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I may be misunderstanding gravitational waves, but doesn't the idea of a gravitational wave imply that spacetime is a field or "medium" through which waves propagate at speed $c$? Was not this idea something disproven by the Michelson-Morley experiment, which discredits the idea that spacetime itself could be a frame of reference? Or would gravitational waves travel more like electromagnetic waves and are consistent in every reference frame thanks to Lorentz transformations?

I guess part of what I'm asking is, from the perspective of gravitational waves, if a massive object is moving through space at a high speed, then is there a gravitational-wave "doppler effect" (thanks to the limit of $c$) or does the gravitational wave propagate in all directions at the same speed relative to the position of the moving object?

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    $\begingroup$ No. Like electromagnetic waves, gravitational waves do not require a medium to travel through. $\endgroup$ – Prahar Oct 21 '15 at 21:21
  • $\begingroup$ @prahar See edited question for clarification of what I'm asking. $\endgroup$ – Danegraphics Oct 21 '15 at 21:30
  • $\begingroup$ The same light cones that describe light rays describe gravitational rays. $\endgroup$ – CR Drost Oct 21 '15 at 21:56
  • $\begingroup$ Assuming gravitational waves exist, so far they haven't been detected, they will propagate at the same speed in all directions. Unless GR is wrong there is no more gravitational ether wind than electromagnetic one. "Spacetime itself" is not a physical entity, especially in GR, which is covariant under any changes of coordinates, and certainly not a medium for any waves. Neither is "field", which is just a distribution of intensities. Early examples of waves were mechanical and spread in a medium, but that does not mean that all waves have to be mechanical and spread in a medium. $\endgroup$ – Conifold Oct 21 '15 at 23:23
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    $\begingroup$ Relative to anything. Under relativistic addition c added to anything is c. As long as the "waves" carry energy they will be moving through vacuum at c relative to any observer and each other. They are physically incapable of moving at any speed other than c, Doppler effect affects frequency, but not speed. en.wikipedia.org/wiki/Relativistic_Doppler_effect You might be implicitly thinking of the classical rule for adding velocities instead of the relativistic one. en.wikipedia.org/wiki/… $\endgroup$ – Conifold Oct 21 '15 at 23:42
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For practical purposes, gravitational waves travel in the same ways as light waves, and definitely at the same speed --- $c$. Like with light, there is no preferred reference frame - so all observers will agree that they are moving at the speed of light (in all directions).

Regarding the Michelson-Morley experiment: the idea that this disproved a 'medium through which waves propagate', is (now) more of a philosophical point than a physical one. What these experiments demonstrated, is that there is no preferred reference frame for light, or more generally, things traveling at the speed of light.

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