Expansion and conservation of energy

[TrickyDicky]

According to quantum field theory there is an intrinsic energy of the vacuum or zero point energy (which is being related to cosmological constant by some cosmologists, i.e.:http://philsci-archive.pitt.edu/archive/00000398/00/cosconstant.pdf ), so if space stretches with expansion, is the energy of this space vacuum being created all the time? if so, is this in conflict with the energy conservation law?

 

[nicksauce]

In GR, energy is only (necessarily) conserved locally. This means that the stress tensor satisfies $$\nabla^{\mu}T_{\mu\nu} = 0$$. The stress tensor that can be used to represent vacuum energy $$T_{\mu\nu} = Cg_{\mu\nu}$$ (for some constant C) certainly satisfies this.

Alternatively, if you want a Newtonian viewpoint, vacuum energy has a negative pressure, and the field does "negative work" to expand the universe. This "negative works" allows for extra energy in the field taking up more volume. It is the exact opposite situation as with photons, where photons have positive pressure and thus do work in expanding the universe, which exactly compensates for the energy loss (redshift) in the photons).

 

[TrickyDicky]

In GR, energy is only (necessarily) conserved locally.

So you are saying GR doesn't have to follow the first law of thermodynamics?
Still expansion is an observed fact not directly derived from GR which is a theory of gravitation.
Maybe someone has a more direct answer to my question?

 

[nicksauce]

So you are saying GR doesn't have to follow the first law of thermodynamics?

Global energy conservation (or the first law of thermodynamics) comes from the time invariance of the Lagrangian, as a consequence of Noether's theorem. In an expanding universe, the Lagrangian is time dependent. There are other problems with energy in GR: One can't identify the energy of the gravitational field properly because all neighbourhoods look locally flat. However, you can still derive all the basic equations of cosmology just by using Netwon's law of gravity, and basic thermodynamics.

Still expansion is an observed fact not directly derived from GR which is a theory of gravitation.

If we believe that the universe is isotropic and homogenous on large scales, and that it is governed by GR on large scales, then it is necessarily true that it will be expanding or contracting. The static solution is unstable, meaning that any small perturbations will cause it to start expanding or contracting. Einstein's failure to realize this is why his cosmological constant was called his "greatest mistake."

 

[TrickyDicky]

Global energy conservation (or the first law of thermodynamics) comes from the time invariance of the Lagrangian, as a consequence of Noether's theorem. In an expanding universe, the Lagrangian is time dependent. There are other problems with energy in GR: One can't identify the energy of the gravitational field properly because all neighbourhoods look locally flat. However, you can still derive all the basic equations of cosmology just by using Netwon's law of gravity, and basic thermodynamics.

That's correct.

Am I to conclude that the expansion of the universe is somewhat in conflict with global energy conservation ,but that it is a fact assumed by the scientific stablishment and either is not seen as a real problem or simply ignored, or seen as small problem and there is people already figuring it out? Or none of the above?

 

[Ich]

Am I to conclude that the expansion of the universe is somewhat in conflict with global energy conservation

There is no such thing as global energy conservation in explicitly time-dependent situations. So how could expansion be in conflict with something that does not exist?

That said, the issue of energy is tricky in GR. There are physical descriptions that restore global energy conservation (google "pseudotensor"). If you have a problem with non-conservation, find comfort in these.

 

[TrickyDicky]

Ok, so there is no conflict because the first law of thermodynamics doesn't apply to time-dependent situations such as expansion, is that it?

I guess what bugs me a little is that most of physics seems to be time-invariant and yet expansion scapes this rule.

 

[Chalnoth]

One way of thinking of it is that energy is only one component of the stress-energy tensor, which is the object upon which gravity acts. Individual components of the stress-energy tensor are not conserved: the quantity as a whole is. And conservation of the totality of the stress-energy tensor (which includes things like momentum, pressure, and shear as well as energy) forces the non-conservation of individual components of the tensor, under the right conditions.

In general, you only get conservation of individual components like energy in a flat space-time. Now, any small enough region of space-time can be described as being flat (which is why it is possible to say that energy is conserved locally, but only if you use coordinates in which the space-time is flat in that local region). But in general you can't describe space-times as being globally flat in this way, so energy conservation is forced to fail due to stress-energy conservation.

 

[yenchin]

Sean Carroll has talked about energy conservation (or the lack thereof) in GR on his blog:
http://blogs.discovermagazine.com/cosmicvariance/2010/02/22/energy-is-not-conserved/

 

[TrickyDicky]

Thanks for the answers.
yenchin , the link is quite interesting, I found it yesterday.