Even better than that! It should experience Doppler shifted waves, and slow down due to CMB (E&M, and whatever equivalent G has) via inverse Compton scattering, or whatever classical wave equivalent you would use to explain the same mechanism.
Obviously, this wouldn't be very important until, well, relativistic speeds, but that's what relativity is about, after all...
I suppose this should perhaps be detectable by particle physics, if the effect depends more on speed than density. Obviously, the much stronger effect from EM has to be subtracted, but perhaps that can be done sufficiently accurately in the lab?
Whereas, if density has to be near-relativistic (say, near the Chandrasekhar limit), that's kind of a non-starter for the lab...
Interesting science, I think, will come out of looking at the higher order terms of common metrics. For instance, you usually just want to know the unaccelerated (inertial) reference frames, so you just cross out all the acceleration terms. But if you've got mass and energy sloshing around (e.g., Woodward effect), and difference out the boring ordinary terms, who knows? You've got to do your homework to find out!
Tim