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It is possible to derive the Einstein Equations (with ##c=1##) via functional variation of an action

$$S=\dfrac{S_H}{16\pi G}+S_M$$

where

$$S_H= \int \sqrt{-g}R_{\mu\nu}g^{\mu\nu}d^4 x$$

and ##S_M## is a corresponding action representing matter. We can decompose ##\delta S_H## into three subsequent actions, i.e.

$$\delta S_H=(\delta S)_1+(\delta S)_2+(\delta S)_3$$

$$(\delta S)_1=\int \sqrt{-g}\big(R_{\mu\nu}\big)\delta g^{\mu\nu} d^4 x$$

$$(\delta S)_2=\int R\delta\sqrt{-g}d^4 x = \int \sqrt{-g}\bigg(-\frac{1}{2}g_{\mu\nu}R\bigg)\delta g^{\mu\nu}d^4 x$$

$$(\delta S)_3=\int \sqrt{-g}g^{\mu\nu}\delta R_{\mu\nu}d^4 x$$

It turns out that ##(\delta S)_3=0##, so we have

$$\delta S_H=\int\sqrt{-g}\bigg(R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}\bigg)\delta g^{\mu\nu}d^4 x$$

And therefore

$$\frac{\delta S_H}{\delta g^{\mu\nu}} = \sqrt{-g}\bigg(R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}\bigg)$$

Finding the extremal values of our original action yields

$$\frac{\delta S}{\delta g^{\mu\nu}}=\frac{\delta S_H}{\delta g^{\mu\nu}}+\frac{\delta S_{M}}{\delta g^{\mu\nu}}=0$$

or

$$\sqrt{-g}\bigg(R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}\bigg)=-16\pi G\frac{\delta S_{M}}{\delta g^{\mu\nu}}$$

It is at this point that we define

$$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta S_{M}}{\delta g^{\mu\nu}}$$

and we find

$$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi G T_{\mu\nu}$$

If we replace the matter action with an action for electromagnetism in GR, we have

$$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta S_{EM}}{\delta g^{\mu\nu}}$$

$$S_{EM} = \int \sqrt{-g}\mathcal{L}_{EM}d^4 x$$

The value of ##\mathcal{L}_{EM}## that yields maxwell's equations when we use the Euler-Lagrange equations with respect to the fields ##A_{\mu}## is

$$\mathcal{L}_{EM} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+A_{\mu}J^{\mu}$$

We can find the electromagnetic stress energy if vary ##S_{EM}## with respect to the metric. However, I'm not sure how to conduct this functional variation. I know the fields in gravitational GR are ##g^{\mu\nu}## and the fields in relativistic EM are ##A_{\mu}##. Should I vary with respect to both and only count the terms that have ##\delta g^{\mu\nu}##, or should I treat ##A_{\mu}## as a set of constants like ##J_{\mu}##?

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# I Derivation of E.M. Stress Energy Tensor

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