Can someone verify this definition for Weyl tensor?

In summary, the Weyl tensor has the same symmetries as the Riemann tensor, and it has an additional interchange symmetry where Racbd = Rcdab. However, different sources may use different conventions, so it's important to stick with one convention when comparing equations.
  • #1
space-time
218
4
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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  • #3
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?
 
  • #4
It should be:
Cabcd = Rabcd - Racgbd + Radgbc + Rbcgad - Rbdgac + (1/3) (gacgbd - gadgbc)R
 
  • #5
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
 
  • #6
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.
 
  • #7
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?
 
  • #8
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.
 

FAQ: Can someone verify this definition for Weyl tensor?

1. What is the Weyl tensor?

The Weyl tensor is a mathematical object in the field of differential geometry that describes the curvature of a space. It is named after the mathematician Hermann Weyl and is closely related to the Riemann curvature tensor.

2. How is the Weyl tensor calculated?

The Weyl tensor is calculated using the Riemann curvature tensor and the metric tensor. It is a combination of the traceless part of the Riemann curvature tensor and the trace of the Ricci curvature tensor.

3. What is the significance of the Weyl tensor in physics?

The Weyl tensor is important in general relativity, where it describes the gravitational field in the absence of matter. It also plays a role in the study of black holes and gravitational waves.

4. Can the Weyl tensor be used to test for the presence of gravitational waves?

Yes, the Weyl tensor can be used to detect gravitational waves. Changes in the Weyl tensor indicate the presence of gravitational waves in a space-time. This has been confirmed by the detection of gravitational waves by the LIGO experiment.

5. Is the Weyl tensor the only way to describe the curvature of space?

No, the Weyl tensor is one of several mathematical tools used to describe the curvature of space. Other commonly used tensors include the Ricci tensor and the Riemann curvature tensor. Each of these tensors provides different insights into the curvature of a space.

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