Now that we believe very strongly in gravitational waves (thanks partly to the Hulse-Taylor binary), we have to accept that one consequence is time-asymmetry in the basic equations of motion.
If any non-inertial motion of matter ... any acceleration of mass involving forces ... results in the generation of gravitational waves that carry away energy in forward time, then any thermal system (such as a container of hot gas) must continually be losing energy in forward time (and gaining energy in reversed time), through g-wave emission (and absorption).
This radiated energy needs to come from somewhere (and in reversed time, the converging absorbed energy needs to go somewhere), so our basic equations of motion need to lose energy in forward time, and gain energy in reversed time.
If one atom emits a photon and another atoms receives it, then since the recoil from both events must result in g-waves carrying away energy, the photon transfer must be lossy. Since the recoil has to generate the same g-wave as a function of the energy of the photon, regardless of the atom's mass, the energy-loss must be a function of the momentum-change of the atom, and of the mometum carried by the photon.
The energy-transfer and Doppler equations need to support g-wave "taxation" of transferred energy, and need to be "lossy" in forward time.
But the SR equations of motion are perfectly time-symmetrical and predict absolutely zero energy-loss or gain.
Therefore, if gravitational waves are real, and carry energy, the SR equation-set cannot be correct. Current physics must be founded on an incorrect set of foundational equations.