Minimal Coupling Needed for Covariant Energy Conservation?

[atyy]

Wouldn't one fail to get covariant conservation of energy without minimal coupling? I've seen a claim like that in http://arxiv.org/abs/gr-qc/0505128 (Eq 11) and in http://arxiv.org/abs/0704.1733 .

I don't see how either of those papers is related to the matter at hand...no one has suggested an action where the Ricci scalar couples non-minimally to any other fields.

Furthermore, I think the answer really depends on how you define the "energy momentum tensor". This is really the topic for an entirely new thread, but...

In those papers, they have written (ignoring the gravity part of the action)

$$\mathcal L' = f(R) \mathcal{L}_m$$
And they have defined the EM tensor as

$$T_{\mu\nu} = - \frac{2}{\sqrt{-g}} \frac{\delta (\sqrt{-g} \mathcal{L}_m)}{\delta g^{\mu\nu}}$$
And frankly, it should be no surprise that this tensor is not conserved. The definition I am more familiar with is to split the total Lagrangian into the gravity part and everything else

$$\mathcal L_{\text{total}} = \mathcal L_{\text{grav}} + \mathcal L'$$
where $\mathcal L'$ is everything else. Then the EM tensor is defined as

$$T_{\mu\nu} = - \frac{2}{\sqrt{-g}} \frac{\delta (\sqrt{-g} \mathcal{L}')}{\delta g^{\mu\nu}} = = - \frac{2}{\sqrt{-g}} \frac{\delta (\sqrt{-g} f(R) \mathcal{L}_m)}{\delta g^{\mu\nu}}$$
for which conservation follows directly as a consequence of the differential Bianchi identity. You can argue about whether the Ricci scalar is "matter", but the point is there should be a conserved tensor of this form.

Thanks. I started a new thread, because I've seen what seemed to me a contradictory claim in Carroll's GR notes (Eq 5.38) - he says diff invariance is enough to get covariant energy conservation. I've never understood whether Carroll's claims and the ones in these papers are really contradictory, and if so which are correct. Let me think about what you wrote, and ask more questions later.

 

[Ben Niehoff]

Well, for one, Carroll wasn't talking about $f(R)$ gravity, which is an alternative research topic you linked to in those papers. But the first of your papers shows that even in $f(R)$ gravity, one gets conservation of the EM tensor, provided you define it as "everything else" as I have above.

 

[atyy]

Well, for one, Carroll wasn't talking about $f(R)$ gravity, which is an alternative research topic you linked to in those papers. But the first of your papers shows that even in $f(R)$ gravity, one gets conservation of the EM tensor, provided you define it as "everything else" as I have above.

Well, Carroll makes the point that one doesn't need the EP to get covariant energy conservation. Since when talking about the EP, one is usually talking about a class of theories which includes f(R) gravity, I think the main difference is the definitions of the EM tensor. I looked at papers citing the ones in the OP, and there doesn't seem to be any controversy about the result.

What are the motivations for the various definitions of the EM tensor? Perhaps one reason for the alternative EM tensor definition is that the motion of test particles can be derived from it, something like in http://arxiv.org/abs/0704.1733 or http://arxiv.org/abs/0811.0913 ?