Tidal Excitation of Relativistic Stars

Abstract

How a relativistic star responds to a time-dependent tide has lacked a first-principles counterpart of the Newtonian normal-mode expansion. In general relativity, the tidal field is gauge sensitive, radiative modes are non-Hermitian and incomplete, and the direct and late-time responses are intrinsically nonmodal. In this work we present, to our knowledge, the first causal relativistic theory that treats all of these components on equal footing. We first develop the spectral decomposition of the retarded Green function of the matched stellar interior--Schwarzschild exterior, which classifies four separate regimes in time domain: acausal region, prompt wave, surface-reflected wave, and mode+tail. The prompt signal is identical to that of a Schwarzschild black hole with the same mass, which is combined with the surface-reflected wave to be referred as the"direct part". With the Green function and a complex bilinear form on the full metric--fluid phase space, we decompose the linear response of a nonrotating perfect-fluid star into direct, quasinormal-mode, and late-time-tail sectors. Crucially, every sector admits an explicit first-order driven evolution equation: discrete bilinear projections determine the dynamically evolving quasinormal-mode amplitudes, while continuum projections of the same tidal source onto branch-cut states determine the direct and tail amplitudes. The result is a complete modal--nonmodal decomposition of the spacetime and fluid perturbations, which is physically described by the dynamics of a discrete and a continuous-frequency set of oscillators. We verify the decomposition numerically under generic time-dependent tidal driving and discuss applications to black hole perturbations.