Variation of quadratic Riemann Curvature tensor

Qatawna blitz
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TL;DR Summary: How can I variate the quadratic Riemann curvature tensor, I tried raising and lowering the indices.

Hi,
Can you help me with this variation, I tried raising and lowering the indices.
1732356039421.png


I tried for months every possible method to reach the following answer without success.

1732356173162.png
 
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Please provide the reference if you want people to help. Your Langrangian suddenly contains two indices u and v, and my crystal ball provides no information about what that means.
 
I think the two indicies uv in the answered lagrangian means variation with respect to guv
It's in the textbook: Introducing Einstein's General Relativity "Ray d'Inverno" @haushofer
 
Hi.

##L_{uv} = \sqrt{-g}[g_{ud}R^{abcd}R_{abcv} + g_{vd}R^{abcd}R_{abcu} - \frac {1}{2}g_{uv}R^{abcd}R_{abcd}]##

I could be wrong but it looks like following expression:

##L_{uv} = \sqrt{-g}[2R_{uv} - \frac {1}{2}g_{uv}R]##
 
Qatawna blitz said:
TL;DR Summary: How can I variate the quadratic Riemann curvature tensor, I tried raising and lowering the indices.

Hi,
Can you help me with this variation, I tried raising and lowering the indices.
View attachment 353814

I tried for months every possible method to reach the following answer without success.

View attachment 353815
Can you post a screenshot with the full problem description?
 
Moderator's note: Thread moved to the advanced physics homework help forum.
 
Qatawna blitz said:
I tried for months every possible method to reach the following answer without success.
Please post what you have tried.
 
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