How do you differentiate $F^{αβ}$ with respect to $g_{μν}$?

Question

I want to experiment with this relation (from Dirac's "General Theory of Relativity"):

$$T^{μν} = -\left(2 \frac{∂L}{∂g_{μν}} + g^{μν} L \right)$$

using the electromagnetic Lagrangian $L = -(16π)^{-1} F_{μν} F^{μν}$. (Signs vary from source to source.) I should produce the electromagnetic stress-energy tensor

$$E^{ρσ} = (4π)^{-1} \left( -{F^ρ}_ν F^{σν} + \frac{1}{4} g^{ρσ} F_{μν}F^{μν} \right).$$

I am almost there, except I am stuck on what to do with the term

$$\frac{∂F^{αβ}}{∂g_{μν}} F_{αβ}$$

which should be equal to $-{F^ρ}_ν F^{σν}$.

How do you differentiate $F^{αβ}$ with respect to $g_{μν}$?

I tried writing $F^{αβ} = g^{\alpha\rho}g^{\beta\sigma}F_{\rho\sigma}$ and using $F_{\rho\sigma} = \kappa_{\rho,\sigma}-\kappa_{\sigma,\rho}$, which gets me somewhere, but I am left with terms like

$$\frac{\partial g^{\beta\sigma}}{\partial g_{\mu\nu}}$$ and I don't know what that is.

Answer 1

If you are going to differentiate $L$ with the respect to the metric, $L$ needs to be rewritten without the metric being implicitly used anywhere. Otherwise, you will not vary the entire dependence on the metric when differentiating.

If you differentiate $L = \frac{1}{16\pi}F^{ab}F^{cd}g_{ac}g_{bd} $, you will get the correct result.