Hey I'm stuck on the last portion of a question
For a plane gravity wave of the form
h_{\mu \nu} = A_{\mu \nu} \exp (i k_{\lambda}x^{\lambda})
Show that the linearised gravitational field equations require that
k^2 h_{\mu\nu} = k_{\nu} w_{\mu} + k_{\mu} w_{\nu}

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

Hence show that for the case k^2=0 that k^{\rho} is an eigenvector of the Riemann tensor in the sense that R_{\sigma \mu \nu \rho}k^{\rho} = 0
Right I've done it up to the point of showing that
<br />
R_{\sigma \mu \nu \rho}k^{\rho} = - k_{\nu} k_{\mu} w_{\sigma}

So I've reduced it to showing that w_{\sigma}=0 which I can't really seem to get out, any hints would be much appreciated

Thanks!