Gravitational Wave Energy Transported by Unit Area

[quasar987]

Homework Statement

I am given the form of the perturbation in the metric:

$$h_{\mu\nu}=\left(\begin{array} {cccc} 0&0&0&0 \\ 0&1&0&0\\ 0&0&-1&0 \\ 0&0&0&0\end{array}\right) \gamma e^{-(z-t)^2}$$

Where gamma<<1. That is to say, $$g_{\mu\nu}(\mathbf{r},t)=\eta_{\mu\nu}+h_{\mu\nu}(\mathbf{r},t)$$ (we use (+---) for eta (Minkowski))

h (or rather, all 16 of its terms) has the form of a plane wave sailing in the z direction at the speed of light c=1.

I am simply asked to find the gravitational energy transported by (transerse) unit area by the wave from t=-infty to t=+infty.

The Attempt at a Solution

I was about to write h as a Fourier integral but I don't know what I'm going to do after that, so is this even a good start?