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?