# Easy Variation Question?

## Main Question or Discussion Point

I am considering the variation of

$\delta ( \sqrt{g} R_{abcd} R^{abcd} )$

and I know the answer is

$- \frac12 \sqrt{g} g_{\mu\nu}R_{abcd} R^{abcd} +\sqrt{g} R_{( \mu}{}^{bcd} R_{\nu ) bcd} + \ldots$

what i do not understand is the coefficient of the last term. For example, when we evaluate the Maxwell Action

$\sqrt{g} F_{ab} F^{ab}$

what we do is to write down as

$\sqrt{g} g^{\mu\nu} g^{\alpha\beta} F_{\mu\alpha} F_{\nu\beta}$

so when we vary the action, we get

$-\frac12 \sqrt{g} g_{\mu\nu} F^2 + 2 \sqrt{g} F_{(\mu}{}^{\sigma} F_{\nu ) \sigma}$

why is it not working with Riemann Tensor? How come there is no factor of 4 on the front of the last term in the variation of Riemann squared action?

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Bill_K
My reference (DeWitt's lectures) does have a factor of 4. Plus there are several terms involving the Ricci tensor.

My reference (DeWitt's lectures) does have a factor of 4. Plus there are several terms involving the Ricci tensor.
do you have a link for these lecture notes?

Bill_K
Sorry, my mistake, the additional factor is 2 not 4. The reference is "Dynamical Theory of Groups and Fields", which is apparently not available online, although many libraries have it. Although DeWitt claims the calculation is easy, it is not! So let me quote his result in full (He must be using the opposite sign convention):

L3 ≡ √g RμνστRμνστ
δL3/δgμν = √g (4Rμνσ -2R;μν -2RμστρRνστρ +½gμνRστρλRστρλ -4RμσντRστ +4RμσRνσ)

:tongue2:

Sorry, my mistake, the additional factor is 2 not 4. The reference is "Dynamical Theory of Groups and Fields", which is apparently not available online, although many libraries have it. Although DeWitt claims the calculation is easy, it is not! So let me quote his result in full (He must be using the opposite sign convention):

L3 ≡ √g RμνστRμνστ
δL3/δgμν = √g (4Rμνσ -2R;μν -2RμστρRνστρ +½gμνRστρλRστρλ -4RμσντRστ +4RμσRνσ)

:tongue2:
You are absolutely right about the factor of 2! i am sorry for the typo. but i still do not understand why the factor is not 4, but 2?

Bill_K