How Does Relativity of Simultaneity Clash w/Thermodynamics?

[Amaterasu21]

TL;DR Summary

Relativity: observers disagree on the order of events, there is no objective past, present or future and no one "great clock" for the Universe. Thermodynamics: entropy of the Universe as a whole always increases, the future is the direction of time with greater entropy, the past the direction of time with lower entropy. How can both be true?

In special relativity, observers can disagree on the order of events - if Alice thinks events A, B and C are simultaneous, Bob can think A happened before B which happened before C, and Carlos thinks C happened before B which happened before A - provided A, B and C are not causally connected, of course.

Yet the second law of thermodynamics states that the entropy of a closed system - including the Universe as a whole - must increase with time, and is the only known physical law with a direction of time and a distinction between past and future. So how does this square with relativity of simultaneity? If the past is the direction in which the Universe's entropy is lower and the future is the direction in which the Universe's entropy is higher, how can observers disagree about the order of events?

...Actually, while I was writing this out, I thought of a possible resolution - if events A, B and C all lead to an entropy increase, then it doesn't matter what order they occur in - the second law says that entropy must increase going from past to future, but not by how much. If (say) event A leads to an entropy decrease, it must be coupled to an event (say B) that leads to a greater entropy increase - which means they must be causally connected and therefore their order can't be disputed. An event which leads to an entropy decrease and is causally unconnected to a greater entropy increase violates the second law and therefore won't happen. Is this the correct resolution?

 

[Dale]

Relativistic thermodynamics is still rather incomplete. As far as I know there is no generally accepted covariant formulation of thermodynamics.

 

[PeterDonis]

In special relativity, observers can disagree on the order of spacelike separated events

See the bolded addition above. It is important.

the second law of thermodynamics states that the entropy of a closed system - including the Universe as a whole - must increase with time

Which, to be consistent with relativity, requires that we specify what "with time" means. There are at least two possibilities (though, as @Dale points out, relativistic thermodynamics is still incomplete and there is no general consensus):

(1) "with time" means "along any timelike worldline". So any observer will see entropy increasing in his future direction of time for events he observes directly. And every observer will agree that every other observer sees entropy increasing along his worldline. But in this viewpoint, observers will not be able to agree on a single "entropy of the whole universe", because they will not be able to agree on what the surfaces of "constant time" are for the universe as a whole, due to relativity of simultaneity.

(2) "with time" means "along a family of spacelike hypersurfaces that are picked out by some symmetry of the spacetime". For example, in cosmology, this family would be the family of surfaces of constant time in standard FRW coordinates--or, to give a more physical definition, the surfaces of constant time for comoving observers, i.e., observers that see the universe as always homogeneous and isotropic, which are symmetries of this spacetime. In this viewpoint, those surfaces of constant time define the relevant ordering of events for purposes of evaluating the entropy of the whole universe.

Note that the "along any timelike worldline" of viewpoint #1 will still be true even if we adopt viewpoint #2; so the question is really whether we stop at viewpoint #1, meaning we don't accept any particular family of spacelike surfaces as surfaces of "constant time" for the whole universe, or whether we go on to try to adopt some family of spacelike surfaces for viewpoint #2. As noted above, there is no general consensus about this (and also there isn't about whether those two viewpoints are the only possible one, or whether there are others).

 

[Amaterasu21]

See the bolded addition above. It is important.
Which, to be consistent with relativity, requires that we specify what "with time" means. There are at least two possibilities (though, as @Dale points out, relativistic thermodynamics is still incomplete and there is no general consensus):

(1) "with time" means "along any timelike worldline". So any observer will see entropy increasing in his future direction of time for events he observes directly. And every observer will agree that every other observer sees entropy increasing along his worldline. But in this viewpoint, observers will not be able to agree on a single "entropy of the whole universe", because they will not be able to agree on what the surfaces of "constant time" are for the universe as a whole, due to relativity of simultaneity.

