. Foremost among them is "How does the information get out?" An increasing number of physicists believe that the evaporation of black holes conserves information, but they don't know precisely how the details of the state which created the black hole get preserved and then encoded in the outgoing Hawking radiation. A lesser-known puzzle, which many people don't even consider a puzzle, hearkens back to a 1994 paper by Stephen Hawking, Gary Horowitz, and Simon Ross. They were trying to use the particular technique called Euclidean Quantum Gravity (in which you temporarily forget that time is any different than space) to calculate rates at which different things could happen, when the stumbled across a puzzle. They calculated the entropy of black holes with electric charge, and in particular of extremal black holes -- configurations where all of the energy really comes from the electric field itself, none from any purported mass that might have fallen into the black hole. And for an extremal black hole, they found an unusual answer: zero! That was a surprise, because it is not what Hawking's original formula (entropy is proportional to area of the event horizon) should give you for such a situation. Most people (including, I think, the authors) believe that this result is not trustworthy, and reflects a breakdown of the particular method used, rather than a deep truth about extremal black holes. But in a field where actual data is sparse on the ground, it's worth keeping puzzles in mind, hoping that some day they will teach you something. Matt Johnson, Lisa Randall and I just submitted a paper in which we revisit this puzzle. We suggest that maybe it's not just a simple breakdown of the methods of Euclidean quantum gravity, but perhaps something interesting is going on.