I am considering the variation of(adsbygoogle = window.adsbygoogle || []).push({});

[itex] \delta ( \sqrt{g} R_{abcd} R^{abcd} ) [/itex]

and I know the answer is

[itex] - \frac12 \sqrt{g} g_{\mu\nu}R_{abcd} R^{abcd} +\sqrt{g} R_{( \mu}{}^{bcd} R_{\nu ) bcd} + \ldots [/itex]

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

[itex] \sqrt{g} F_{ab} F^{ab} [/itex]

what we do is to write down as

[itex] \sqrt{g} g^{\mu\nu} g^{\alpha\beta} F_{\mu\alpha} F_{\nu\beta}[/itex]

so when we vary the action, we get

[itex] -\frac12 \sqrt{g} g_{\mu\nu} F^2 + 2 \sqrt{g} F_{(\mu}{}^{\sigma} F_{\nu ) \sigma}[/itex]

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?

Thanks in advance

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Easy Variation Question?

**Physics Forums | Science Articles, Homework Help, Discussion**