# A Minimal coupling in general relativity

1. Apr 16, 2017

### spaghetti3451

Consider the Einstein-Maxwell action (setting units $G_{N}=1$),

$$S = \frac{1}{16\pi}\int d^{4}x\sqrt{-g}\ (R-F^{\mu\nu}F_{\mu\nu})$$

where

$$F_{\mu\nu} = \nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu} = \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.$$

This describes gravity coupled to electromagnetism. The equations of motion derived from this action are

$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi T_{\mu\nu}$$
$$\nabla_{\mu}F^{\mu\nu} = 0.$$

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Why does the electromagnetic field tensor $F_{\mu\nu}$ reduce to $\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ even in curved spacetime?

Would this not mean that the equation $\nabla_{\mu}F^{\mu\nu} = 0$ would also reduce to $\partial_{\mu}F^{\mu\nu} = 0$ even in curved spacetime?

2. Apr 16, 2017

### dextercioby

The first why is simply some calculation. Do it and convince yourself.

3. Apr 16, 2017

### spaghetti3451

Can you help me get started?

4. Apr 16, 2017

### dextercioby

What is $\nabla_{\mu}A_{\nu}$ equal to ?