Riemann Tensor Formula in Terms of Metric & Derivatives

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Discussion Overview

The discussion centers around the formulation of the Riemann Tensor in terms of the metric and its first and second derivatives, specifically seeking a version that does not explicitly include the Christoffel symbols. The conversation touches on theoretical aspects of general relativity and the implications of such a formulation.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant requests a version of the Riemann Tensor expressed solely in terms of the metric and its derivatives.
  • Another participant expresses skepticism about the reliability of finding such an expression online and questions the purpose behind the request.
  • A participant acknowledges the possibility of deriving the desired form by substituting the metric into the standard Riemann Tensor formula but expresses concern about potential errors in doing so.
  • One participant claims they can demonstrate that Birkhoff's Theorem is incorrect and suggests that this proof would require the specific form of the Riemann Tensor being discussed.
  • Another participant challenges this claim, asserting that the first participant has made an error and that personal theories are not supported on the forum.

Areas of Agreement / Disagreement

Participants do not reach a consensus. There is disagreement regarding the validity of the claim about Birkhoff's Theorem, and the discussion remains unresolved with respect to the formulation of the Riemann Tensor.

Contextual Notes

The discussion includes references to potential errors in calculations and the reliability of external sources for mathematical expressions. There is also a mention of the forum's policy against supporting personal theories.

Jogging-Joe
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TL;DR
Could someone please write out or post a link to the Riemann Tensor written out solely in terms of the metric and its first and second derivatives--i.e. with the Christoffel symbol gammas and their first derivatives not explicitly appearing in the formula.
Could someone please write out or post a link to the Riemann Tensor written out solely in terms of the metric and its first and second derivatives--i.e. with the Christoffel symbol gammas and their first derivatives not explicitly appearing in the formula.

Thanks.
 
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This link has both the Riemann tensor and the Christoffel symbols. I am not inclined to retype everything just to substitute one into the other. Where are you headed with this? Even if you could find such an expression prechurned on the internet, it is one of those things I would not trust to be error free.
 
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Thanks.

I know that a person can get the Riemann Tensor in the form I want by substituting the metric and its first derivatives into the Christofell gammas in the standard Riemann Tensor formula.. As you noted, it is easy to make errors and that is why I would not trust me doing it, and would want a second opinion calculation. Someone must have put it in that form, somewhere.
 
Jogging-Joe said:
Thanks.

I know that a person can get the Riemann Tensor in the form I want by substituting the metric and its first derivatives into the Christofell gammas in the standard Riemann Tensor formula.. As you noted, it is easy to make errors and that is why I would not trust me doing it, and would want a second opinion calculation. Someone must have put it in that form, somewhere.
But why do you want it in that form?
 
I can show that Birkhoff's Theorem is wrong, and I think I might be able to prove it in a second independent way, which would require me to use that particular form of the Riemann Tensor.
 
Jogging-Joe said:
I can show that Birkhoff's Theorem is wrong

No, you can't. You may think you can, but you've made an error somewhere.

Jogging-Joe said:
I think I might be able to prove it in a second independent way, which would require me to use that particular form of the Riemann Tensor.

Sorry, we don't help people with personal theories on this site, particularly personal theories which we already know are wrong (see above).

Thread closed.
 
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