Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

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?

share|improve this question
    
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
1  
@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.

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.