Einstein Tensor; What is wrong here?

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Discussion Overview

The discussion revolves around the manipulation of the Einstein tensor and the Ricci tensor in the context of General Relativity. Participants are examining the validity of certain mathematical steps and assumptions related to the contraction of tensors and the properties of the metric tensor.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a series of equations starting with the identity \(\nabla^{\mu}R_{\mu\nu}=\nabla^{\mu}R_{\mu\nu}\) and manipulates it to derive a relationship involving the metric tensor and Ricci scalar, questioning the correctness of their result in relation to General Relativity.
  • Another participant suggests that the inserted factor in the equations should involve different indices to avoid confusion, indicating that \(\mu\) and \(\nu\) are already used in the context.
  • A participant expresses confusion regarding the assertion that indices are "taken," questioning whether one can assume the existence of the metric tensor without prior proof in the specific case discussed.
  • Another participant reiterates the concern about overloading indices, emphasizing that an index should not appear more than twice in any term, which relates to the validity of the manipulations being discussed.
  • One participant points out that there is a double sum in the term that was inserted, arguing that this affects the ability to contract the Ricci tensor correctly without including the metric tensor.

Areas of Agreement / Disagreement

Participants express differing views on the validity of the mathematical manipulations and the assumptions made regarding the indices and the metric tensor. There is no consensus on the correctness of the steps taken or the implications of the assertions made.

Contextual Notes

Participants highlight potential issues with index notation and the assumptions underlying the manipulation of tensors, indicating that the discussion is deeply rooted in the technical details of tensor calculus and General Relativity.

nobraner
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Start with

[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]=[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]

Insert

[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]=[itex]\nabla^{\mu}[/itex][itex]\frac{g_{\mu\nu}g^{\mu\nu}}{4}[/itex]R[itex]_{\mu\nu}[/itex]

Contract the Ricci Tensor

[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex] = [itex]\nabla^{\mu}[/itex][itex]\frac{g_{\mu\nu}}{4}[/itex]R

Thus

[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]=[itex]\frac{1}{4}[/itex][itex]\nabla^{\mu}[/itex][itex]{g_{\mu\nu}}[/itex]R

But General Relativity says

[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]=[itex]\frac{1}{2}[/itex][itex]\nabla^{\mu}[/itex][itex]{g_{\mu\nu}}[/itex]R

What is wrong here?
 
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nobraner said:
Start with

[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]=[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]

Insert

[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]=[itex]\nabla^{\mu}[/itex][itex]\frac{g_{\mu\nu}g^{\mu\nu}}{4}[/itex]R[itex]_{\mu\nu}[/itex]

[snip]

What is wrong here?

Your inserted factor should be
[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]=[itex]\nabla^{\mu}[/itex][itex]\frac{g_{\sigma\tau}g^{\sigma\tau}}{4}[/itex]R[itex]_{\mu\nu}[/itex] (since [itex]\mu[/itex] and [itex]\nu[/itex] are already "taken").


Note that since your proposed proof makes no use of the unique properties of Ricci, it would seem that your result would work for any symmetric tensor. So, you must look at it with suspicion.
 
I don't understand your declaration that [itex]\mu \nu[/itex] are already taken. Does that mean we can never assume that such a metric as

g[itex]^{\mu\nu}[/itex]

exists without first proving that it is so for the specific case of

[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]=[itex]\frac{1}{4}[/itex][itex]\nabla^{\mu}[/itex]g[itex]_{\mu\nu}[/itex]R
 
nobraner said:
I don't understand your declaration that [itex]\mu \nu[/itex] are already taken. Does that mean we can never assume that such a metric as

g[itex]^{\mu\nu}[/itex]

exists without first proving that it is so for the specific case of

[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]=[itex]\frac{1}{4}[/itex][itex]\nabla^{\mu}[/itex]g[itex]_{\mu\nu}[/itex]R

You're overloading your indices. An index shouldn't appear more than twice in any term.
 
nobraner said:
I don't understand your declaration that [itex]\mu \nu[/itex] are already taken. Does that mean we can never assume that such a metric as

g[itex]^{\mu\nu}[/itex]

exists without first proving that it is so for the specific case of

[itex]\nabla^{\mu}[/itex]R[itex]_{\mu\nu}[/itex]=[itex]\frac{1}{4}[/itex][itex]\nabla^{\mu}[/itex]g[itex]_{\mu\nu}[/itex]R

There is a double sum in the term that you inserted. Therefore you cannot contract the Ricci leaving out the g[itex]_{\mu\nu}[/itex]
 

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