Gravitational waves from pulsating stars: Evolving the perturbation equations for a relativistic star

We consider the perturbations of a relativistic star as an initial-value problem. Having discussed the formulation of the problem (the perturbation equations and the appropriate boundary conditions at the center and the surface of the star) in detail, we evolve the equations numerically from several different sets of initial data. In all the considered cases, we find that the resulting gravitational waves carry the signature of several of the star’s pulsation modes. Typically, the fluid $f$ mode, the first two $p$ modes, and the slowest damped gravitational $w$ mode are present in the signal. If such mode signals, from coalescing neutron stars or following a supernova, can be detected by future gravitational-wave antennae, one can hope to infer detailed information about neutron stars. Since a perturbation evolution should adequately describe the late time behavior of a dynamically excited neutron star, the present work can also be used as a benchmark test for future fully nonlinear simulations.