Planck length compression of gravitational waves due to gravitational time dilation

I've made an error in my thinking somewhere concerning gravitational waves in the following thought experiment, and would appreciate knowing where I went wrong:

Consider two overlapping gravitational waves, with regions both ahead and behind the overlapping area. Relative to a far away observer, the overlapping area would be traveling slightly below c, due to gravitational time dilation.

As a result, the region behind the overlapping area of the two gravitational waves would eventually 'catch up' to the overlapping area. However, once any portion of that region enters the overlapping area of the two waves, relative to the far away observer, it would appear to travel at the same speed as the overlapping region, slightly slower than c.

Considering that, according to current theory, it is impossible to observe events below the Planck length and time (and further, that such notions become meaningless, in that quantum gravitational effects become important, and classical notions of space and time break down) the observer would, according to this assumption, only be able to observe the effects of the overlapping area on the incoming region of the gravitational wave (and vice versa) after a Planck time (at absolute minimum) or when the incoming region of the wave had traveled a Planck length into the overlapping area.

Therefore, when such conditions are satisfied, the far away observer sees that the incoming portion of the wave has traveled a Planck length into the overlapping region, and is now traveling at the same speed as the overlapping area of the two waves. This process repeats until the entire region of the waves situated initially behind the overlapping area of the two gravitational waves has been 'compressed' into a Planck length.

Eventually said region 'splits off' from the initial overlapping area, once the gravitational time dilation of the 'compressed' region becomes significant enough to observe, as all portions of the incoming wave that was initially situated behind the overlapping region affect each other, producing a collective slow down relative to the observer.

• "Relative to a far away observer, the overlapping area would be traveling slightly below c, due to gravitational time dilation." can you please elaborate on this? May 19, 2020 at 3:46
• The idea is that, because gravitational waves are themselves affected by gravity, the waves that overlap experience each other's gravitational acceleration. As a result of experiencing said acceleration, they would experience time dilation relative to a faraway observer, and their propagation speed in that reference frame would be affected. May 19, 2020 at 4:04
• This idea is described somewhat on the wiki page for gravitational time dilation, under "Important features of gravitational time dilation", where they discuss observers tracking remote light signals. Link: en.wikipedia.org/wiki/Gravitational_time_dilation May 19, 2020 at 4:14
• "gravitational waves are themselves affected by gravity, the waves that overlap experience each other's gravitational acceleration." You may be confusing a static gravitational field, with a GW. Now the wiki article is about static gravitational fields. GWs cause stretching and squeezeing of spacetime itself, but do not cause gravitational time dilation like a static field would. A static field has a certain center of mass, and a gravitational potential. This gravitational potential is relatively static. This is why you can compare to clocks being in two static gravitational fields. May 19, 2020 at 15:49
• You cannot do that with two GWs, since they change so fast, they do not have a relatively static effect of gravitational potential. Two GWs do effect each other, but that is called damping. May 19, 2020 at 15:50