(2) "with time" means "along a family of spacelike hypersurfaces that are picked out by some symmetry of the spacetime". For example, in cosmology, this family would be the family of surfaces of constant time in standard FRW coordinates--or, to give a more physical definition, the surfaces of constant time for comoving observers, i.e., observers that see the universe as always homogeneous and isotropic, which are symmetries of this spacetime. In this viewpoint, those surfaces of constant time define the relevant ordering of events for purposes of evaluating the entropy of the whole universe.

Note that the "along any timelike worldline" of viewpoint #1 will still be true even if we adopt viewpoint #2; so the question is really whether we stop at viewpoint #1, meaning we don't accept any particular family of spacelike surfaces as surfaces of "constant time" for the whole universe, or whether we go on to try to adopt some family of spacelike surfaces for viewpoint #2. As noted above, there is no general consensus about this (and also there isn't about whether those two viewpoints are the only possible one, or whether there are others).

Thank you, that makes a lot of sense!

 

[vanhees71]

Relativistic thermodynamics is still rather incomplete. As far as I know there is no generally accepted covariant formulation of thermodynamics.

Hm, using special-relativistic thermodynamics and transport theory all the time, I've the impression that the mess has settled at least since the late 1960ies (van Kampen et al). It's all covariant, beginning from the covariant definition of phase-space distribution functions. The Boltzmann H-theorem is also no quibble due to this covariant formulation. For a review, see my lecture notes here:

https://itp.uni-frankfurt.de/~hees/publ/kolkata.pdf

For a very detailed and clear treatment, see the book by Cercignani cited therein as well as

K. S. Thorne and R. D. Blandford, Modern Classical Physics:
Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical
Physics, Princeton University Press, Princeton, Oxfordshire
(2017). I'm not too famliar with the general-relativistic case though. Are there still issue concerning the H-theorem and the thermodynamic arrow of time vs. the causal arrow of time? Maybe it's an issue for some spacetimes?

 

[Dale]

Thanks, I will check it out. I still see disagreements in the literature so I am not convinced how widespread the acceptance of this approach is, but at least there is some progress being made.

 

[vanhees71]

Can you point me to some paper concerning disagreements?

As I said, I'm pretty sure that there are no issues concerning thermodynamics/transport within special relativity, and at least this scheme, where all thermodynamical quantities like temperature, chemical potentials, internal energy and other thermodynamical potentials (or rather their densities) are defined in the fluid (local) rest frame and thus scalars/scalar fields.

There was a lot of confusion in the early days with different definitions of the thermodynamic quantities involving famous physicists like Planck. As it turns out at in most of the cases the physics content agrees at the end when you interpret the quantities defined in each consistent systematic formulation of thermodynamics right. It's only quite inconvenient to use some of these definitions, where e.g., temperature is treated as some strange time-like quantity but with no spatial component to make complete it to a four-vector, through which procedure the manifest covariance gets lost. There are also some such systems of formulations that lead to wrong conclusions.

For an early review one can take Pauli's famous encyclopedia article.

As I said before, then in the 60ies-70ies the issue was taken up again (maybe also in connection with the then new interest in GR), and the manifestly covariant formalism has been developed. A nice review on the different formulations and their probable shortcomings/advantages from this "modern" point of view can be found here:

N. G. van Kampen, Relativistic thermodynamics of moving
systems, Phys. Rev. 173, 295 (1968),
https://doi.org/10.1103/PhysRev.173.295.

The then apparently new covariant approach is in Sect. 9.

Newer developments in my community (relativistic heavy-ion collisions) involve also derivations of relativistic transport equations from non-equilibrium QFT, e.g., using the Schwinger-Keldysh real-time formalism, which is also manifestly covariant. For a pedagogical review, see

W. Cassing, From Kadanoff-Baym dynamics to off-shell
parton transport, Eur. Phys. J. ST 168, 3 (2009), 0808.0715,
https://doi.org/10.1140/epjst
https://arxiv.org/abs/0808.0715

I'm not aware that there is any disagreement within at least this community of practitioners of relativistic thermodynamics, hydrodynamics (including viscous hydro, which was still an issue when I entered the field in the mid 1990ies but which are now resolved and well developed also applying higher-order moments), transport theory, and manybody theory.

