*(Note: I’ve made some updates because I’ve refined the physics.)*

Now let us turn to de Sitter black holes. I mean here any black hole for which the asymptotic spacetime is de Sitter spacetime, which is to say it has positive cosmological constant. This is of course also interesting since one of the most natural (to some minds) possible explanations for the accelerating expansion of our universe is a cosmological constant, so maybe all black holes in our universe are de Sitter black holes in some sense. This is also interesting because you often read here about explorations of physics involving negative cosmological constant, so this is a big change!

One of the things people find puzzling about applying the standard black hole thermodynamics is that there are two places where the standard techniques tell you there should be a temperature associated with them. There’s the black hole horizon itself, and there’s also the cosmological horizon. These each have temperature, and they are not necessarily the same. For the Schwarzschild-de Sitter black hole, for example, (so, no spins or charges… just a mass with an horizon associated with it, like in flat space), the black hole’s temperature is always larger than that of the cosmological horizon. In fact, it runs from very large (where the black hole is small) all the way (as the black hole grows) to zero, where the two horizons coincide.

You might wonder, as many have, how to make sense of the two temperatures. This cannot, for a start, be an equilibrium thermodynamics system. Should there be dynamics where the two temperatures try to equalise? Is there heat flow from one horizon to another, perhaps? Maybe there’s some missing ingredient needed to make sense of this – do we have any right to be writing down temperatures (an equilibrium thermodynamics concept, really) when the system is not in equilibrium? (Actually, you could ask that about Schwarzschild in flat space – you compute the temperature and then discover that it depends upon the mass in such a way that the system wants to move to a different temperature. But I digress.)

The point of my recent work is that it is entirely within the realm of physics we have to hand to make sense of this. The simple system described in the previous post – the three level maser – has certain key *interconnected* features that seem relevant:

- admits two distinct temperatures and
- a maximum energy, and
- a natural instability (population inversion) and a channel for doing work – the maser output.

My point is that these features are all present for de Sitter black holes too, starting with the two temperatures. But you won’t see the rest by staring at just the Schwarzschild case, you need to add rotation, or charge (or both). As we shall see, the ability to reduce angular momentum, or to reduce charge, will be the work channel. I’ll come back to the maximum energy (a crucial piece of the story) issue in a bit.

Let’s take the case of having some rotation – the Kerr de Sitter black hole. The rotation parameter is often called a, and the cosmological constant is the Greek symbol capital Lambda. There is a parameter called m in the solution as well. Some combination of these (read the paper) gives angular momentum J and mass M. I spoke the names of the parameters a, Lambda and m because there is a nice representation of the entire parameter space of physical solutions that occupies the (a/m, m^2 Lambda) plane. It is given in the figure to the left. The left hand line segment is the case of Schwarzschild, running from small mass (large temperature) to large mass where the two horizons coincide, giving zero temperature. Along the bottom is the physics of the flat space case with the a/m=1 case being the largest spin/mass ratio, the famous extremal case, again with temperature T=0. This connects to the right hand line segment which is the generalization of the extremal case to non-zero Lambda. This T=0 locus connects to the T=0 locus (coincident horizons) running along the top. Red interior region is where the black hole has greater temperature than the cosmological horizon. Blue the opposite. The region separating them is called the lukewarm region.

The proposal here is that there is a sensible thermodynamics that is comprised of thermally driving the Kerr-dS system by connecting to heat baths at a high and low temperature. There is a channel for doing work, and the resulting system is a continuous heat engine entirely analogous to the maser. I’ve put a diagram on the right that should remind you of the diagram I drew in the previous post, organising the three-level maser’s physics. The point here is to show the key elements that they share. (The shading is to indicate a continuum of states, emphasising that I don’t think that de Sitter is a simple three level system. I just need that highest energy level that’s indicated, which I will discuss below.)

The first law of thermodynamics for this system comes in two parts, one for the cosmological horizon, and one for the black hole horizon. It turns out that when you write the sum of the two (see Dolan et. al. – more refs in paper) you get (roughly), *at constant mass/energy*:

T_H dS_H = T_C dS_C – (Omega_H + Omega_C) dJ

where H refers to black hole horizon and C refers to cosmological horizon. (Omega is the angular velocity at an horizon.) If I stay in the red region of the diagram for now (see more in the paper), T_H and T_C also refer to the hot and cold reservoir temperatures.

Now, the term on the left hand side should be interpreted as the heat flowing into the system, via the hot reservoir, making the black hole horizon increase in area. Heat flowing out of the system is via the cold reservoir, and makes the cosmological horizon ~~decrease~~ increase in area. Work is the decrease of angular momentum J. So, overall, the first Law is:

Q_H = Q_C + W.

This is the conversion of input heat Q_H into work W, with exhaust heat (Q_C), so very familiar from engines! But instead of going in a loop, returning the internal energy to its starting value every time (a reciprocating engine), this is a continuous engine, and so the energy of the system stays fixed (hence the constant mass/energy remark above), while there is heat flowing in and out and work being done every instant. The case of J=0 is the Schwarzschild situation, and we have just heat flow in and heat flow out: Q_H = Q_C, and so the mass/energy stays fixed.

I should pause here to head off a bit of confusion. This is not a closed system, with a flow of energy from the black hole out to the cosmological horizon, making it evaporate away. That’s something I want to come to, but not what we’re doing here. This is a driven system, with heat flowing to and from the external reservoirs making our continuous heat engine. I expect that the evaporation of the black hole is more like what happens when you turn off the maser setup and let everything decay away.

