The Contracted Bianchi Identity and the Einstein Tensor

We start with the Bianchi Identity for the Riemann-Christoffel Curvature tensor, then contract on two indices. We arrive at a relationship between the contravariant derivative of the Ricci tensor and the covariant derivative of the scalar curvature. We then define the Einstein tensor and use the contracted Bianchi identity to prove that the trace of the contravariant derivative and the Einstein tensor is zero, which implies that the Einstein tensor can equate to the stress energy tensor consistently! #mikethemathematician, #profdabkowski, #mikedabkowski, #tensoranalysis