Ricci Tensor: Covariant Derivative & Its Significance

[dsaun777]

I read recently that Einstein initially tried the Ricci tensor alone as the left hand side his field equation but the covariant derivative wasn't zero as the energy tensor was. What is the covariant derivative of the Ricci tensor if not zero?

 

[PeterDonis]

the covariant derivative wasn't zero as the energy tensor was

It is true that the covariant divergence (not derivative) of the Ricci tensor is in general not zero. However, it is not true that the covariant divergence of the stress-energy tensor is zero. More precisely, there is no way of showing it to be zero, independently of the Einstein Field Equation. In other words, in GR as it was finally formulated, we deduce that the covariant divergence of the SET is zero because we know the covariant divergence of the Einstein tensor is zero, not the other way around.

What is the covariant derivative of the Ricci tensor if not zero?

The exact nonzero value of the covariant divergence of the Ricci tensor (in spacetimes where it is not zero) depends on the spacetime. In vacuum solutions, such as Schwarzschild spacetime, the Ricci tensor itself is identically zero (that's part of what it means to be a vacuum solution), so its covariant divergence is also zero.

 

[dsaun777]

I thought the covariant divergence of energy tensor was an implied result of the continuity equation which led to him seeking a curvature term that had that also had the same result.

 

[PeterDonis]

I thought the covariant divergence of energy tensor was an implied result of the continuity equation

Why do you think that? Where did you get the idea from?

 

[dsaun777]

Why do you think that? Where did you get the idea from?

My mom told me.

 

[PeterDonis]

My mom told me.

Are you serious? Is your mom a physicist?

 

[Dale]

My mom told me.

Your mom sounds awesome!