- #1
Andrew Kim
- 12
- 6
From Carroll (2004)
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}##?
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}##?