Energy Loss and Motion of Compact Binary Systems in Einstein–Æther Gravity

In this study, we analyze Einstein–Æther (Æ) theories of gravity using the post-Minkowskian approach. These theories extend general relativity by introducing a dynamical, long-range vector field with a fixed norm that defines a preferred time direction at each spacetime point. We focus on the three-parameter subclass of Æ-theories in which the speed of tensor gravitational waves equals the speed of light, a condition confirmed to extraordinary precision by the joint observation of gravitational waves and gamma-ray bursts from the neutron-star merger GW170817.

We applied the complete post-Minkowskian formalism, together with the Direct Integration of the Relaxed Einstein Equations (DIRE) method, to derive the equations of motion and gravitational radiation from orbiting compact bodies to high orders in a post-Newtonian expansion. We adopted the method pioneered by Eardley for incorporating the effects of the Æther field on the internal structure and mass of compact, self-gravitating bodies, introducing dimensionless “sensitivities” that quantify how each body’s mass varies in response to changes in the Æther field.

We present the equations of motion for compact binaries through 2.5 post-Newtonian (PN) order. In particular, we compare and contrast the resulting radiation-reaction terms at 1.5PN (dipole) and 2.5PN orders with previous results based on Noether-charge calculations of the energy flux. These results reveal new features of gravitational-wave emission in Lorentz-violating gravity theories and provide the theoretical basis for testing Einstein–Æther gravity with present and future gravitational-wave observations.