After finding the Einstein Tensor

[Reedeegi]

Before I begin, I stress that this is NOT a homework question but rather a self-study question.

In any case, I calculated the Einstein Tensor of a body with the metric diag[{2GM/r-1,0,0,0},{0,1+2GM/r,0,0},{0,0,1+2GM/r,0},{0,0,0,1+2GM/r}]. What does the Einstein Tensor represent? Does its determinant/trace/or matrix operations have any physical meaning?

 

[atyy]

The covariant derivative of the Einstein tensor is zero, which by the Einstein field equation means the covariant derivative of the stress-energy-momentum tensor is zero, which is related to energy-momentum conservation in flat spacetime.

 

[VKint]

Basically, the Einstein tensor is a trace-reversed version of the Ricci tensor (a contraction of the Riemann curvature tensor). Its most salient feature is that it is conserved, in the following sense: $$G^{ab}_{;b} = 0$$. In other words, the "divergence" of the Einstein tensor vanishes.

 

[rrogers]

As was mentioned indirectly; the Einstein tensor is physically identified with the physical stress-energy tensor. The physical properties associated with the Stress-Energy tensor are explained in various books. The wikipedia article seems adequate:
http://en.wikipedia.org/wiki/Stress-energy_tensor

Ray