# Gauge invariance in gravitational field

I have read that the linearized equation for the metric fluctuations $h_{\mu\nu}$, namely:

$$\partial^2h^{\mu\nu}-\partial_{\alpha}(\partial^{\mu}h^{\nu\alpha}+\partial^{\nu}h^{\mu\alpha}) +\partial^{\mu}\partial^{\nu}h=0$$

is invariant under the gauge transformations:

$$h^{\mu\nu}(x) \quad\rightarrow\quad h^{\mu\nu}(x)+\partial^{\mu}\epsilon^{\nu}(x)+\partial^{\nu}\epsilon^{\mu}(x)$$

This is supposed to be the reparametrization invariance, that is, the invariance under any choice of coordinate system. The problem is that I can't see how that transformation accounts for a change of coordinate system.

• Do you know how the components of the metric tensor (or any 2-tensor) transform under a change of coordinates? – Robin Ekman Apr 6 '16 at 2:13
• Yes I do: $g'^{\mu\nu}=\Lambda^{\mu}_{\sigma}\Lambda^{\nu}_{\lambda}g^{\sigma\lambda}$ – Kirov Apr 6 '16 at 7:55