Energy conservation in General relativity

[Starship]

This is how highly complex but how is energy conserved in general relativity?

Baez says that energy is conserved by installing energy pseudo-tensors into Einstein's field equations but is that it?

 

[dextercioby]

The matter energy-momentum 4tensor must satisfy the requirement that the covariant 4divergence must be 0.However,though this is the natural covariant extension of the flat-case conservation law

$$\mbox{SR}:\partial^{\mu} \Theta_{\mu\nu}^{\mbox{matter}} =0\rightarrow \mbox{GR} :\nabla^{\mu} \Theta_{\mu\nu}^{\mbox{matter}} =0$$

,the latter is not longer a conservation law...The real conservation appears only in the case when the whole energy & momentum of both matter & gravity field are considered.

The fact that the gravitational field energy momentum 4tensor is in fact a pseudo-tensor is really annoying,both in the classical description (frame dependence of gravit.energy,for example),both in the quantum one...

Daniel.

 

[Garth]

There has been quite a lot of discussion about this on these Forums.

Basically GR conserves energy-momentum, which is generally different to energy conservation, and in general energy is not conserved in GR.

Energy is a frame dependent concept, and as there are no preferred frames in GR there is no clear definition of energy or whether it is conserved or not.

However where space-time tends asymptotically towards flatness energy may be defined and treated as if conserved.

Garth

 

[Starship]

So this problem arises in the context of quantum gravity (when trying to localize the energy of the gravitational field)?

László B. Szabados writes that it is possible to avoid this by using backround metrics, tetrads or higher derivatives currents...

http://relativity.livingreviews.org/Articles/lrr-2004-4/articlesu3.html

 

[pervect]

The problem arises in classical gravity, too, not just quantum gravity.

If you re=read the sci.physics.faq (I presume from your previous comments that you've read it), you'll see that there are two main possibillities for true energy conservation - if one can satisfy either one of them, one has a defined energy.

The first is to have a static metric. (Another way of saying this is that one has time-translation symmmetry, or a timelike Killing vector). This leads directly to a conserved energy by Noether's theorem, and the fact that GR can be written as an "action theory" (see the Einstein Hilbert action for the technical details).

The second possibility is to have an asymptotically flat space-time. It's this possibility which really makes energy "pseudo-tensors" work. You'll find a good discussion of this in Wald's "General Relativity". As has also been mentioned, you'll find a ton of stuff archived on this board, the subject has been talked about a lot before here.

The second possibility is a litle more arcane, but it boils down to the fact that Gauss's law can be made to give on a conserved mass if one has an asymptotically flat space-time. This requires that gravity drop off as 1/r^2 in the limit as r-> infinity, and that space-time is flat at infinity and not expanding.

Note that an expanding universe isn't asymptotically flat, so these energy conservations don't apply directly to our universe when it is looked at on a cosmological scale.

Aside from the above, GR has a differential form of the energy conservation law, which basically says (as the sci.physics FAQ says) that no energy is created in an infinitesimal volume of space-time. However, this form of the conservation law isn't good enough to prevent a finite volume from not conserving energy. Our exanding universe illustrates this - the cosmological energy loss is proportioanl to pressure * dv. Because the pressure in our universe is so low, the cosmological energy loss is also very low, and energy is "almost" conserved. Because the energy loss is proportional to volume, the differential conservation law can be (and is) satisfied, even though there isn't an actually conserved constant "energy of the universe".

 

[Crosson]

Our exanding universe illustrates this - the cosmological energy loss is proportioanl to pressure * dv

If we are in a (flat) closed universe, then there will one day be a big crunch. When the universe turns around and starts going the other way, wouldn't it be appropriate to say that it "stored potential energy" when it expanded?

Have you considered the possibility that the work done by the universe as it expands is equal to the loss in energy of the redshifted photons? Each photon has E = hf, and the CMB phtons are at a redshift of 1000. That means they have given up 99.9% of their energy (to no one knows where).

 

[pervect]

If we are in a (flat) closed universe, then there will one day be a big crunch. When the universe turns around and starts going the other way, wouldn't it be appropriate to say that it "stored potential energy" when it expanded?

Have you considered the possibility that the work done by the universe as it expands is equal to the loss in energy of the redshifted photons? Each photon has E = hf, and the CMB phtons are at a redshift of 1000. That means they have given up 99.9% of their energy (to no one knows where).

You might be able to make this idea work, but it isn't a standard idea from anything I've read on the topic (energy in GR).

Some of the hurdles in making the idea work - how much energy does a unit volume of empty space contain (is it a consistent amount), does this energy generate gravitation or is it in any other way experimentally detectable (assuming that it does contribute to gravitation at least yields an experimental prediction that the stored energy should be related to the cosmological constant, and puts the idea on firmer philosophical ground, but is likely to yield a theory that isn't vanilla GR.)

Another isssue is how do you do the bookeeping to measure said volume. The volume I was talking about earlier was the volume in one specific coordinate system. Defining the volume in terms that have meaning in an arbitrary coordinate system isn't going to be easy if it's possible at all.

The standard notion of energy in GR is firmly grounded in the mathematical structure of asymptotically flat space-times at infinity, and thus only defined in asymptotically flat space-times. There are actually two sorts of energy defined (the Bondi energy and the ADM energy), which vary in terms of which infinity one looks at (there is null infinity, and spacelike infinity - timelike infinity doesn't give any meaningful notion of energy). The two sorts of energy turn out to be closely related, though it takes a lot of work to establish this. For a really full discussion of this, you can see Wald's "General Relativity", but it's rather advanced stuff.

[add]
Actually defnining the volume in a particular coordinate system isn't the issue, the problem is finding some way to keep the invariant mass of the system conserved in all coordinate systems - i.e. to write down some expression for the energy-momentum 4-vector of the system in a coordinate independent fashion. Note that while some non-flat metrics, like isotropic expanding cosmologies, have fairly "natural" coordinate systems, an arbitrary non-asymptotically flat metric will not necessarily have any particular "natural" coordinate system.