Energy Tensor, covariant derivate, variation respect to the metric [duplicate]

This question is an exact duplicate of:

I'm doing the variation of a Lagrangian respect to the metric, but I am having problem with a particular terminus. My action is:

$$S=\int d^4x \sqrt{-g}[ (\nabla_\mu A^\mu)^2]$$

My lagrangian is:

$$\mathcal{L}=(\nabla_\mu A^\mu)^2$$

to get the energy tensor the formula is $$T_{\mu \nu} = -2 \frac{\partial \mathcal{L}}{\partial{g^{\mu \nu }}}-g_{\mu \nu}\mathcal{L}$$

The question is, is correct this procedure? $$T_{\mu \nu} = -2 \frac{\partial{(g^{\alpha n }\nabla_\alpha A_n)^2}}{\partial{g^{\mu \nu }}}-g_{\mu \nu}\mathcal{L}$$ $$=-4(g^{\alpha n } \delta^\alpha _\mu \delta^n_\mu \nabla_\alpha A_n \nabla_\alpha A_n) -g_{\mu \nu}\mathcal{L}$$ $$= -4(\nabla_\alpha A^\alpha \nabla_\mu A_\nu)-g_{\mu \nu}\mathcal{L}$$

$A_\mu$ is like a electromagnetic field and $(\nabla_\mu A^\mu)^2$ is like a kinetic term

marked as duplicate by Kyle Kanos, Qmechanic♦Jul 15 '14 at 23:03

This question was marked as an exact duplicate of an existing question.

• I am new in this forum, the last post was put "in hold" i supposed that i did have that ask most clearly – Por qué Jul 15 '14 at 22:35
• Indeed, but you are expected to edit your old question (there's this row "share - edit - flag" below each post), which can, if it is sufficiently improved, be reopened, not to post a now one. However, even in this version you are basically asking us "Please check my work", which is something most users here consider not a good conceptual question as per our homework policy (which applies to any "calculational problem" of this kind regardless of whether it is actual homework.) – ACuriousMind Jul 15 '14 at 22:42