# Confusing expression in GR

1. Jul 20, 2012

### unchained1978

Can anyone help me simplify the expression $\frac{\delta g_{\mu\nu}}{\delta g^{\kappa\lambda}}$? I haven't seen a term like this before and I don't know how to proceed. It seems like it might be the product of some kronecker delta's but I'm not sure.

2. Jul 21, 2012

### tom.stoer

You get a pair of Kronecker deltas

3. Jul 21, 2012

### Bill_K

Write the Kronecker delta as δμν = gμα gνα. take the variation: 0 = δgμα gνα + gμα δgνα. Now multiply by gμβ to solve for δgνα.

4. Jul 21, 2012

### w4k4b4lool4

This is a functional derivative, defined as follows:
$$\frac{\delta F[g(x)]}{\delta g(y)} \equiv \lim_{\epsilon\rightarrow 0}\frac{F[g(x)+\epsilon \delta(x-y)]-F[g(x)]}{\epsilon}$$
$$\frac{\delta g_{\mu\nu}(x)}{\delta g_{\rho\sigma}(y)} =\frac{1}{2}\Big(\delta_{\mu}^{\rho}\delta_{\nu}^{\sigma}+\delta_{\nu}^{\rho}\delta_{\mu}^{\sigma} \Big) \delta^D(x-y),$$