Conservation of Energy in General Relativity

[tionis]

The CMB used to be gamma rays, right? And now it's microwaves - more redder energy: so where did the rest of the energy go?

 

[PAllen]

In GR, global total energy can only be defined asymptotically flat spacetimes. Cosmological solutions do not have this property, and total energy is undefined for them. It is better to think of 'conservation of total energy of the universe' as undefined rather than violated. This is because there are ways to look at any individual CMB photon and say there is no change in energy - it is no different than a gamma ray emitted from a source at rest in one frame, and absorbed by a detector moving near c away from the source (arbitrary redshift possible). However, there is still no way to get from here to globally conserved energy - you just have that the 'loss of energy' for any specific photon is a coordinate dependent statement.

 

[yenchin]

A related nice read: http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

 

[tionis]

In GR, global total energy can only be defined asymptotically flat spacetimes. Cosmological solutions do not have this property, and total energy is undefined for them. It is better to think of 'conservation of total energy of the universe' as undefined rather than violated. This is because there are ways to look at any individual CMB photon and say there is no change in energy - it is no different than a gamma ray emitted from a source at rest in one frame, and absorbed by a detector moving near c away from the source (arbitrary redshift possible). However, there is still no way to get from here to globally conserved energy - you just have that the 'loss of energy' for any specific photon is a coordinate dependent statement.

A related nice read: http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

Drat, I was really psyched 'cause I thought I was going to break Einstein's work in two lol.

 

[PAllen]

A related nice read: http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

I feel hardly qualified to question Sean Carroll, but I wonder about some statements made here.

He seems to suggest there is some well formulated energy that is not conserved (rather than not definable for Cosmology, like the ADM energy). Does anyone have any idea what he is referring to?

- It can't be Bondi energy as that ignores the energy of light and also requires asymptotic infinity assumptions which do not apply to our universe.

- It can't be Komar mass, because the universe is not static.

So what is it? Some pseudo-tensor integrated in comoving coordinates, taken as a meaningful physical quantity?

Since he doesn't reference, even by footnote, any more technical discussion, I can't guess what he is referring to.

Does anyone perhaps familiar with his body work or opinions have any idea?

 

[pervect]

I'd interpret Seans remarks as saying that any conserved energy must be associated, by Noether's theorem, with some sort of time translation symmetry.

Sean glosses over some technical issues here - the time translation symmetry associated with ADM or Bondi mass is not particularly obvious. In general one wind up defining a group of supertranslations, like the BMS group that is infinite dimensional, an since Noether's theorem requires a finite dimensional group to have a preserved energy, the supertranslation group has no associated energy. But the story doesn't end quite there. If you impose some additional conditions on the behavior at infinity, you can come up with a subgroup of this infinite dimensional group that IS finite dimensional. This finite dimensional subgroup does gives you a conserved energy - but you needed the structure at infinity to single out this preferred subgroup.

For ADM mass, the associated group is the SPI group, I believe.

Google finds "Group Theory and General Relativity: Representations of the Lorentz Group" which talks about this in some detail, at least for the BMS group. (google for BMS group and look for this link).

 

[PAllen]

Actually, the arguments about why you should not expect a conserved energy were fine for me and well expressed. What bothered me were Carroll's comments about adding up energy densities in volumes arriving at some purported energy (that is defined but not conserved). I am not aware of an accepted way to do even this in GR, and wondered what he was referring to. Attempts in this direction that I know of suffer from the fact that such a quantity is not tensor based, but is coordinate dependent. My suspicion is that he is treating co-moving coordinates as privileged for the purpose of computing a non-conserved but defined total energy of the universe. Some hint that this is what he is doing, and that this is not a consensus viewpoint, are in order IMO.

(Everyone agrees you can't defined a conserved total energy for plausible cosmological solutions. From there, if you insist that something like total energy of the universe only makes sense if it is coordinate independent, you conclude that is undefinable. If you allow privileging coordinates (or at least a privileged foliation), only then can you talk about a specific total energy that is computable and that changes with the cosmological time parameter. The meaningfulness of this is disputed, not consensus).

 

[tionis]

I feel hardly qualified to question Sean Carroll, but I wonder about some statements made here.

He seems to suggest there is some well formulated energy that is not conserved (rather than not definable for Cosmology, like the ADM energy). Does anyone have any idea what he is referring to?

- It can't be Bondi energy as that ignores the energy of light and also requires asymptotic infinity assumptions which do not apply to our universe.

