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From my understanding, the equivalence principle says that it is impossible to know if you are moving or are stationary (and everything else is moving around you).

Do gravitational waves violate this? If I am moving, detectors would be able to tell that gravitational waves are emanating from me. If, in fact, I am stationary and something else is moving, the waves would be propagating toward me. There would be an objective way to determine who is moving.

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Your question suggests that you may be confusing the Equivalence Principle with the Principle of Relativity. However, following your title, I have provided an answer regarding the Equivalence Principle only.

The Equivalence Principle tells you that local observations cannot distinguish between acceleration or a uniform gravitational field. The locality and uniformity restrictions are key (and, in fact, complementary) A gravitational wave is, but its nature, not a uniform gravitation field; it has a spatial scale (the wavelength) over which its distortion of spacetime varies.

When a gravitational field is nonuniform, it can easily be distinguished from an accelerational effect by making measurements over an extended region. However, such observations are, of course, not local. The locality criterion essentially means that you cannot distinguish gravitation from a fictitious translational force unless you compare measurement that are made over a region of space that is comparable to the scale on which the gravitational field varies. For a gravitational wave, this means that to observe its wave structure, you need to make measurements over a region that is at least comparable in size to the wavelength.

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The problem is that gravitational waves don't have a wavelength (or frequency). Those are frame dependent. So, you measure a frequency of 18.5 Hz. So what? How does that tie down a reference frame?

Also: If you detect the waves, they are moving towards you. Directly towards you, at the speed of light. No matter what.

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