Curvature and Stress-Energy: Solving the Einstein Equation with Tensor Densities

[Phrak]

The metric tensor is expressed in 10 independent elements. From this is obtained the Einstein equation once given 7 to 9 or so requirements imposed on the the connection and covariant derivative.

In my mind the Christoffel based covariant derivative is an ugly thing, good for a first attempt at understanding the connection between mass and gravity, but not the last word.

Instead: Can the relationship between curvature and stress-energy (The Einstein equuation,or something like it) be expressed in terms of oriented tensor densities with lower indices sans the goofey connections?

 

[Mentz114]

Have you come across Einstein-Cartan gravity or any of the many versions of gauge gravity ? A web search will find plenty, and I attach one paper that I have to hand.

 

[Phrak]

As far as I know Einstein-Cartan gravity is Einstein gravity where the requirement that the connection be torsion-free (symmetric in it's lower indices) is relaxed. I was unable to tell, scanning the .pdf, whether this more general theory can be expressed without connections.

All gauge gravity sites I've visited seem to indicate that gauge/gravity are 10 dimensional brane theories. I was rather more interested in 4 dimensional General Relativity, or it's variants.

I may have misunderstood the directions in which you are pointing.

 

[Mentz114]

All gauge gravity sites I've visited seem to indicate that gauge/gravity are 10 dimensional brane theories. I was rather more interested in 4 dimensional General Relativity, or it's variants.

Not what I meant at all. Here's a couple more references. I don't think GR is possible without connections because in GR gravity->curvature->parallel transport->connection ( but not necessarily in that order).
If the gravitational field is in the torsion, the Weitzenbock connection is used ( but there's no geodesics).

arXiv:gr-qc/0011087v1
arXiv:gr-qc/9602013 v1

Look for more work by the authors of these papers.