# After finding the Einstein Tensor

## Main Question or Discussion Point

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?

## Answers and Replies

Related Special and General Relativity News on Phys.org
atyy
Science Advisor
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.

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.

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