These days, everybody is talking about entropy. In fact, there is so much talk about entropy I am waiting for a Hollywood starlet to name her daughter after it. To help that case, today a contribution about the entropy of black holes.

To begin with let us recall what entropy is. It’s a measure for the number of micro-states compatible with a given macro-state. The macro-state could for example be given by one billion particles with a total energy E in a bag of size V. You then have plenty of possibilities to place the particles in the bag and to assign a velocity to them. Each of these possibilities is a micro-state. The entropy then is the logarithm of that number. Don’t worry if you don’t know what a logarithm is, it’s not so relevant for the following. The one thing you should know about the total entropy of a system is that it can’t decrease in time. That’s the second law of thermodynamics.

It is generally believed that black holes carry entropy. The need for that isn’t hard to understand: if you throw something into a black hole, its entropy shouldn’t just vanish since this would violate the second law. So an entropy must be assigned to the black hole. More precisely, the entropy is proportional to the surface area of the black holes, since this can be shown to be a quantity which only increases if black holes join, and this is also in agreement with the entropy one derives for a black hole from Hawking radiation. So, black holes have an entropy. But what does that mean? What are the microstates of the black hole? Or where are they? And why doesn’t the entropy depend on what was thrown into the black hole?

While virtually nobody in his right mind doubts black hole have an entropy, the interpretation of that entropy is less clear. There are two camps: On the one side those who believe the black hole entropy counts indeed the number of micro-states inside the black hole. I guess you will find most string theorists on this side, since this point of view is supported by their approach. On the other side are those who believe the black hole entropy counts the number of states that can interact with the surrounding. And since the defining feature of black holes is that the interior is causally disconnected from the exterior, these are thus the states that are assigned to the horizon itself. These both interpretations of the black hole entropy are known as the volume- and surface-interpretations respectively. You find a discussion of these both points of view in Ted Jacobson’s paper “On the nature of black hole entropy” [gr-qc/9908031] and in the trialogue “Black hole entropy: inside or out?” [hep-th/0501103].

A recent contribution to this issue comes from Steve Hsu and David Reeb in their paper

Black hole entropy, curved space and monsters
Phys. Lett. B 658:244-248 (2008)
arXiv:0706.3239v2

Steve is a neighbor here on blogspot over at Information Processing. In their paper Steve and David examine the question how much matter one can stuff into a volume bounded by a given surface, and how much entropy this matter can carry. In flat space-time the relation between the volume of an area and its surface is trivial, it’s just Euclidean geometry. But not so if space-time is strongly curved!

To see this, consider the often made analogy of a curved space to a rubber sheet. Draw a circle on it. That’s your surface. But it’s a rubber sheet, meaning you can deform the sheet inside the circle arbitrarily. You could for example form it to a bag and stuff a lot of gold into it.

This pictorial terminology is sadly not my invention: these kind of solutions have been known to be possible in General Relativity for a long time, and have been dubbed “bags of gold” by Wheeler already in the early 70s. Their defining property is that they have a potentially arbitrarily large interior volume, but a small surface area.

Steve and David in their paper now construct a weird kind of solution they dub “monsters,” which exemplifies what one can do with these bags. To understand what a monster is, consider some stuff (eg coins of gold) dispersed in space-time, such that the background is to good approximation flat. Now pick up these coins and put them closely together - so close that they almost, but not entirely, form a black hole. What you achieve in this way is that you get a strong gravitational field and a deviation of the volume-surface relation from flat space. That process of picking up and redistributing the coins should not be thought of as a process that is actually dynamically happening, but just as a way to create the initial conditions*. If you create these initial conditions carefully you can achieve most importantly two things:

1. You can get the asymptotic mass a far away observer would measure (ADM mass) to be arbitrarily small, no matter how many coins you have had. The reason for this is that the strong gravitational field contributes with a negative binding energy.
2. You can similarly get an arbitrarily large entropy inside a sphere with fixed surface area, think of the coins as the particles forming a particular micro-state. The reason is that the volume can get arbitrarily large, and you can stuff all the coins in, even though the surface area and the asymptotic mass might remain small.

The authors also show in their paper that if you create the monster state and let it evolve in time, it inevitably forms a black hole. Since it can have been arbitrarily close to being a black hole, it is plausible to expect that almost all of this entropy goes into the black hole. If the volume interpretation of the black hole entropy was correct, this would be in conflict with it. Weirder than that, the monster solution must have come out of a white hole in the past. This solution is thus very similar to an expanding and re-collapsing closed FRW universe embedded in empty space.

Despite these monster solutions existing in GR, there remains the question however whether they do exist in reality, since they are somewhat pathological and constructed. Though it might be possible to argue these states will never be formed from any sensible initial condition, in a quantum theory the situation is more tricky since everything that can happen does happen - even though it might be very improbable. That means the monsters could be spontaneously formed through tunneling processes. That might however in practice not happen even once during the lifetime of the universe.

Steve was visiting PI in November and gave a very clear talk about the monsters, that is recommendable if you want to know more details. You can find it at PIRSA 08110026 and the slides are here.

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* You shouldn’t take the picture too literally though, much like in the often used example with the marble on the rubber-sheet it is slightly misleading as there isn’t actually something “on” the spacetime (the sheet) that extends into an additional dimension.