News from the Front, XVII: Super-Entropic Instability

I’m quite excited because of some new results I got recently, which appeared on the ArXiv today. I’ve found a new (and I think, possibly important) instability in quantum gravity.

Said more carefully, I’ve found a sibling to Hawking’s celebrated instability that manifests itself as black hole evaporation. This new instability also results in evaporation, driven by Hawking radiation, and it can appear for black holes that might not seem unstable to evaporation in ordinary circumstances (i.e., there’s no Hawking channel to decay), but turn out to be unstable upon closer examination, in a larger context. That context is the extended gravitational thermodynamics you’ve read me talking about here in several previous posts (see e.g. here and here). In that framework, the cosmological constant is dynamical and enters the thermodynamics as a pressure variable, p. It has a conjugate, V, which is a quantity that can be derived once you know the pressure and the mass of the black hole.

Well, Hawking evaporation is a catastrophic quantum phenomenon that follows from the fact that the radiation temperature of a Schwarzschild black hole (the simplest one you can think of) goes inversely with the mass. So the black hole radiates and loses energy, reducing its mass. But that means that it will radiate at even higher temperature, driving its mass down even more. So it will radiate even more, and so on. So it is an instability in the sense that the system drives itself even further away from where it started at every moment. Like a pencil falling over from balancing on a point.

This is the original quantum instability for gravitational systems. It’s, as you probably know, very important. (Although in our universe, the temperature of radiation is so tiny for astrophysical black holes (they have large mass) that the effect is washed out by the local temperature of the universe… But if the univverse ever had microscopic black holes, they’d have radiated in this way…)

So very nice, so very 1970s. What have I found recently?

A nice way of expressing the above instability is to simply say that the system’s specific heat is negative. You get energy out of the system and the temperature goes up. Very strange.

In extended thermodynamics, the instability I talked about above translates into a specific heat at constant pressure, p. If you’ve done any thermodynamics you might know that there is a sibling (arguably more important) called the specific heat at constant volume. You might, as I did, start wondering about properties of this quantity for various reasons (see my previous post on why and what that led to). It turns out that it is harder to extract, but you can do it. In fact, there is a really simple example for which it can be done exactly, and so I did, and I noticed to my surprise that it was negative, and with a similar kind of temperature dependence as the familiar Hawking example! It is unstable to evaporation!

All other examples I’d looked at had C_V (is it is called) either zero or positive, in which case there’s no instability. I was going to leave it as a footnote in my last paper, and update the paper before (also about specific heats) that with a little note about the computation, but it kept bugging me as a nice result that should not be buried. (Also, the exact example is a charged version of the so-called “BTZ blackhole”, which is a famous and important solution that lots of people care about and work on, so that’s also maybe a reason to not bury the observation.)

I’d intended to point out in the footnote that the instability is analogous to Hawking’s when I realised that (a) while that’s true, it is a completely new phenomenon in a new context, and (b) it could be the first example of many, and if so, could be a useful tool for studying other solutions of gravity in the extended thermo context, helping determine if they are stable or not.

A little bit of thought later, I realised that the instability links nicely to another observation in the literature that had not received any explanation. The solution I was working on is what’s called a “super-entropic” black hole. Such black holes have more entropy than a certain bound observed for most black hole solutions: For the value of their thermodynamic volume, they have more entropy than a Schwarzschild black hole with the same thermodynamic volume would have. They are said to violate the “reverse isoperimetric inequality”¹, if you want to tell that to your friends in the pub tonight to impress them. That such black holes solutions can exist had been noted (charged BTZ is one of them – the simplest, arguably), but it was never clear whether such a bound is a physical restraint or not.

Well, it did not take me more than a few lines of algebra to show that the negativity of the specific heat is exactly controlled by the super-entropicity of the black hole! So then I had a natural question. Are all super entropic black holes unstable? How do I prove that? Well, the first thing to do when you have a conjecture is to test it on a few examples. I looked up some more complicated examples in the literature² and did some numerical extraction of the specific heat and lo and behold, they have the instability too! (In fact I think an infinite family of them does). So all of a sudden my exciting little result got enlarged to include a conjecture about a larger issue: Super-entropic black holes are unstable. The very nature of the setup means that a general proof is hard to construct, but I have a few arguments as to why the two phenomena are likely connected.

Anyway I wrote it all up in a tiny tiny little paper you can read in almost one breath, because I know you’re busy. Abstract pictured above. Paper here.

-cvj

P.S. Yes, of course I’ve wondered if the positive cosmological constant version of this might be useful for studying our own actual universe. We shall see.

¹See this paper by Cvetic et. al. for more on that.
²They were constructed by Hennigar et. al. here.