Gravitational waves travel at the speed of light. A photon travelling in exactly the same direction as a gravitational wave will therefore remain in exactly the same position relative to the wave - at the peak, the trough, or somewhere in between. So my question is: if we're observing light emitted from the same direction as a source of gravitational waves (or even light emitted from the GW source), would we be able to detect any difference compared to light observed later, after the GW strength had dropped below detectable level?
The answers to another question (Effect of Gravitational Waves on light?) mention "riding" a GW and that "for the impacted time, [the GW] should impact speed of light, time, wavelength etc", but none of the answers provide specific detail on what the actual effect would be.
If the stretching/squeezing of spacetime was in the same direction as the GW, I'd expect the light to be alternatively redshifted and blueshifted in pulses of the same frequency as the GW. [Whether we could develop the technology to detect this is another matter]. However, as Paul T notes in his answer, the stretching/squeezing occurs in the transverse directions. I'm at a loss to picture what this does to the light when we observe it.
So I'm interested to know what the effect of light "riding" gravitational waves would be, and whether any effect would actually be observable.
I'm assuming the effect would be ridiculously tiny; but on the other hand, if LIGO can detect an oscillation in spacetime the size of a proton...