(a) In the transverse traceless gauge, a plane gravitational wave propagating in the $z$ direction is described by a perturbation $h_{\alpha \beta}$ of the Minkowski metric $\eta_{\alpha \beta}=$ $\operatorname{diag}(-1,1,1,1)$ in Cartesian coordinates $x^{\alpha}=(t, x, y, z)$, where

$$h_{\alpha \beta}=H_{\alpha \beta} e^{i k_{\mu} x^{\mu}}, \quad \text { where } \quad k^{\mu}=\omega(1,0,0,1)$$

and $H_{\alpha \beta}$ is a constant matrix. Spacetime indices in this question are raised or lowered with the Minkowski metric.

The energy-momentum tensor of a gravitational wave is defined to be

$$\tau_{\mu \nu}=\frac{1}{32 \pi}\left(\partial_{\mu} h^{\alpha \beta}\right)\left(\partial_{\nu} h_{\alpha \beta}\right)$$

Show that $\partial^{\nu} \tau_{\mu \nu}=\frac{1}{2} \partial_{\mu} \tau_{\nu}^{\nu}$ and hence, or otherwise, show that energy and momentum are conserved.

(b) A point mass $m$ undergoes harmonic motion along the $z$-axis with frequency $\omega$ and amplitude $L$. Compute the energy flux emitted in gravitational radiation.

[Hint: The quadrupole formula for time-averaged energy flux radiated in gravitational waves is

\left\langle\frac{d E}{d t}\right\rangle=\frac{1}{5}\left\langle\dddot{Q}_{i j} \dddot{Q}_{i j}\right\rangle

where $Q_{i j}$ is the reduced quadrupole tensor.]