Gravitational-wave effects in the most general vector-tensor theory

In this paper, we establish a model-independent framework based on the Isaacson picture to analyze the gravitational-wave effects in the most general vector-tensor theory that yields second-order field equations. Within this framework, we derive two basic sets of equations for the Isaacson picture. These equations enable the analysis of gravitational wave polarization modes, the dispersion relation of each mode, the effective energy-momentum tensor of gravitational waves, and the memory effects. These features are closely tied to observable phenomena and have attracted considerable attention. They are expected to be detected by the next generation of gravitational wave observatories designed to test potential modifications to general relativity. Using this framework, we present the explicit expression for the effective energy-momentum tensor of gravitational waves in the most general second-order vector-tensor theory and perform a complete analysis of their polarization modes.