# Thermal equilibrium in GR

1. Feb 19, 2010

### evilcman

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 possess 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?

2. Feb 19, 2010

### bcrowell

Staff Emeritus
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.

3. Feb 19, 2010

### Stingray

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.

4. Feb 19, 2010

### bcrowell

Staff Emeritus
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.

5. Feb 19, 2010

### Stingray

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.

6. Feb 19, 2010

### bcrowell

Staff Emeritus
7. Feb 19, 2010

### yuiop

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.