Gravitational waves induce changes in the $h_{00}$ (time) component of the metric? I'm rather stumped by a subtle point regarding metric perturbations of GW. I'm well aware the GW are able to produce changes in the flat space metric, They are transverse and have planes of polarizations (namely $\times$ and $+$). However a friend and I are arguing over if GW can also produce warps in time, His argument is that in the quadruple approximation there is no $h_{00}$ component, $h_{00}$ is not a solution of the wave equation. 
However he says once you leave the quadrupole approx regime such assumption can be valid. I believe that it is possible for a GW to induce both length variations and time variations just like what goes on in special relativity. Who is right? And can anyone provide a more solid explanation?
Thanks.
 A: Group theoretically you can split the graviton $h_{\mu \nu}$, where as usual $g_{\mu \nu}=\eta_{\mu \nu}+h_{\mu \nu}$,  in 3 irreducible representations of $SO(3)$: 
$$h_{00} \rightarrow Spin \, 0 \; (Scalar)$$
$$h_{i0}\rightarrow Spin \, 1 \; (Vector)$$
$$h_{\mu \nu}\rightarrow Spin \, 2\; (Tensor)$$
Before choosing a gauge, look at the equations of motion: the equation for the spatial components is something like $\partial_i \partial_j h_{00}= \dots$, where the derivatives are spatial. So there aren't time derivatives acting on $h_{00}$. This means that $h_{00}$ is not a true degree of freedom of the gravitational field, but can be determined from the other components. It turns out that the only propagating degrees of freedom are the tensorial ones. In four dimension this means 6 d.o.f, but after gauge fixing only 2 survive.
Notice that this a feature of General Relativity. In alternatives theories of gravitation (for instance with additional terms $R^2, R^4, \dots$ in the lagrangian, or with more fields $\phi, \psi, \dots..$) the scalar d.o.f can became dynamical.
This discussion is parallel to the one in electrodynamics, with the vector potential $A^{\mu}$.
Reference: S. Carroll, Spacetime and Geometry
