gravitational waves are strictly transversal (in the linear regime at least), also their amplitudes are tiny even for cosmic scale events like supernovas or binary black holes (at least far away, maybe we should ask some physicists located a bit closer to the center of the galaxy), but lets put all those facts aside for a second and consider a gravitational source big enough to generate gravitational waves with amplitudes of the order of the galaxy. For instance consider a planar wave like in my mediocre drawing:
$$ h_{\alpha \beta} e^{i (k_{y} y – \omega t)} $$
where
$$ h_{\alpha \beta} \approx 1 $$
so the perturbation is in the nonlinear regime
i draw two far away objects in three different time slices (this is why they are repeated 3 times), the topmost is the objects without the gravitational wave, the one in the middle represents the objects in the crest of the gravitational wave, and the one in the bottom represents the objects in the valley of the wave.
So, my point is that people would only have to travel an arbitrarily small distance when the wave is on the valley (assuming circular polarization) even if the "normal" distance (i.e: $h_{\mu \nu} = 0$) is several light-years away
Besides being slightly impractical to set up such a mammoth gravitational source, this kind of warp drive is valid from a physical standpoint? Are there any physical limits to gravitational wave amplitudes in such nonlinear regime?
Best Answer
I don't think you could use this as a warp drive unless you could collimate the gravity waves. If you consider a spaceship moving at constant velocity through a gravity wave, the ship will be accelerated then decelerated again as the wave passed through but it's average velocity would be unchanged. The only way you could get a net effect from the wave is if you could move from a region of high amplitude to low amplitude within half a cycle of the wave. I can't think of any (plausible) geometry that would allow this. Possibly you could do it very close to a black hole binary, where the gravity wave generation doesn't look like a point source.