# The Einstein and Ricci tensors

## Main Question or Discussion Point

I'm trying to understand the Einstein field equations conceptually, and one of the things that I'd like to understand is why Einstein decided that the left side of the GR equation should be the Einstein tensor instead of the Ricci tensor, I heard that initially he entertained the idea of equating the Ricci tensor with the stress-energy tensor part, but he finally, in November 1915 came up with the final form. What were the physical reasons to decide that the Ricci tensor alone couldn't account for the curvature of the stress-energy tensor gravitational field?

Thanks

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phyzguy
Because energy is conserved, the stress energy tensor has zero divergence. Since he wanted to equate the stress-energy tensor on the right-hand side with the curvature of space-time on the left-hand side, he needed a tensor on the left-hand side which also had zero divergence. The Ricci tensor has a non-zero divergence, so he created the Einstein tensor, which does have zero divergence.

Because energy is conserved, the stress energy tensor has zero divergence. Since he wanted to equate the stress-energy tensor on the right-hand side with the curvature of space-time on the left-hand side, he needed a tensor on the left-hand side which also had zero divergence. The Ricci tensor has a non-zero divergence, so he created the Einstein tensor, which does have zero divergence.
Thanks, I suspected it was related to energy conservation issues.
Can you give a an explanation for laymen of what it means to have zero divergence, and define divergence in this context?
Thanks.

Zero divergence means locally conserved. Like a fluid. That the quantity flowing into any small volume is equal to the quantity flowing back out of the volume.

In differential geometry, the Bianchi identity says that what we now call the Einstein tensor is a quantity for which the divergence is exactly zero always. By making a law equating this to the mass-energy tensor, you automatically ensure the mass-energy will be locally conserved according to the theory. But I assume other candidate divergence-free curvature tensors exist; there was probably further motivation for choosing the Einstein tensor specifically..