Are some gravitational wavelengths forbidden by causality? Consider a gravitational wave in linearized gravity $d_{\mu \nu}(X_{\eta}) = D_{\mu \nu} e^{i X_{\eta} K^{\eta}}$ with $K^{\eta} = (-\omega t, \textbf{k})$. Let $d=| \textbf{D}|$ the scalar maximum amplitude of the wave, measured in distance units.
A gravitational wave makes in a single oscillation of period:
$$T= \frac{ 2\pi}{ \omega}$$
an orthogonal displacement between points separated by a distance $2d$. It would seem to me that if an amplitude of a gravitational wave exceeds the distance that rays could travel at the speed of light, then you could outrun a Light-cone, at least in principle.
If the above analysis is correct and it allows you to outrun a light-cone, maybe there is some solution to the breakdown of causality: 


*

*We ignore causality, or develop a quantum theory of causality that accounts for FTL tinkering

*Certain wavelengths in gravitational waves are forbidden


If we apply the Occam's razor, it would seem that the second option should be the first option to study. Is it possible that this expression holds for gravitational waves?
$$ \frac{d \omega}{\pi} < c $$
Basically disallowing wavelengths that would allow wavefronts to communicate spacelike-separated points of space faster than a light-cone could in the same time.
 A: The physical effects of gravitational wave (GW) is best understood in the transverse, traceless gauge. So, if a linearly polarized GW is propagating in the z-direction, it can have only $h_{11}= - h_{22}$ and $h_{12}= h_{21}$ as non-zero components which are orthogonal to the direction of propagation, along with the condition that the magnitudes of these components are much much less than unity.
In a time interval $T= 2 \pi/\omega$ (where $\omega$ is the angular frequency of a monochromatic GW), the separation $\Delta L$ between two test particles intercepting the GW can change by an amount of the order of $h L$, where $h$ and $L$ are the magnitude of GW amplitude and initial separation between the particles (provided $L$ is much much less than the wavelength of the GW, which is $2\pi c/\omega$).
Hence, $\Delta L/T= h L/T = h L \omega/2 \pi$, which is much much less than $h c$.
Since, $h$ is much much less than unity, $\Delta L/T$ is much much less than $c$. So, the rate of change of separation between the test particles can never exceed the speed of light.
