Temperature as frequency spectrum of stress-energy tensor?

Question

I am currently learning general relativity, and in the textbooks that I am reading, temperature seems to be treated as a scalar field, extraneous to the geometry of spacetime.

This is puzzling me, because I would naively believe that purely geometric data, namely the stress-energy tensor, should be enough to determine temperature, from the following consideration:

In cold matter, nearby particles are moving at low speeds relative to each other, and thus the stress-energy tensor must vary "slowly" across that matter, i.e. it must be "low-frequency"; in hot matter, nearby particles are moving at higher speeds relative to each other, which means that four-momentum varies significantly from one particle to the next, i.e. the stress-energy tensor must vary "quickly" across that distribution of matter, i.e. it must be "high-frequency".

So it occurred to me that perhaps temperature might be just a crude measure of how "high-frequency" the stress-energy tensor field is, across a small region of spacetime around a given event?

Note that this wouldn't contradict statements about how no "new" scalar field can constructed from spacetime geometry, because these statements are about purely local constructions (e.g. contract the Ricci curvature tensor at each point to define scalar curvature), and the construction that I am talking about here is not purely local, as one needs to perform frequency analysis of the stress-energy tensor over a small region of spacetime to define how "high-frequency" it is. Accordingly, temperature is not purely local, since in a region of spacetime small enough to contain only one particle, temperature should be undefined (correct?).

Could anyone here either point out the first mistake in my reasoning here, or mention some literature that I should read on this topic?

In particular, I don't really know precisely how to make sense of "the frequency spectrum of the stress-energy tensor over a region of spacetime", or even what kind of mathematical object that would be, and would be very interested to read any text clarifying that.

My other question, then, would be: if temperature is just a crude measure of how high frequencies some kind of spectrum of the stress-energy tensor contains, then there must exist situations where talking of temperature as a scalar is inappropriate, and one has to account for a "superposition of temperatures"? I suppose that the implicit approximation usually made is to account only for the highest frequency in the stress-energy spectrum as temperature, but surely if there is another, slightly lower frequency with a sufficiently higher amplitude there, it should be able to have significant temperature-like effects to forbid ignoring it as a temperature?

Answer 1

"Thermodynamics of continuum" is the discipline that defines the temperature as a classical field i.e. a function of the coordinates $(x,y,z,t)$ that parameterize a continuum (solid, liquid, gas, plasma) and its evolution in time.

(Temperature cannot be a quantum field because it doesn't really correspond to any operator on the Hilbert space. In other words, whenever we take the thermodynamic limit so that the temperature is well-defined, the limiting procedure automatically removes the quantumness of the ensemble of atoms, too.)

Such a field can never be "one of the elementary fields" that may fully define a theory. The reason is that in general, the temperature isn't well-defined. It is only well-defined at equilibrium. So instead of thinking of temperature as a function of the configuration (or quantum state), one should think in the opposite way: the configuration or the quantum state (or its local behavior) is a function of the temperature!

So the temperature is something that determines the state of a physical object or all local properties of the state of a medium. The probability distribution or the density matrix behave like $C\exp(-E/kT)$ where $E$ is either the classical energy or the Hamiltonian in quantum mechanics.

All these comments are compatible with the fact that the temperature cannot be quite well-defined in arbitrary small regions of space, for arbitrarily small ensembles of atoms etc. The temperature isn't a well-defined function of any state; on the contrary, some special states may be defined as functions of the temperature. But they are really very special states, only. Physically, they correspond to states at equilibrium or (local) quasi-equilibrium.

One may also visualize the temperature as the (inverse) periodicity of the Euclidean time circle (the dual interpretation is that the imaginary total energy takes on quantized values i.e. integers in some way, but that's useless physically because the total energy isn't imaginary) but it's still true that one should start with the temperature and get a (special) state, instead of imagining that one starts with a general state and calculates the temperature. If the temperature is treated via this "thermal circle", we are abandoning the ordinary time coordinate in favor of the periodic Euclidean time coordinate. The normal time disappears because the states with well-defined temperatures are equilibrium states so they can't have any well-defined dependence on time. Local quasi-equilibrium states may evolve (heat conduction described by diffusion equation etc.) but all these equations are just derived approximate equations capturing some thermal properties of the medium "almost right".

The periodicity of the Euclidean time is $\Delta t_E=\hbar\beta=\hbar /kT$ so formally, if the energy could be imagined to be imaginary, it would be quantized in the units of $2\pi \hbar / \Delta t_E = 2\pi\hbar / \hbar\beta = 2\pi kT$. What this "quantization" really means is ill-defined but of course the characteristic scale of energy connected with the temperature $T$ is a number of order $kT$ where $k$ is the Boltzmann constant. I suspect this is the only sensible description similar to the "high-frequency" comments by the OP.

Answer 2

For temperature to make sense the system must be in approximate local thermal equilibrium. In this case the energy momentum tensor is (approximately) of the ideal fluid form $$T_{\mu\nu}=(\epsilon+P)u_\mu u_\nu + P g_{\mu\nu}\, . $$ This means that that $T_{00}$ in the local rest frame is the energy density of the matter (indeed, this fact is not modified by dissipative corrections to the ideal fluid form). If you know the equation of state $\epsilon=\epsilon(T)$ then you can construct $T$ from $\epsilon$. If you do not know the equation of state then you can reconstruct it using thermodynamic identities like $dP=sdT$. Note that dimensionally $\epsilon\sim T^4$, so $T$ cannot transform as a simple scalar field.