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If we look at the Einstein vacuum equations, that is without matter (there is the possibility or curvature without matter), for instance we may consider gravitational waves. The question is: Is there some link between the Riemann curvature tensor, and/or the Weyl tensor, and some gravitational "physical" quantities (as stress-energy tensor or total energy) (one may take the example of gravitational waves). Of course, at first glance, there is no covariant gravitational stress-energy tensor, so it seems there is no relation, but maybe things are more subtle?

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Are you asking why the equations relate only the Ricci tensor and the stress-energy, but not the Weyl tensor? But why would expect something like that? For example only two of the classic Maxwell equations couple with the sources. This may not be a good analogy but I still think you could say something about your motivation. –  MBN Sep 6 '13 at 16:14
@MBN : The motivation is that, without matter, there is still gravitational energy in some way (even if it is non localizable), so I was asking about any interesting relation between Riemann/Weyl tensor, or operations on them, and gravitational energy quantities. –  Trimok Sep 6 '13 at 16:18
In vacuum, the stress-energy tensor is $0$ by definition. Hence, for cosmological constant $0$, the Riemann tensor equals the Weyl tensor, $R_{abcd}=C_{abcd}$. Some ideas of getting matter from vacuum were explored by Wheeler (search for "geons"). You can also take a look, with a grain of salt, at fqxi.org/data/essay-contest-files/… –  Cristi Stoica Sep 6 '13 at 18:29
@CristiStoica : Thanks for the link. –  Trimok Sep 7 '13 at 7:54
@CristiStoica You may know this, but Wheeler wasn't the first with his "geons": William Kingdon Clifford had the idea that matter itself might be curvature in a spacetime manifold. I must say, though, that geons strike me as a thoroughly Wheeler-original idea! –  WetSavannaAnimal aka Rod Vance Sep 7 '13 at 13:09

1 Answer 1

Is there some link between the Riemann curvature tensor [...] and some gravitational "physical" quantities*

Maybe you could clarify what you want that would qualify as "physical." Curvature is observable, and IMO is physical. Projects like LIGO are designed to detect gravitational waves. Gravity Probe B was a project that accomplished its purpose of essentially verifying GR's predictions of spacetime curvature in the neighborhood of a gravitating, spinning body. In the simplest terms, curvature can be measured by transporting a gyroscope around a closed path. This is essentially what GPB did.

Of course, at first glance, there is no covariant gravitational stress-energy tensor

But that's only a prohibition on defining a local measure of gravitational-wave energy. For example, in an asymptotically flat spacetime, the ADM energy includes energy being radiated away to null infinity by gravitational waves. If LIGO-like projects succeed, they will measure the energy of gravitational waves.

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I am interested in any detailed example of a relation between gravitational physical quantites like energy, radiated energy, stress-energy tensor, and Riemann/Weyl tensor, with Einstein vacuum equations. –  Trimok Sep 6 '13 at 16:05
OK, then I think the second half of my question should provide an example of such a relation. However, I would object to this definition of "physical" as excluding curvature, for the reasons given in the first half of my answer. –  Ben Crowell Sep 6 '13 at 16:07

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