So it happened again. I got musing to myself about something and decided to do a quick computation to check it out, and it took me down an interesting rabbit hole, which resulted in me writing a nice little paper at the end of last week that appeared today on the arxiv. I think the physics is really really nice. Let me tell you a bit about it. It is in the same area of ideas that I mentioned last time, concerning that paper I wrote last month. So let me pick up the story there, since I did not really touch on the core of the story. *[Note: for non-experts, the following will get somewhat technical and full of terms and ideas that I will not explain. Sorry.]*

One of the things that might have struck you (if you’re an expert in the area) from my proposal to make heat engines out of black holes that do real mechanical work like the engines you read about in physics textbooks is that there ought to be no actual mechanical work since there’s no pistons – no pistons changing volumes and so forth. That is (or rather, was) a missing ingredient in the standard thermodynamics of black holes in quantum gravity. Well, that all changed a short few years ago with the work of a number of authors, particularly with the clear suggestion of David Kastor, Sourya Ray, and Jennie Traschen, and work by Brian Dolan, with a fair bit of followup investigations by various other authors including some I’ll mention below. (Update: Two reviews, with different foci, can be found in here and here.) The general idea is that if you allow the cosmological constant [tex]\Lambda[/tex] to be a thermodynamical variable as well (and there is a long history of authors considering this in various contexts), where it naturally acts like a pressure [tex]p = -\Lambda/8\pi G[/tex], (*G* is Newton’s constant, and I’m setting various other constants to unity in the usual way) then you naturally include a conjugate to that variable that should be the pressure.

For a simple static black hole like Schwarzschild, the volume turns out to the the naive volume you get by taking the radius of the black hole and forming [tex]V=4\pi r_h^3/3[/tex] where [tex]r_h[/tex] is the horizon radius. But… and this is an important point… but that seems to be just a coincidence. Other example have been worked out now, and it is known that this naive geometrical formula for the volume is not correct. (See the paper by Cvetic, Gibbons, Kubiznak, and Pope.)

I’ll call this the “extended black hole thermodynamics”, and the really nice observation by Kastor, Ray, and Traschen is that the usual identification of the mass of a black hole *m/G* with the internal energy *U* in the thermodynamics is incorrect when the cosmological constant is dynamical. It is in fact the enthalpy *H=U+pV* !

This is super-interesting for a lot of reasons, and one of them is that you can think of enthalpy (if you remember your thermo) as the internal energy of the system plus the energy it took to form the system by pushing against the environment (under pressure *p*) to form a volume *V*. This will be important for what I say below when I suggest that the formalism be extended. For the static black hole, that *pV* where the naive thermodynamic volume is used, geometrically looks like you started with empty space and then pushed out a volume *V* to form the black hole, and that’s the story the equations tell. Like I said though, in general it is not so simple – the thermodynamic volume is not the naive geometric volume. In fact, one of the things I present in my recent paper is an extreme case of all this – a non-zero thermodynamic volume when the geometric volume is in fact zero!

Anyway, now you can see that it is this setting that made my Holographic Heat Engines paper very natural. The pressures and volumes in my heat engines were in this context. Morover, the idea of the holographic heat engines, very naturally, comes to life in the context of negative cosmological constant since surely (I suggested in the paper) this extended thermodynamics must have consequences in AdS/CFT and other related holographic dualities? (I’m sure I was not the first to wonder about such consequences. For example, Kastor, Ray, and Traschen, and also Dolan, had begun to try to make connections of this sort in comments in their papers.) Well, you can trace dynamical pressure to dynamically changing *N* in the large *N* gauge theory story, and in my paper I note that there is already a dynamical mechanism that changes *N* – holographic RG flow in the full dual gauged supergravity – and so one can implement holographic heat engines in this way… Fun stuff.

But (although that’s just the tip of what I think is an interesting and important iceberg), I wanted to talk about the paper that I wrote just a few days ago, that is now on the arxiv. I do several things in it, all of which I am excited about, and I won’t be able to do them justice here, but of course that’s why I wrote the thing. You can go and read it.

The chief suggestion overall is that I think that we can’t stop at black holes. It’s bigger than this. The enthalpy suggestion *must* work for any spacetime solution in the quantum gravity story if you have cosmological constant dynamical. This seems natural to me in view of holographic dualities. Moreover, it is not just black holes for which we formulate thermodynamics in gravity. There are other spacetimes that can be assigned intrinsic temperatures, entropies, and masses. The masses of those spacetimes should be declared the enthalpy. Those spacetimes therefore have thermodynamic volumes, and the formula for what that volume is should be worked out. This fits with a picture I have in mind that I think is rather illuminating. One should be able to think of any spacetime solution in terms of the formation process – the internal energy *and* the energy of formation. Now for a lot of solutions, in a random context, that is not going to give you much, but I think that there will be several examples where this tells you a lot about the thermodynamics.

So I worked out two examples in detail, using two systems that are dear to my heart: The Taub-NUT and Taub-Bolt spacetimes. These are such interesting spacetimes for lots of reasons I don’t have time to go into, but the core thing you should know is that they intrinsically define a temperature, a mass, entropy and other thermodynamic quantities due to their geometric properties. (Part of the reason they are dear to my heart is that I keep ending up working on them in one context or another. I had the honour of writing the first paper to introduce Taub-NUT into a string theory context, in a non-trivial way, as the target space of a rather intricate conformal field theory construction 20 years ago. I still love that paper.)

Taub-NUT, the simpler of the two, does not even have something resembling much in the way of an horizon, and overall you should not mistake their thermodynamic properties with those of black holes. They are quite different, in a number of senses. So they are great examples of my suggestion – their masses should be enthalpies in the extended thermodynamics, and the natural question then becomes – What are the thermodynamic volumes for Taub-NUT and Taub-Bolt? I get some very pretty results. One of them is that I’ve discovered more examples where the thermodynamic volume is different from the geometric volume. Taub-NUT is an extreme such example because it has nothing to play the role of the geometric volume, while the thermodynamic volume does exist, and I calculate it. In fact, it is *negative*!

This latter fact puzzled me for a while until I realised that it is very natural. Instead of the system pushing against the environment (the universe) to form, the environment (universe) pushed against it. This is my next point in the paper – once you enlarge the context dynamically, lots of puzzling things (like a negative volume) make a lot of sense, and I explain carefully in the paper presenting my picture of thinking of all spacetime solutions in terms of the story behind their dynamical formation from “nothing”.

I think this is the beginning of a very interesting story which will broaden much of what we know about quantum gravity, and perhaps enrich string theory and even give insights on cosmology, given the importance of the cosmological constant to the whole story.

Have a look at my paper here, if interested. And yes, I know the title is a bit Harry-Potter-esque. (Oh, and the nice diagram I posted recently (see “The Road“) that looks like a road features in the paper too*).

Enjoy!

-cvj

*And I just realised that I did not remove the placeholder caption “the road” from the paper. I should do that when I revise the manuscript later on…

If I may ask, what’s the significance of the musical instrument on the first page of your papers?

Well, I’m not sure there is any significance. I like the trumpet. I used to play a lot, and made a symbol to put on my papers many years ago. I’m just continuing in that tradition I started back then. That’s all really.

Thanks for asking!

-cvj

I think that your paper is stunning. I am still trying to get my head around the aspect of volume in your paper. In particular, trying to understand the rather subtle difference to the geometric volume. Congragulations.

Thank you!

-cvj

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