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## Main Question or Discussion Point

I am considering the variation of

[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

[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