[raattgift]

> Further, I think relativity still has an "arrow of time". Thermodynamic quantities can transform under reps of Poincare group - "Relativistic Thermodynamics" is certainly a thing. Landau and Lifshitz has a few chapters on it I believe. So the idea of time "passing" (entropy increasing at least carries over.

I'd be grateful if you could show me where in §27 (PDF pages 94-96) of http://people.physics.tamu.edu/kcolletti1/classes/fall15/sta... you get any of that from.

> "Relativistic Thermodynamics" is certainly a thing

Is it a thing which really deals with an arrow of time, or is it a thing which lets one study the temperature of a moving object, the temperature of an object in a gravitational field [Tolman], or the temperature of an object in which individual velocities (of fluid elements or particles) are large [Israel & Stewart]?

[foxes]

Actually, yes that book doesn't mention that entropy is Lorentz invariant, but I think there are papers on it [0].

However it doesn't actually look like they reach a decisive conclusion - it is still for debate.

[0] http://iopscience.iop.org/article/10.1088/0305-4470/38/13/00... [1] https://arxiv.org/pdf/1802.07650.pdf

[raattgift]

This seems to be going off into the weeds.

I think what you might be trying to argue is that in vacuum spacetime with a horizon there is a Gibbons-Hawking temperature T_{GH}. A Eulerian observer with a purely timelike worldline could measure T_{GH} at various points along its worldline. In de Sitter space the temperature is proportional to the Hubble parameter H.

Unfortunately, there is afaik no consistent or complete microphysical description of the Gibbons-Hawking effect. Additionally, it's so small that the presence of a relic matter field (like the cosmic microwave background) completely swamps it by many orders of magnitude (for H_0 it's 30 orders). So T_{GH}, purely general-relativistic, does not seem like a useful arrow of time in our universe until the latest of epochs maybe.

Look instead to the Boltzmann entropy of the matter; we can simply set a low-entropy boundary condition somewhere in spacetime, and then we do not even need to have an expanding universe to have higher entropy in spacetime at increasing distance from that boundary. Alternatively, we could go through some hoops to try to remove the boundary, as Hartle & Hawking among others.

Carroll has a list of relevant slides at https://www.slideshare.net/seanmcarroll/what-we-dont-know-ab...

None of this has much if any contact with relativistic thermodynamics. In cosmology we are almost always deliberately considering a family of privileged observers and ignoring the others (even though they exist). Our defined cosmological observers see matter (incl. radiation and any horizon radiation) as homogeneous and isotropic. Any expansion or contraction of space is an adiabatic process. \Lambda is nearly zero, but still sufficient to generate enormous spacetime curvature (the purely timelike worldlines of our observers are not parallel).

Finally, calculating in the cosmological frame is a convenience, nothing more.