Energy-Momentum Tensor for Electromagnetism in curved space

[Pouramat]

Homework Statement

Consider
$$
\mathcal L = \sqrt{-g} (\frac{-1}{4}F^{\mu \nu}F_{\mu \nu} + A_\mu J^\mu)
$$
a) Find $T^{\mu \nu}$
b) Adding a new term to the Lagrangian $\mathcal L' = \beta R^{\mu \nu} g^{\rho \sigma} F_{\mu \rho} F_{\nu \sigma}$. How are Maxwell's eqs altered? How about Einstein's Eqs? Is the current still conserved?

Relevant Equations

N/A

a) I'd separated the Lagrangian into:
$$
\mathcal L = \mathcal L_{Max}+\mathcal L_{int}
$$
in which $\mathcal L_{Max} =\frac{-1}{4}\sqrt{-g} F^{\mu \nu}F_{\mu \nu}$ and $\mathcal L_{int} =\sqrt{-g} A_\mu J^\mu$
Thus:
$$
T^{\mu \nu}_{Max}= F^{\mu \lambda}{F^{\nu}}_{\lambda}-\frac{1}{4}F^{\alpha \beta}F_{\alpha \beta}
$$
But I don't know how to treat the second term in Lagrangian $\mathcal L_{int} =\sqrt{-g} A_\mu J^\mu$
b) I could'nt solve it...

 

[NateD]

A:The energy-momentum tensor for the interacting part of the Lagrangian is defined by$$T^{\mu\nu}_{\text{int}} = \frac{2}{\sqrt{-g}} \frac{\delta(\sqrt{-g}\mathcal{L}_{\text{int}})}{\delta g_{\mu\nu}}$$Hence,\begin{align}T^{\mu\nu}_{\text{int}} &= \frac{2}{\sqrt{-g}} \frac{\delta(\sqrt{-g}A_\mu J^\mu)}{\delta g_{\mu\nu}} \\&= \frac{1}{\sqrt{-g}} \left( \frac{\partial (\sqrt{-g}A_\mu J^\mu)}{\partial g_{\mu\nu}} - \frac{\partial }{\partial x^\nu} \frac{\partial (\sqrt{-g}A_\mu J^\mu)}{\partial (\partial x^\mu)} \right) \\&= \frac{1}{\sqrt{-g}} \left( \frac{1}{2}\sqrt{-g} \left(g_{\mu\nu} A_\alpha J^\alpha + g_{\mu\nu} A_\alpha J^\alpha - g_{\alpha\nu}A_\mu J^\alpha - g_{\mu\alpha}A_\nu J^\alpha \right) \right. \\&\qquad \qquad \left. - \frac{\partial (\sqrt{-g}J^\mu A_\nu)}{\partial x^\nu} \right) \\&= \frac{1}{2} \left(g_{\mu\nu}A_\alpha J^\alpha - g_{\alpha\nu}A_\mu J^\alpha - g_{\mu\alpha}A_\nu J^\alpha \right) - \frac{1}{\sqrt{-g}} \frac{\partial (\sqrt{-g}J^