Is there a Doppler shift of gravity waves?

Question

Are gravity waves affected by distance/velocity, like light from distant objects is red-shifted? How could this be detected and what would be the effect on objects when the wave reached them?

Answer 1

Yes, totally!, It is possible that gravitational waves occuring from quasi-symmetric binary system of masses has doppler shift, just like any other wave traveling at c.

Some analogues of gravitational waves instead of photons:

1. Cosmological redshift - This can occur in gravitational waves since the Universe (is a consequence of FRLW model) is expanding, and since gravitational waves are just waves occur on spacetime, therefore they too are stretched just like electromagnetic fields along with the spacetime therefore increasing their wavelength and decreasing their frequency and energy. Technically the Young's modulus (the ratio of tensile stress (σ) to tensile strain (ε)) also plays role in this since the spacetime's stiffness is continuously increasing at a speeding rates, thus straining the spacetime even more, therefore the frequency and energy of the gravitational radiation is decreasing and wavelength is increasing, but this isn't a huge factor since the Young's modulus is less than 14 times the stiffness of Jello.

2. Gravitational redshift - The Gravitational wave is affected by a massive object just like photons because of the positive curvature warping on spacetime applied by the body a.k.a gravity

3. Relativistic Doppler shift - This occurs in spacetime because each succesive wave crust is emitted from a source further than the previous crest therefore increasing the time to reach the observer which causes the distance between wave crusts to increase ultimately decreasing the frequency.

All of these occur because the medium is spacetime on which these gravitational waves occur is just like any other surface

One evidence of gravitational redshift is the anthropotic principle, If those gravitational waves would not have been redshifted or inverse square law, life as we know it would not exist at all (including Earth because our planet would have been teared apart instead of moving less than the width of proton)

We can find out the redshift by using Gravitational-Wave Interferometers like LIGO.
And as Prof. Rob said in his answer, the doppler shift of gravitational waves is indistinguishable from changing the mass therefore cosmologists use the Standard Siren Technique combined with the cosmological model to find out the redshift in the luminosity-distance.

Note: Because of the similarities between gravitation waves and electromagnetic radiation, it is also called as gravitational radiation

Hope it answers your question , Thanks!