For a plane gravity wave of the form

$\displaystyle h_{\mu \nu} = A_{\mu \nu} \exp (i k_{\lambda}x^{\lambda})$

Show that the linearised gravitational field equations require that

$\displaystyle k^2 h_{\mu\nu} = k_{\nu} w_{\mu} + k_{\mu} w_{\nu} $

Where $\displaystyle k^2 = k^{\rho}_{\rho}$

and $\displaystyle w_{\mu} = k^{\rho}(h_{\mu \rho} -\frac{1}{2} \eta_{\mu \rho} h^{\lamda}_{\lambda})$

Hence show that for the case $\displaystyle k^2=0$ that $\displaystyle k^{\rho}$ is an eigenvector of the Riemann tensor in the sense that $\displaystyle R_{\sigma \mu \nu \rho}k^{\rho} = 0$