- It can't be Komar mass, because the universe is not static.

So what is it? Some pseudo-tensor integrated in comoving coordinates, taken as a meaningful physical quantity?

Since he doesn't reference, even by footnote, any more technical discussion, I can't guess what he is referring to.

Does anyone perhaps familiar with his body work or opinions have any idea?

PAllen, hi!

I corresponded with Prof. Carroll this morning and this is what he said about your questions:

''Just the integral of the energy density over a Robertson-Walker spacelike hypersurface (with the factor \sqrt{-g} = a^3 to account for the expansion of the universe). That's what people think of when they say "the energy of the universe" in a cosmological context, and my point was simply that it's not conserved, there's no reason to expect it to be conserved, and there's no way to fix it up to make it conserved. (Except, of course, to take the true Hamiltonian, which will be identically zero in cosmology and doesn't really reflect what people are thinking about when they worry over conservation of energy.) -Sean''

 

[PAllen]

PAllen, hi!

I corresponded with Prof. Carroll this morning and this is what he said about your questions:

''Just the integral of the energy density over a Robertson-Walker spacelike hypersurface (with the factor \sqrt{-g} = a^3 to account for the expansion of the universe). That's what people think of when they say "the energy of the universe" in a cosmological context, and my point was simply that it's not conserved, there's no reason to expect it to be conserved, and there's no way to fix it up to make it conserved. (Except, of course, to take the true Hamiltonian, which will be identically zero in cosmology and doesn't really reflect what people are thinking about when they worry over conservation of energy.) -Sean''

Thanks. That is what I expected, and the only nuance of disagreement I would have is that just because "that's what people think of when they say the energy of the universe", doesn't make it meaningful to do so. Because you would come to different conclusions for different ways to slice up spacetime, I prefer to say the energy of the universe is undefined unless it is of the special class of geometries in which ADM energy can be defined (and then, it is conserved). ADM energy is coordinate independent. Oh, and for all plausible cosmologies, ADM energy cannot be defined.

 

[anorlunda]

Don't neglect the elephant in the room -- dark energy. No general relativity required.

Dark energy seems to account for 70% of the universe's current rate of expansion. Most notable, since it is not diluted by expansion, then every moment the volume of the universe increases, the dark energy increases with it. Not even a hint of energy conservation there. The energy of the universe is growing exponentially.Dark energy, vacuum energy, cosmological constant, all the same thing. We don't understand what dark energy is, but that fact that it exists and that it does not dilute seems to be well established.

PERVECT said, "I'd interpret Sean's remarks as saying that any conserved energy must be associated, by Noether's theorem, with some sort of time translation symmetry." That's the simplest answer. In cosmology, time translation is not symmetrical, so the entire concept of conservation of energy does not apply.

 

[tom.stoer]

Dark energy seems to account for 70% of the universe's current rate of expansion. Most notable, since it is not diluted by expansion, then every moment the volume of the universe increases, the dark energy increases with it. Not even a hint of energy conservation there. The energy of the universe is growing exponentially.

That's not the point. You can introduce the cosmological constant into the field equations w/o relating it to energy at all. Locally you will still have a covariantly conserved energy-momentum tensor. The point is that you cannot define the concept "energy" as a volume integral in general, therefore energy is not even defined (and it's pointless to discuss whether something that is not defined can be conserved)

 

[zaybu]

An article by Michael Weiss and John Baez might help.

http://cybermax.tripod.com/Energy.html

Another one by Philip E. Gibbs

http://www.prespacetime.com/index.php/pst/article/viewFile/89/85

 

[PAllen]

An article by Michael Weiss and John Baez might help.

http://cybermax.tripod.com/Energy.html

Another one by Philip E. Gibbs

http://www.prespacetime.com/index.php/pst/article/viewFile/89/85

Good references. The first, I would say, is a good summary of the consensus view, which I was presenting (Carroll's view, that there is a well defined energy that is not conserved is, IMO, not a consensus view).

Gibb's view is attractive, and may be right, but as he repeats many times, he has failed to convince many others, so it represents part of the diversity of opinion on whether more can be achieved than the consensus.

To repeat again the issue with Carroll's presentation, the energy that is defined and not conserved that he presents explicitly leaves out gravitational energy. He totally unconvincingly tries to ignore this. But the Taylor-Hulse pulsar shows you can't - there is observable energy radiated in GW. Further, in asymptotically flat spacetime, as two bodies spiral closer, the (Bondi) mass of the two body system is smaller as they get closer, while the ADM energy (including the radiated GW and EM if there is charged matter) stays constant. This matches ordinary physical expectation from conversion of gravitational potential to radiation.