So the bottom line is that, as we’ve set it up, we’ve got a system with heat being converted to work, with some possible heat exhaust. In such a case, because the total entropy must increase (or not change, if I do a reversible process), there is a particular bound on the efficiency of conversion to work, W/Q_H = 1- Q_C/Q_H. It is the Carnot limit, 1-T_C/T_H.

In the paper, I actually formulate this bound in a strong way and a weak way. I prefer the strong, and conjecture that it might hold for de Sitter holes quite generally. It says that trajectories through parameter space must obey this bound instantaneously, which is to say that it is a condition on the instantaneous flow of work and heat, which amounts to a condition on the derivative of J with respect to entropy. I won’t write it here, but you can go and look in the paper. The weaker bound might be more appropriate though – we shall see.

The general statement I make is that even if you have a more complicated black hole solution in de Sitter, there will be a notion of this continuous heat engine setup, and it’ll be **constant mass curves**, allowing for heat to flow in and out, with work making up any deficit. The work can come from reducing angular momentum, or other conserved quantities like charge, etc. I’ve noticed (this an update) that Kerr and Reissner-Nordstrom are in a very definite sense too simple to show all the structure that is possible. The constant mass curves (examples right, for Kerr-dS) are such that, when you examine the heat flows, for positive work and positive Q_H, the heat flow Q_H is negative, which means both reservoirs are contributing energy to convert to work. So they avoid the Carnot bound, cleverly, by having exactly the same input heat as work. This is not a violation of the second law (as I thought at first) because the system is not going in a closed cycle. You *can* turn heat into work from (effectively) one hot reservoir with efficiency unity as long as you don’t end up in the same state at the end. Consider the isothermal expansion of an ideal gas, for example. But I believe this is only a special circumstance. I think more complicated black holes will allow for more general behaviour, and it would be nice to find examples.

It is worth noting something key here. Whether you care about the maser aspect or not, the scheme of running as a continuous heat engine is robust. The combined first law of thermodynamics that I wrote above applies as you move around parameter space, and the various terms have their interpretation as heat flow in or out of either horizon, and work done by the system. These interpretations should be correct regardless of the underlying microscopic interpretation. But the second law of thermodynamics should apply as well, and this means that it limits the conversion of heat into work (depending upon he direction of the heat flows). This is the origin of the Carnot limit on heat engines, whether continuous or reciprocating (i.e. the complementary holographic heat engine construction for AdS holes).

So, this scheme I’ve presented ought to be useful for ruling out (isomass) trajectories in de Sitter black hole parameter space that correspond to conversion of heat into work with too high efficiency. I think it would be interesting to study more such systems.

Finally, turning back to the maser story, I ought to address the crucial existence of a maximum energy. It is a maximum energy that individual underlying degrees of freedom, whatever they are, carry. There’s a way of seeing that without knowing the full underlying description, and in this gravity context I uncovered this phenomenon in an earlier project. The point is that I uncovered something rather striking about certain gravity solutions. If you compute the temperature dependence of the specific heat associated with the total entropy, it grows from zero temperature, peaks, and then falls off to zero at higher temperature. This phenomenon in condensed matter physics is called a Schottky peak (see figure on the right), and it is a well-known signal that none of the accessible degrees of freedom can occupy energies beyond some maximum energy scale. Even if you don’t know the underlying microscopic physics in detail (and in general, we certainly don’t, for de Sitter physics), this behaviour of C(T) is a clear signal of this. Well, I found this for certain charged and rotating black holes in anti-de Sitter spacetime, in a certain sector, and the other point of my recent paper was to show that rotating and charged de Sitter black holes have it as well – maybe even more directly because it is in standard black hole thermodynamics that it shows up or de Sitter, you don’t have to go to the so-called “extended” thermodynamics to see it as you do for anti-de Sitter. It’s there even if you don’t have rotation and charge, by the way (another group of researchers independently showed that too), and to my mind, this is a signal that there’s a fundamental “highest energy” cutoff lurking in the physics of cosmological constant that may be as universal as holography turned out to be for negative cosmological constant…

But I’m getting off track here. The really key point is that there’s the extra element that we need – a maximum energy that degrees of freedom can occupy (see that maser-like diagram above). Looking back at that list of key things that a maser system needs, and you’ll see that we’re got all of them in play for de Sitter black holes.

**While I don’t have a proof, I don’t think that’s an accident!**

Ok, I’d better stop here and get back to work. Some takeaways for future research include:

- Uncovering Schottky behaviour for other black hole systems, and finding ways to model it.
- Working out the “maser” story for other black holes besides Kerr (I’ve done the charged, Reissner-Nordstrom, case), including finding non-trivial examples that have flows of heat in and out.
- Recall my comments above about this scheme applying more generally to trajectories in de sitter black hole parameter space, maser interpretations aside. There’s broader work to be done here, exploring/constraining other trajectories in other schemes of heat flow and work.
- Exploiting and seeing the broader consequences of the observation that de Sitter black holes have a maximum energy for the degrees of freedom. I suspect that this might even tell us something about the actual universe we live in! I now realize that this connects to remarks about de Sitter entropy that Tom Banks used to say all those years ago c. 2000.

There’s a longer list in the paper. (Look for an update on Monday) Enjoy!

-cvj