Analyzing gravitational wave effects in general modified gravity: an example based on the most general vector-tensor theory

The Isaacson picture provides a general method for describing the effects of gravitational waves, established on two sets of basic equations. The first set of equations describes the propagation of gravitational waves within a given background, while the second set elucidates the interaction between the effective energy-momentum tensor of gravitational waves and the background metric. These two sets of equations are typically derived by expanding the field equations to second-order perturbations. In addition to the method of perturbing field equations, there is an alternative method: obtaining the two sets of basic equations by expanding the action to the second-order of perturbations. In this paper, we elaborate on this method, establishing its foundations more rigorously. Especially, the second-order perturbation action, with the Minkowski metric as the background, contains all necessary information to construct the Isaacson picture far from the source. We outline the derivation of the two sets of basic equations to describe the gravitational wave effects in the Isaacson picture, based on the second-order perturbation action. This will enable us to analyze the following effects: polarizations of gravitational waves, the wave speed for each polarization mode, the effective energy-momentum tensor of gravitational waves, and the nonlinear memory effect of gravitational waves. We illustrate this method by applying it to the most general second-order vector-tensor theory including parity-breaking terms. We further analyze the polarization modes of gravitational waves in this theory. We highlight that this method facilitates model-independent research on various gravitational wave effects in general modified gravity theories. These effects are anticipated to be identified by the upcoming generation of gravitational wave detectors, aimed at testing potential modifications to gravity theory.