Can someone verify this definition for Weyl tensor?

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Discussion Overview

The discussion revolves around the definition of the Weyl tensor in the context of general relativity, focusing on its mathematical formulation and notation. Participants seek to verify the accuracy of a specific expression for the Weyl tensor and explore variations in its representation across different sources.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents a definition of the Weyl tensor, questioning its accuracy based on differing representations found online.
  • Another participant suggests a different formulation of the Weyl tensor, omitting a factor of (1/2) present in the original definition.
  • Some participants note that the discrepancies may arise from different conventions used in various sources.
  • A participant points out that the Wikipedia entry for the Weyl tensor does not mention the interchange symmetry present in the Riemann tensor, prompting a question about whether the Weyl tensor shares this symmetry.
  • Another participant confirms that the Weyl tensor does indeed have the same interchange symmetry as the Riemann tensor.

Areas of Agreement / Disagreement

Participants express disagreement regarding the inclusion of the factor (1/2) in the definition of the Weyl tensor, with no consensus reached on which version is correct. The discussion about the symmetry properties of the Weyl tensor also remains open, although one participant affirms its similarity to the Riemann tensor.

Contextual Notes

Participants highlight the potential for confusion due to varying conventions in the representation of the Weyl tensor across different texts and sources. The discussion does not resolve the specific mathematical discrepancies or the implications of these conventions.

space-time
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I just want to make sure I have this right because when I go to different sites, it seems to look different every time.

This is the Weyl tensor:

Cabcd = Rabcd + (1/2) [- Racgbd + Radgbc + Rbcgad - Rbdgac + (1/3) (gacgbd - gadgbc)R]

Is this correct?
 
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Shyan said:
Take a look at here!

I am unfamiliar with the notation with all those brackets and parentheses around the indices nor am I familiar with indices being on the left side of the large letter as opposed to the right.

That is why I looked for a statement of the Weyl tensor in just plain tensor notation as seen in the OP rather than using all those brackets and such. Can you please simply verify its accuracy in the form that I have it in the OP?
 
It should be:
Cabcd = Rabcd - Racgbd + Radgbc + Rbcgad - Rbdgac + (1/3) (gacgbd - gadgbc)R
 
Shyan said:
It should be:
Cabcd = Rabcd - Racgbd + Radgbc + Rbcgad - Rbdgac + (1/3) (gacgbd - gadgbc)R

It seems that everything in the form I wrote it in the OP matches the form you wrote it in except your form does not include (1/2). Is there a possible reason why the site that I got my form from would include (1/2) or why you did not include it?

This wasn't the site that I originally got it from, but this site still includes (1/2) when talking about 4 dimensions.

http://ion.uwinnipeg.ca/~vincent/4500.6-001/Cosmology/WeylTensor.htm
 
space-time said:
It seems that everything in the form I wrote it in the OP matches the form you wrote it in except your form does not include (1/2). Is there a possible reason why the site that I got my form from would include (1/2) or why you did not include it?

This wasn't the site that I originally got it from, but this site still includes (1/2) when talking about 4 dimensions.

http://ion.uwinnipeg.ca/~vincent/4500.6-001/Cosmology/WeylTensor.htm

compare equation 8.7 in your link with the equation in wikipedia, a 2 is missing.
I think its only a matter of different conventions. It doesn't matter which one you choose as long as you stick with it.
 
Shyan said:
compare equation 8.7 in your link with the equation in wikipedia, a 2 is missing.
I think its only a matter of different conventions. It doesn't matter which one you choose as long as you stick with it.

Thanks for the help. One more question. The Wiki says that the Weyl tensor has the same symmetries as the Riemann tensor, but I notice that the wiki for the Weyl tensor does not mention the interchange symmetry that the Riemann tensor has:

Rabcd = Rcdab

Does the Weyl tensor have that same symmetry?
 
space-time said:
Thanks for the help. One more question. The Wiki says that the Weyl tensor has the same symmetries as the Riemann tensor, but I notice that the wiki for the Weyl tensor does not mention the interchange symmetry that the Riemann tensor has:

Rabcd = Rcdab

Does the Weyl tensor have that same symmetry?
Yeah, it has.
 

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