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.

Extremal limits and black hole entropy
Authors: Sean M. Carroll, Matthew C. Johnson, Lisa Randall

Abstract: Taking the extremal limit of a non-extremal Reissner-Nordström black hole (by externally varying the mass or charge), the region between the inner and outer event horizons experiences an interesting fate — while this region is absent in the extremal case, it does not disappear in the extremal limit but rather approaches a patch of $AdS_2times S^2$. In other words, the approach to extremality is not continuous, as the non-extremal Reissner-Nordström solution splits into two spacetimes at extremality: an extremal black hole and a disconnected $AdS$ space. We suggest that the unusual nature of this limit may help in understanding the entropy of extremal black holes.

Let’s unpack this a little bit. A fascinating property of a charged (“Reissner-Nordström”) black hole is that it has more than one event horizon. In an ordinary uncharged (“Schwarzschild”) black hole, there is a single horizon, corresponding to the point of no return — once you pass the event horizon, you can never escape back to the rest of the world. Inside, there is a singularity, and you are forced to crunch into the singularity in a finite time. In the charged black hole, what we call the outer event horizon (r₊ on the diagram) is the point of no return. But in between the outer event horizon and the singularity is an inner event horizon (r_– on the diagram). It is, again, a point of no return — once you cross the inner horizon, you can’t get back out — but you are not forced to hit the singularity in the middle. Inside the inner horizon, you can avoid the singularity if you wish. On the diagram, we’ve portrayed this by arrows, indicating the direction of “moving forward in time.” Inside the outer horizon, moving forward in time is moving toward the inner horizon, and can’t be helped; but outside the black hole, and inside the inner horizon, moving forward in time looks pretty conventional, and you’re not forced anywhere.

An uncharged black hole has no inner horizon; it should come as no surprise, then, that as you increase the charge (keeping the mass of the black hole fixed), the two horizons at r₊ and r_– come together. In an extremal black hole, where all of the energy comes from the electric field itself, the two horizons coincide: we have

r₊ = r_–        (extremality).

Everyone has known that forever. But Matt, Lisa and I stumbled across an interesting miracle of curved spacetime, which I think some people recognized but certainly isn’t widely appreciated: as you creep up on extremality, increasing the charge of the black hole while keeping its mass fixed, and therefore driving r_– ever closer to r₊, the spacetime volume of the region in between them does not go to zero. It approaches some finite size and stays there.

So we have a region of spacetime of fixed volume, which doesn’t shrink to zero as you increase the charge, but which suddenly disappears entirely when you hit exactly the extremal value. In other words, the limit is discontinuous.

Among ourselves, we referred to this region as “Whoville.” (That didn’t make it into the paper — another reason why blogs are better.) The spacetime geometry of this region looks like the product of a piece of two-dimensional anti-de Sitter spacetime and a two-dimensional sphere.

And … so? Well, mostly we just wanted to point to the interesting feature of the discontinuous limit. But it’s difficult to resist connecting this puzzle with the puzzle of the vanishing entropy. Hawking, Horowitz and Ross found that the entropy of a charged black hole approached a smooth limit as the charge was increased, but discontinuously went to zero exactly at extremality. We found that the volume of spacetime between the two horizons approached a smooth limit as the charge was increased, but discontinuously disappeared exactly at extremality. It certainly suggests a new angle on the behavior of the entropy: maybe HHR were right after all, and the entropy of a precisely extremal black hole really is zero — because the entropy that should be there has escaped into Whoville, this new anti-de Sitter spacetime that is a different part of the discontinuous limit.

Maybe; I wouldn’t stake my life on it at this point, but it’s certainly a viable possibility worth taking seriously. The best argument against it is the kind of microstate counting from string theory pioneered by Strominger and Vafa — they were always looking precisely at the extremal case, and found a non-zero entropy. On the other hand, that technique did always have a step in which you equated the states at very weak coupling (where you could do the counting, but there was no black hole) to those at strong coupling (where you couldn’t directly count, but there is an extremal black hole). It’s long been recognized that one could worry about phase transitions as the coupling was varied — but it got the right answer, so why worry too much? Perhaps the states disappear into Whoville, rather than really describing an extremal black hole.

In the meantime, it’s been fun to think about these things. Despite having written a textbook on general relativity, I had never written a paper directly about black hole physics. Always good to learn new things, and I look forward to learning more.