Vacuum Energy in SR: A Relativity Question

[kcdodd]

I have just a very general question regarding the idea of vacuum energy. It seems that it exists it would have to occur such that it would be the same in every reference frame to satisfy the principle of relativity, yes? Would that suggest it would be part of a stress-energy tensor which is invariant under transformations?

 

[PeterDonis]

I have just a very general question regarding the idea of vacuum energy. It seems that it exists it would have to occur such that it would be the same in every reference frame to satisfy the principle of relativity, yes?

Yes.

Would that suggest it would be part of a stress-energy tensor which is invariant under transformations?

If by "invariant under transformations" you mean "a multiple of the metric tensor", then yes. (The term "invariant under transformations" by itself could really describe any tensor, but I think you meant something much more specific.) See Ned Wright's page on vacuum energy density the cosmological constant here:

http://www.astro.ucla.edu/~wright/cosmo_constant.html

There's a link a little way down to a proof that the stress-energy tensor of the vacuum, if it is not zero, must be a multiple of the metric tensor.

 

[kcdodd]

Yes that is what I meant. I disagree with his statement at the beginning that it must be a constant. It should of course depend on the boundary conditions of the "vacuum" if it is going to be a result of quantum mechanics.

Now, something I find strange. It seems to me that if one had a volume filled with particles of some field the pressure would be positive, not negative. For instance, a volume filled with electromagnetic fields not in the ground state definitely produce a positive pressure on the walls, so why would the supposed ground state be different.

 

[PeterDonis]

Yes that is what I meant. I disagree with his statement at the beginning that it must be a constant. It should of course depend on the boundary conditions of the "vacuum" if it is going to be a result of quantum mechanics.

Not sure what you mean by "the boundary conditions of the vacuum", but they don't sound like things that would vary from event to event in spacetime, so they would seem to imply a constant vacuum energy density. However, a scalar field that varies in spacetime can play a role in the phase transition from a "false vacuum" to a "true vacuum"; see below.

It seems to me that if one had a volume filled with particles of some field the pressure would be positive, not negative. For instance, a volume filled with electromagnetic fields not in the ground state definitely produce a positive pressure on the walls, so why would the supposed ground state be different.

Because the vacuum is not "filled with particles of some field". It's not a ground state with respect to a particular field only; it's a ground state with respect to the overall Hamiltonian, which includes *all* the fields.

Actually, as Wright mentions, it's more complicated than this, because there can be different vacuums. He talks about "false vacuum" and "true vacuum", which are the terms usually used to describe what happened when the inflationary epoch of the early universe ended, with the vacuum that existed during inflation being the false vacuum and the one we are living in now the true vacuum. But of course if we are still seeing a nonzero vacuum energy density, then our vacuum must still in some sense be a "false vacuum" for Wright's argument to work; there must be some further "true vacuum" with still lower ground state energy, which our universe simply hasn't had time to quantum-tunnel into yet.

The quantum-tunneling process from "false vacuum" to "true vacuum", in the simplest version, involves a scalar field, as described briefly on Wikipedia here...

http://en.wikipedia.org/wiki/False_vacuum

...or by Alan Guth here:

http://ned.ipac.caltech.edu/level5/Guth/Guth3.html

Since the scalar field can vary, so can the tunneling process. So in that sense you are right, Wright's statement that vacuum energy must be constant is an oversimplification. What he should have said is that, within a particular "bubble" of false vacuum, the vacuum energy will be constant, because the value of the scalar field that is locally minimized in the false vacuum does *not* vary. But the process of tunneling from this local minimum to the "global" minimum of the true vacuum does not have to happen equally everywhere.

However, AFAIK, none of these complications affect the fact that vacuum energy has negative pressure; that's shown by the simple energy conservation argument Wright gives (Guth gives the same argument on his page).