As I said before, I'm not familiar with this status in general relativity. I guess, there may be problems for general spacetimes. The only application of transport theory within GR, I've read about, is the application of transport equations in cosmology (Robertson-Walker-Friedmann-Lemaitre spacetimes) and in simulations of neutron-star mergers in connection with gravitational-wave observations ("kilo novae"), but I'm far from being an expert in these fields.

 

[Dale]

Can you point me to some paper concerning disagreements?

Certainly.

Farías, C., Pinto, V.A. & Moya, P.S. What is the temperature of a moving body?. Sci Rep 7, 17657 (2017). https://doi.org/10.1038/s41598-017-17526-4 "The construction of a relativistic thermodynamics theory is still controversial after more than 110 years. To the date there is no agreement on which set of relativistic transformations of thermodynamic quantities is the correct one"

Afshin Montakhab, Malihe Ghodrat, and Mahmood Barati. Phys. Rev. E 79, 031124 – Published 30 March 2009. Statistical thermodynamics of a two-dimensional relativistic gas. https://doi.org/10.1103/PhysRevE.79.031124 "We also address the controversial topic of temperature transformation. "

Tadas K.Nakamura. "Covariant thermodynamics of an object with finite volume". Physics Letters A. Volume 352, Issue 3, 27 March 2006, Pages 175-177. https://doi.org/10.1016/j.physleta.2005.11.070 "Theory of relativistic thermodynamics has a long and controversial history (see, eg, [1] and references therein). The controversy seems to have been settled more or less by the end of 1960s [2], however, papers are still being published these days "

Malihe Ghodrat Afshin Montakhab. "Molecular dynamics simulation of a relativistic gas: Thermostatistical properties", Computer Physics Communications. Volume 182, Issue 9, September 2011, Pages 1909-1913. https://doi.org/10.1016/j.cpc.2011.01.018 "We briefly outline some of its thermostatistical properties which help resolve controversial issues in relativistic thermodynamics"

C S Lopez-Monsalvo. Covariant Thermodynamics and Relativity. https://arxiv.org/abs/1107.1005 "Does a moving body appear cold? This remarkably simple question, raised by the late Professor P. T. Landsberg some forty years ago, highlights a profound missing link in our current understanding of classical thermodynamics and relativity"

As I said, I'm pretty sure that there are no issues concerning thermodynamics/transport within special relativity, and at least this scheme, where all thermodynamical quantities like temperature, chemical potentials, internal energy and other thermodynamical potentials (or rather their densities) are defined in the fluid (local) rest frame and thus scalars/scalar fields.

Again, as I said above, I still see disagreements in the literature so I am not convinced how widespread the acceptance of this approach is, but at least there is some progress being made.

 

[vanhees71]

Very interesting. I'll have a look at the papers. Of course, it's indeed an issue when it comes to transport simulations (both in the Monte-Carlo and the molecular-dynamics approach) to ensure causality and Poincare covariance, but it's all possible to formulate it in a manifestly covariant way and also to implement it as such. For a review about one relativistic transport code (GiBUU) used in relativistic heavy-ion physics, see also

https://arxiv.org/abs/1106.1344

There are many transport codes in this field. At least in Frankfurt, we have 4 being used in practice: UrQMD, GiBUU, BAMPS, SMASH. All use the manifestly covariant approach, where temperature is a scalar (field).

I'm very surprised that the old non-covariant ideas a la Planck vs. Ott are still discussed. Particularly if you want to generalize these to GR, I'd think you have great conceptual problems. As I said, I have to read the papers to see what the authors conclude about these problems.

 

[Vanadium 50]

The Farías paper is a bit odd, but the first half (where they discuss the history) is quite good.

Anyway, one might say "but of course the phase space transformation formalism is the right way to look at it." But one might also say "but of course internal energy transforms as energy" or various other "of courses".

Fundamentally, the problem is how one determines if two objects are at the same temperature. You can touch them and see that there is no heat flow in either direction. Fine - but what do you do when you have well-separated objects? And moving at high relative velocity too? (That can't help) Non-relativistic thermodynamics was invented around a particular concept (two objects at the same T) that really doesn't exist in relativity.