# Divergence theorem in curved space

1. Apr 11, 2010

### utesfan100

I have been contemplating my confusion about my intuition regarding GR and believe I have tracked down the primary source of confusion.

The classical theories I have been taught assumed flat space with independent time and used the divergence theorem to derive inverse squared laws for fields. Despite GR holding a local form of the differential form of Gauss's law, every attempt to use the integral form is shot down immediately as not conforming to GR.

Am I correct to state that in a curved space the divergence theorem may not apply in a straight forward manner?

It appears that to use the divergence theorem in a curved space time one needs to use something called Killing vectors to define a curvilinear set of local radial lines to define a closed curve around a simply connected region in the curved space. I would think this would allow "local" to be extended from the infinitesimal range of the stress-energy tensor to a finite bounded volume.

Is this the proper interpretation for what the Landau-Lifgarbagez pseudotensor does for GR? ( http://en.wikipedia.org/wiki/Landau-Lifgarbagez_pseudotensor [Broken] )

Since the Einstein Tensor is proportional to the Stress Energy tensor and also definable in terms of the metric alone it appears that these three tensors are three ways of describing the same mathematical object. Since the Landau-Lifgarbagez pseudotensor is defined in terms of the Einstein Tensor and the metric alone it can be defined in terms of the metric alone, and thus is another description of the same mathematical object.

Am I wrong to think that infinitesimally a pseudotensor represents an apparent force that can be eliminated by a change of reference frame? It would then seem reasonable that the Landau-Lifgarbagez pseudotensor represents the net unresolvable apparent force in a finitely bounded region caused by the spacial curvature in that finite volume.

Is this a correct way to think about gravity?

Last edited by a moderator: May 4, 2017
2. Apr 11, 2010

### atyy

3. Apr 12, 2010

### utesfan100

It references "many psuedo tensors" to do this task, while I have been exposed to only two. What other options exist?

4. Apr 12, 2010

### atyy

I'm not sure, maybe try http://arxiv.org/abs/hep-th/9310025