Relativistic excitation of compact stars

In this work, we study the excitation of a compact star under the influence of external gravitational driving in the relativistic regime. Using a model setup in which a wave with constant frequency is injected from past null infinity and scattered by the star to future null infinity, we show that the scattering coefficient encodes rich information of the star. For example, the analytical structure of the scattering coefficient implies that the decay rate of a mode generally plays the role of ``star excitation factor,'' similar to the ``black hole excitation factor'' previously defined for describing black hole mode excitations. With this star excitation factor, we derive the relativistic transient mode excitation as a binary system crosses a generic mode resonance of a companion star during the inspiral stage. This application is useful because previous description of resonant mode excitation of stars still relies on the mode and driving force decomposition based on the Newtonian formalism. The relativistic transient mode energy may differ from the Newtonian prediction by one order of magnitude. In addition, we show that the scattering phase is intimately related to the total energy of spacetime and matter under the driving of a steady input wave from infinity. We also derive the relevant tidal energy of a star under steady driving and compare that with the dynamic tide formula. We estimate that the difference may lead to $\mathcal{O}(0.5)$ radian phase modulation in the late stage of the binary neutron star inspiral waveform.