Hi, Let [itex]T_{\alpha\beta}[/itex] be the stress-energy momentum tensor. What does [itex]g_{\alpha\beta}T^{\alpha\beta}[/itex] mean? I have always thought of the Ricci tensor and the SEM as the same thing essentially, but the Ricci scalar essentially assigns a number to the curvature of the manifold, what does [itex]T[/itex] say? Thanks,
I don't know in GR, but Nordstrom's scalar gravitation, the first consistent relativistic theory of gravity, can be reformulated using the Ricci Scalar and the trace SEM. It doesn't match observation, but it's historically interesting. See Eq 16 of http://arxiv.org/abs/gr-qc/0405030