Can someone verify this definition for Weyl tensor?

[space-time]

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:

C_abcd = R_abcd + (1/2) [- R_acg_bd + R_adg_bc + R_bcg_ad - R_bdg_ac + (1/3) (g_acg_bd - g_adg_bc)R]

Is this correct?

 

[ShayanJ]

Take a look at here!

 

[space-time]

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?

 

[ShayanJ]

It should be:
C_abcd = R_abcd - R_acg_bd + R_adg_bc + R_bcg_ad - R_bdg_ac + (1/3) (g_acg_bd - g_adg_bc)R

 

[space-time]

It should be:
C_abcd = R_abcd - R_acg_bd + R_adg_bc + R_bcg_ad - R_bdg_ac + (1/3) (g_acg_bd - g_adg_bc)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

 

[ShayanJ]

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.

 

[space-time]

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:

R_abcd = R_cdab

Does the Weyl tensor have that same symmetry?

 

[ShayanJ]

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:

R_abcd = R_cdab

Does the Weyl tensor have that same symmetry?

Yeah, it has.