- #1
sergenyalcin
- 3
- 0
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