Is there a thermal equilibrium in General Relativity?

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Discussion Overview

The discussion revolves around the concept of thermal equilibrium in the context of General Relativity (GR), particularly in relation to the Friedmann-Robertson-Walker (FRW) cosmological model. Participants explore the implications of the lack of a timelike Killing vector and its relationship to equilibrium states, as well as the nature of gravitational entropy.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant notes that the absence of a timelike Killing vector in the FRW model suggests that the universe cannot be in thermal equilibrium, but they seek clarification on how this conclusion is reached.
  • Another participant proposes that equilibrium in GR might refer to maximizing the entropy of the gravitational field, suggesting that a state of equilibrium could consist of a single large black hole.
  • Some participants argue that the lack of a timelike Killing vector indicates that the system is not stationary, making it difficult to conceive of equilibrium in a constantly changing context.
  • A counterexample is provided, where a moon orbiting a planet lacks a timelike Killing vector yet could be considered in equilibrium, indicating that the relationship between Killing vectors and equilibrium may not be straightforward.
  • One participant expresses uncertainty about the meaning of thermal equilibrium in GR, suggesting that it is a subtle topic even in special relativity.
  • Another participant references a paper that may provide additional insights into the entropy of the universe.
  • One participant challenges the idea that black holes represent a final equilibrium state, arguing that their evaporation due to Hawking radiation complicates the notion of equilibrium in an expanding universe.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the lack of a timelike Killing vector for thermal equilibrium, with no consensus reached on the nature of equilibrium in GR or the role of black holes in this context.

Contextual Notes

The discussion highlights the complexity of defining thermal equilibrium in GR, particularly in relation to gravitational entropy and the behavior of black holes, without resolving the underlying assumptions or definitions involved.

evilcman
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I am reading the book The Early Universe by Kolb and Turner and found the following:
"In the strictest mathematical sense the universe cannot be in thermal equilibrium, as the FRW cosmological model does not possesses a time-like Killing vector".

It is just a matter of calculations to show that the Killing-equation does not have a
time-like solution in the Robertson-Walker metric, but I don't know how this implies anything
about thermal equilibrium. The books I have do not treat relativistic thermodynamics,
can someone explain this or point me to a good reference?
 
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They probably mean equilibrium not in the sense of maximizing the entropy of the matter fields but of maximizing the entropy of the gravitational field.

I think the equilibrium state of GR consists of one big black hole. It's the opposite of an ordinary ideal gas, where the equilibrium has the particles spread out uniformly.
 
The lack of a timelike Killing vector means that the system is not stationary in any reasonable sense. It is hard to imagine how something could be in equilibrium if it is constantly changing.
 
Stingray said:
The lack of a timelike Killing vector means that the system is not stationary in any reasonable sense. It is hard to imagine how something could be in equilibrium if it is constantly changing.

Well, suppose you have a system consisting of a moon orbiting around a planet. It doesn't have a timelike Killing vector, but I think it's reasonable to say that it's in equilibrium.
 
I agree, but it's much more difficult in the cosmological cases when you do have a full complement of (spatial) translational and rotational Killing vectors. In any case, I have no idea what thermal equilibrium means in general relativity, and my impression is that nobody else does either. It's rather subtle even in special relativity.
 
bcrowell said:
They probably mean equilibrium not in the sense of maximizing the entropy of the matter fields but of maximizing the entropy of the gravitational field.

I think the equilibrium state of GR consists of one big black hole. It's the opposite of an ordinary ideal gas, where the equilibrium has the particles spread out uniformly.

I don't think black holes represent an equilibrium as in being the final state of an evolving system, because as the universe expands and the cosmic microwave background temperature falls, the black holes evaporate as per Hawking's radiation leaving a universe with dispersed particles/ radiation.
 

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