The energy-momentum tensor

[BlackHole]

In general relativity the gravitational field energy-momentum 4-tensor is a pseudo-tensor. Basically GR conserves energy-momentum, which is generally different to energy conservation, which general relativity conserves only in asymptotically flat space-times. 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.

 

http://arxiv.org/PS_cache/gr-qc/pdf/0410/0410004.pdf

http://arxiv.org/PS_cache/hep-th/pdf/0308/0308070.pdf

http://www.pma.caltech.edu/Courses/ph136/yr2002/chap01/0201.2.pdf

 

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

 

[math] \mbox{SR}:\partial^{\mu} \Theta_{\mu\nu}^{\mbox{matter}} =0\rightarrow \mbox{GR} :\nabla^{\mu} \Theta_{\mu\nu}^{\mbox{matter}} =0 [/math]

[□h=-16πT]

And...?

[BlackHole]

Gravity must draw upon some energy/power source in order to do work. It cannot draw upon 4-dimensional spacetime. I think it's the earth's core. I also think that a "force" which curves our vacuum cannot be weak. Einstein said that mass inside stars is converted into energy. I think there is a deeper reality to it.

[timo]

Why is the energy momentum tensor a pseudo-tensor and not a tensor? Not only that this was completely new for me I´d furthermore doubt it since the energy momentum tensor forms one side of a very basic tensorial equation called Einstein equation. That´s, at least, if a pseudo tensor is what I currently have in mind for this term: A tensor-like structure with non-tensorial transformation laws for it´s entries.

 

What makes you come to the conclusion that because of local conservation of energy "gravity must draw upon some energy/power source in order to do work"?

[□h=-16πT]

Yes, I was wondering why you have called the stress-energy tensor a pseudo-tensor.

[BlackHole]

Why is the energy momentum tensor a pseudo-tensor and not a tensor? Not only that this was completely new for me I´d furthermore doubt it since the energy momentum tensor forms one side of a very basic tensorial equation called Einstein equation. That´s' date=' at least, if a pseudo tensor is what I currently have in mind for this term: A tensor-like structure with non-tensorial transformation laws for it´s entries.

 

What makes you come to the conclusion that because of local conservation of energy "gravity must draw upon some energy/power source in order to do work"?[/quote']

 

The gravitational field energy momentum tensor is a unique pseudotensor because actually there is none in general relativity. The difficult question is whether gravitational waves carry energy themselves.

[timo]

Sorry, I didn´t understand your answer. So with that "gravitational field energy momentum tensor" you are not talking about the matter-side of the Einstein Equations? And what do you mean by "The gravitational field energy momentum tensor is a unique pseudotensor because actually there is none in general relativity"?

[BlackHole]

Sorry, I didn´t understand your answer. So with that "gravitational field energy momentum tensor" you are not talking about the matter-side of the Einstein Equations? And what do you mean by "The gravitational field energy momentum tensor is a unique pseudotensor because actually there is none in general relativity"?

 

General relativity does not seem to conserve energy, only energy-momentum because energy alone is frame dependent but general relativity treats all frames of reference as equivalent. I found this paper very helpful but it's rather advanced stuff. I really have problems to digest it.

[timo]

The problem with conservation of energy in GR is not that energy is a frame-dependent concept - you could still assign a definite value in any coordinate system that would be conserved if the stress-energy-momentum tensor was. The problem -if I remember that correctly- is rather how to define conservation of energy. You also have a continuity equation here (that´s your 2nd formula in your 1st post) but problems occur when you want to derive the integral form of it (sorry for being so vague, I´d have to look up a few things to be more precise and correct).

 

It´s funny that you posted research papers in your first post without any comments on them so I couldn´t understand their connections to your statement and that you now post a link to an introduction to special relativity and call it "rather advanced stuff" . Perhaps it would help if you said what in particular your problems with the text are so maybe I or others can clear things up for you.

[BlackHole]

The problem with conservation of energy in GR is not that energy is a frame-dependent concept - you could still assign a definite value in any coordinate system that would be conserved if the stress-energy-momentum tensor was. The problem -if I remember that correctly- is rather how to define conservation of energy.

 

I thought that the real conservation appears only in the case when the whole energy-momentum of both matter-gravity field are considered. The components of a 4-vector change with frame, but its magnitude remains invariant. So the magnitude of the energy-momentum 4-vector is invariant, though its composition as energy & momentum change with frame.

 

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 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".

 

The second possibility is to have an asymptotically flat space-time. It's this possibility which really makes energy "pseudo-tensors" work.

 

[math]M = 2 \int_{\Sigma} (T_{ab} - \frac{1}{2} T g_{ab}) n^a \xi^b dV[/math]

 

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 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 proportional to pressure. 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".