Modulations of Gravitational Waves due to Non-static Gravitational Lenses

Gravitational waves (GWs) offer a new observational window into the universe, providing insights into compact objects and cosmic structures. Gravitational lensing, commonly studied in electromagnetic waves, also affects GWs, introducing magnification, time delays, and multiple images. While existing studies focus on static lenses, many astrophysical lenses are dynamic, with time-varying mass distributions such as moving stars or orbiting binaries. We develop a general theoretical framework to describe non-static lenses and demonstrate how they modulate GW signals, inducing unique time-varying amplitude modulations and spectral broadening. By examining uniformly moving and orbiting binary lenses, we show that these modulations provide new observational signatures, enhancing our understanding of lensing objects and the dynamics of the universe. Our findings have important implications for GW astronomy, offering novel ways to probe lens dynamics and improve the interpretations of GW signals.