Spinor Info

I remain troubled by this business with black holes.

In particular, the zeroth law. Many authors, such as Wald, say that the zeroth law states that a body’s temperature is constant at equilibrium. I find this formulation less than satisfactory. Thermodynamics is about equilibrium systems to begin with, so it’s not like you have a choice to measure temperatures in a non-equilibrium system; temperature is not even defined there! A proper formulation for the zeroth law is between systems: the idea that an equilibrium exists between systems 1 and 2 expressed in the form of a function f(p₁, V₁, p₂, V₂) being zero. Between systems 2 and 3, we have g(p₂, V₂, p₃, V₃) = 0, and between systems 3 and 1, we have h(p₃, V₃, p₁, V₁) = 0. The zeroth law says that if f(p₁, V₁, p₂, V₂) = 0 and g(p₂, V₂, p₃, V₃) = 0, then h(p₃, V₃, p₁, V₁) = 0. From this, the concept of empirical temperature can be obtained. I don’t see the analog of this for black holes… can we compare two black holes on the basis of J and Ω (which take the place of V and p) and say that they are in “equilibrium”? That makes no sense to me.

On the other hand, if you have a Pfaffian in the form of dA + B dC, there always exists an integrating denominator X (in this simple case, one doesn’t even need Carathéodory’s principle and assume the existence of irreversible processes) such that X dY = dA + B dC. So simply writing down dM – Ω dJ already gives rise to an equation in the form X dY = dM – Ω dJ. That κ and A serve nicely as X and Y may be no more than an interesting coincidence.

But then there is the area theorem such that dA > 0 (just like dS > 0). Is that another coincidence?

And then there is Hawking radiation. The temperature of which is proportional to the surface gravity, T = κ/2π, which is what leads to the identification S = A/4. Too many coincidences?

I don’t know. I can see why this black hole thermodynamics business is not outright stupid, but I remain troubled.

I’m reading Robert Wald’s book, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, and I am puzzled. According to Wald, the black hole equivalent of the First Law reads (for a Kerr black hole):

(1/8π)κdA = dM – ΩdJ,

where κ is the surface gravity, A is the area of the event horizon, M is the mass, Ω is the angular velocity of the event horizon, and J is the black hole’s angular momentum.

The analogy with thermodynamics is obvious if one write the First Law as

TdS = dU + pdV,

where T is the temperature, S is the entropy, U is the internal energy, p is the pressure, and V is the volume. Further, as per the black hole area theorem, which Wald proves, A always increases, in analogy with the thermodynamical entropy.

But… if I am to take this analogy seriously, then I am reminded of the fact that in a thermodynamical system the temperature is determined as a function of pressure and volume, i.e., there is a function f such that T = f(p, V). Is there an analogue of this in black hole physics? Is the surface gravity κ fully determined as a function of Ω and J? It is not obvious to me that this is the case, and Wald doesn’t say. Yet without it, there is no zeroth law and no thermodynamics. He does mention the zeroth law in the context of a single black hole having uniform surface gravity, but that’s not good enough. It doesn’t tell me how the surface gravity can be calculated from Ω and J alone, nor does it tell me anything about more than one black hole being involved, whereas in thermodynamics, the zeroth law is about multiple thermodynamical systems being in thermal equilibrium.

Another puzzling aspect is that the area theorem has often been quoted as “proof” that a black hole cannot evaporate. Yet again, if I take the analogy with thermodynamics seriously, the Second Law applies only to closed systems that exchange neither matter nor energy with their environment; it is, in fact, quite possible to reduce S in an open system, otherwise your fridge would not work. So if a black hole can exchange energy and matter with its environment, perhaps it can evaporate after all.

Moreover, for the analogy to be complete, we’d also be required to have

8π∂M/dA = κ,
∂M/∂J = Ω,

just as in ordinary thermodynamics, we have T = ∂U/∂S and p = –∂U/∂V. So, do these relationships hold for black holes?

I guess I’ll go to ArXiv and read some recent papers on black hole thermodynamics.

I’m reading a 40-year old book, Methods of Thermodynamics by Howard Reiss. I think I bought it after reading a recommendation on Amazon.com, describing this book as one of the few that takes the idea of axiomatic thermodynamics seriously, and treats it without mixing in concepts from statistical physics or quantum mechanics.

It is a very good book. Not only does it deliver on its promise, it also raises some issues that would not have occurred to me otherwise. For instance, the idea that a so-called equation of state does not fully describe the state of a material, even an ideal gas. You cannot derive U = C_vT from the equation of state. You cannot that the internal energy U is a linear function of the temperature T, it has to be postulated.

One thing you can derive from the ideal gas equation of state alone is that an adiabatic expansion must be isothermal. As an ideal gas expands and its volume increases while its pressure decreases, its temperature remains constant. It also made me think again about the cosmological equation of state… cosmologists often play with idealized cases (e.g., dust-filled universe, radiation-filled universe) but until now, I never considered the possibility that even in these idealized cases, the equations of state do not full describe the stuff that they supposedly represent.

In two days, I got two notices of papers being accepted, among them our paper about the possible relationship between modified gravity and the origin of inertia. I am most pleased, because the journal accepting it (MNRAS Letters) is quite prestigious and the paper was a potentially controversial one. The other paper is about Pioneer, and was accepted by Physical Review D. Needless to say, I am pleased.