The remaining open issue is whether you can generalize these exact results to spacetimes that are not static or asymptotically flat. The majority view is that you cannot (or it hasn't been successfully done yet). Gibb's presents an interesting argument that you can, that is not yet accepted.

 

[PeterDonis]

Gibb's presents an interesting argument that you can, that is not yet accepted.

One interesting consequence that appears to me to follow from his final formulation for the conserved Noether current (the one that is simplified by using the EFE) is that, if you can find a coordinate chart on a spacetime such that the timelike basis vector is independent of *all* the coordinates (i.e., its partial derivative with respect to all four coordinates is zero), then the total energy of that spacetime, by his definition, is zero. (This follows immediately from the formula I mentioned since it includes a multiplicative factor that depends on the partial derivatives.) An equivalent way of formulating the condition is that there must be a chart in which the metric coefficient $g_{00}$ is constant.

One obvious case that satisfies the above condition is any FRW spacetime in the standard FRW chart--i.e., not just the closed FRW spacetime that Gibbs specifically mentions in his paper, but *any* of the FRW spacetimes, since the metric in all of them obviously meets the condition of constant $g_{00}$.

 

[PAllen]

One interesting consequence that appears to me to follow from his final formulation for the conserved Noether current (the one that is simplified by using the EFE) is that, if you can find a coordinate chart on a spacetime such that the timelike basis vector is independent of *all* the coordinates (i.e., its partial derivative with respect to all four coordinates is zero), then the total energy of that spacetime, by his definition, is zero. (This follows immediately from the formula I mentioned since it includes a multiplicative factor that depends on the partial derivatives.) An equivalent way of formulating the condition is that there must be a chart in which the metric coefficient $g_{00}$ is constant.

One obvious case that satisfies the above condition is any FRW spacetime in the standard FRW chart--i.e., not just the closed FRW spacetime that Gibbs specifically mentions in his paper, but *any* of the FRW spacetimes, since the metric in all of them obviously meets the condition of constant $g_{00}$.

Yes, that's right. If you go the the very bottom (p. 1084) of the article (not the arxiv paper), you see an explcicit formula for conserved energy in all expanding cosmologies, for any cosmological constant, and it is (and is stated to be) zero. This is one of the points of debate on this: Gibbs argues that in no way makes it trivial (clearly, of conservation, it doesn't matter what value you set for total energy, when there are negative (potential) as well a positive contributions. Others argue that this makes the whole exercise trivial.

 

[Naty1]

Any comments on the following perspectives?

The Gibbs paper says, page 11:

Some physicists have claimed that energy conservation is violated when you look at the cosmic background radiation. This radiation consists of photons that are redshifted as the universe expands. The total number of photons remains constant but their individual energy decreases because it is proportional to their frequency ( E = hf ) and the frequency decreases due to redshift. This implies that the total energy in the radiation field decreases, but if energy is conserved, where does it go? The answer is that it goes into the gravitational field, but to make this answer convincing we need some equations.

Brian Greene:

...As the universe expands, the inflation field gains energy from gravity [negative pressure]

and Wikipedia has this

...Self-creation cosmology (SCC) theories are gravitational theories in which the mass of the universe is created out of its self-contained gravitational and scalar fields, as opposed to the theory of continuous creation cosmology..

http://en.wikipedia.org/wiki/Self-creation_cosmology#The_new_.282002.29_theory

"The scalar field is a source for the matter-energy field if and only if the matter-energy field is a source for the scalar field."
As the source for the scalar field is the trace of the stress-energy tensor, the PMI is delivered by coupling this trace to the divergence of the stress-energy tensor.

 

[Naty1]

Since no one has replied, I will. I read this

An article by Michael Weiss and John Baez ...

http://cybermax.tripod.com/Energy.html

And it probably gives as good a set of explanations as any.

 

[yenchin]

Since no one has replied, I will. I read this

An article by Michael Weiss and John Baez ...

http://cybermax.tripod.com/Energy.html

And it probably gives as good a set of explanations as any.

This article was already mentioned above. :-)

 

[Naty1]

This article was already mentioned above. :-)

Read more: https://www.physicsforums.com

yes, that's how I happened to see it...my point is that it's a good article...
I referenced it in case anyone wondered what other perspectives might be
and like me had not read it.