# Weyl & Riemann curvature tensors and gravitational “physical” quantities in Einstein vacuum equations

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?

-
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
show 1 more comment