# I Dark Energy - something from nothing?

1. Jun 15, 2016

### Schnellmann

My understanding is that one explanation of dark energy is that it is linked to vacuum energy, which is linked to virtual particles that pop in and out of existence. What I don't understand is that it space is continually expanding then the vacuum energy would also seem to be growing as well...isn't that sort of like getting something for nothing?

2. Jun 15, 2016

### Chalnoth

Energy isn't conserved in General Relativity. Here's a nice blog post that goes through some of this:
http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

3. Jun 15, 2016

### timmdeeg

No, it is something for something. One can show that to expand a small box the work done by the vacuum pressure is just sufficient to maintain constant energy density. Conversely, if the box is filled with a gas its adiabatic expansion goes at the expense of the internal energy of the gas. The reason is that a gas has positive pressure, in contrast, vacuum exerts negative pressure.

4. Jun 17, 2016

### Schnellmann

Thanks for the link. I'm still not sure whether the article is staying that energy is conserved under GR but only if you look at the universe as a whole or just that it is not conserved. The latter seems to suggest that net energy can be created. The article is correct that cosmologists haven't spread the word as I've read quite a bit as an interested layman and not come across the idea before. Given that conservation of energy is drummed into us at school level physics it would be worthwhile making it more known.

5. Jun 17, 2016

### Garth

Energy is a frame dependent concept.

For example: The Apollo space ship hurtling towards the Earth had a large kinetic energy as measured in the Earth's frame of reference which had to be dissipated by the heat shield when it decelerated before splashing into the ocean. However as measured by the astronauts themselves, the Command Module was stationary in their frame of reference so it had zero kinetic energy as measured in their frame of reference.

In General Relativity, as in SR, there is no preferred frame of reference so the concept of energy has no general meaning. To define energy you have to define a specific frame of reference you are measuring it in.

In GR and SR it is energy-momentum that is conserved, not energy alone.

I hope this helps,
Garth

Last edited: Jun 17, 2016
6. Jun 17, 2016

### timmdeeg

The OP, see post #1, doesn't understand that if "space is continually expanding then the vacuum energy would also seem to be growing as well ...".
One could also ask, why is the vacuum energy density constant even though the volume increases. The energy density * is not frame-dependent. It's another story that in GR the energy isn't conserved.

EDIT * vacuum energy density. See post 7.

Last edited: Jun 17, 2016
7. Jun 17, 2016

### Garth

Actually energy-density is frame dependent; it is the energy-momentum tensor that is frame independent and conserved w.r.t. covariant differentiation:
$T^{\mu}{\nu}{;\mu} = 0$.

The energy-density is just the T00 component of this tensor $T{\mu}{\nu}$ . In order to extract this component of the energy-density tensor you have to operate on $T{\mu}{\nu}$ with an observer's velocity four-vector $U^{\mu}$ . You then obtain the energy density as measured in that observer's frame of reference.

When we talk of the universe's energy-density we automatically assume we are measuring it in the frame of reference of a co-moving observer.

Vacuum energy-density is constant because that is the nature of vacuum energy, if you don't like that then you set it to zero; unfortunately observations of the universe seem to indicate that it is not zero, and so it appears that "we are getting something from nothing".

But why it should have such a small value compared with QM expectation is another matter and where the real question lies.....

Garth

Last edited: Jun 17, 2016
8. Jun 17, 2016

### timmdeeg

Thanks, I've added vacuum, vacuum density was meant.

9. Jun 17, 2016

### Chalnoth

It's more accurate to simply say that energy isn't conserved, but the way it changes over time is predictable.

The issue is that you can only define a total energy for the universe if we live in a closed universe. It's impossible to define the total energy for an open or flat universe. If you do have a closed universe, because the energy in the matter fields changes over time in a predictable way, it's possible to write down a potential energy that balances this. But ultimately I'm not sure it helps anybody to understand what's going on.

The issue at hand is that conservation laws are a result of symmetries (read up on Noether's theorem if you're curious how this works). Conservation of energy is a result of symmetries in time. That is to say, if you can write down the physical rules for how a system behaves, and those physical rules don't change over time, then that system conserves energy. General Relativity throws a wrench into this because time is no longer a separate parameter: the behavior of time depends upon the gravitational field. Thus it no longer really makes sense for energy to be conserved. In the particular case of our expanding universe, the fact that it is expanding makes for a change over time which means energy conservation doesn't apply.

10. Jun 18, 2016

### Staff: Mentor

It's saying that energy is not conserved. More precisely, it is saying that "energy" as a global concept, as in "the total energy of the universe", is not conserved.

11. Jun 19, 2016

### Chalnoth

It's not really conserved locally either.

You can write down something in General Relativity that looks like local conservation of energy. In flat space-time, it's an equation that states that the energy flow in/out of a region equals the change in energy of the region. But when space-time is curved, there's an additional term due to that curvature. That additional term makes it so that the change in energy of the region can be different from the flow in/out of that region.

12. Jun 19, 2016

### Staff: Mentor

If you are referring to the fact that the covariant derivative of the stress-energy tensor is zero, that is the local conservation of energy in GR. The additional terms in the covariant derivative due to curvature (more precisely, connection coefficients) do not mean stress-energy is not locally conserved; they mean that in a curved spacetime, you need to correct for curvature effects in order to count the density of stress-energy properly so that you can verify that it is conserved. In other words, what is conserved locally is the physical stress-energy; whereas what the ordinary divergence (without the connection coefficient terms) of the SET evaluates is the local conservation of "coordinate stress-energy", which only has a direct physical meaning in flat spacetime.

No, it doesn't, it just corrects the math so you are looking at physical densities and flows instead of coordinate densities and flows. See above.