Ricci Tensor: Covariant Derivative & Its Significance

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Discussion Overview

The discussion revolves around the Ricci tensor, its covariant derivative, and its significance in the context of Einstein's field equations in General Relativity. Participants explore the relationship between the Ricci tensor and the stress-energy tensor, particularly focusing on the implications of their covariant divergences.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that Einstein initially considered the Ricci tensor for his field equations but found that its covariant derivative was not zero, unlike the energy tensor.
  • Another participant clarifies that the covariant divergence of the Ricci tensor is generally not zero and emphasizes that the covariant divergence of the stress-energy tensor is not independently shown to be zero without the context of the Einstein Field Equation.
  • A participant mentions that in vacuum solutions, such as Schwarzschild spacetime, the Ricci tensor is zero, leading to a zero covariant divergence.
  • One participant expresses the belief that the covariant divergence of the energy tensor is an implied result of the continuity equation, prompting a question about the source of this belief.
  • A humorous exchange occurs regarding the source of one participant's belief, referencing their mother, which leads to light-hearted comments about her expertise.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the covariant divergences of the Ricci tensor and the stress-energy tensor, with no consensus reached on the interpretations or implications of these divergences.

Contextual Notes

The discussion includes assumptions about the definitions and implications of covariant derivatives and divergences, as well as the context of General Relativity, which may not be fully articulated by all participants.

dsaun777
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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?
 
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dsaun777 said:
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.

dsaun777 said:
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.
 
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.
 
dsaun777 said:
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?
 
PeterDonis said:
Why do you think that? Where did you get the idea from?
My mom told me.
 
dsaun777 said:
My mom told me.

Are you serious? Is your mom a physicist?
 
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dsaun777 said:
My mom told me.
Your mom sounds awesome!
 

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