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We compute the internal modes of a nonspinning neutron star and its tidalmetric perturbation in general relativity, and determine the effect ofrelativistic corrections to the modes on mode coupling. Claims have been madethat a new hydrodynamic instability can occur in a neutron star in a binaryneutron star system triggered by the nonlinear coupling of the companion'stidal field to pairs of p-modes and g-modes in it as the binary inspiralstoward merger. This "PG" instability may be significant since it can influencethe binary's inspiral phase by extracting orbital energy, thereby potentiallycausing large deviations in their gravitational waveforms from those predictedby theoretical models that do not account for it. This can result in incorrectparameter estimation, at best, or mergers going undetected, at worst, owing tothe use of deficient waveform models. On the other hand, better modeling ofthis instability and its effect on binary orbits can unravel a new phenomenonand shed light on stellar instabilities, via gravitational wave observations.So far, all mode-tide coupling instability studies have been formulated inNewtonian perturbation theory. Neutron stars are compact objects, sorelativistic corrections might be important. We present and test a new code tocalculate the relativistic eigenmodes of nonrotating relativistic stars. We usethese relativistic tide and neutron star eigenmodes to compute the mode-tidecoupling strength (MTCS) for a few selected equations of state. The MTCS thuscalculated can be at most tens of percent different from its purely Newtonianvalue, but we confirm the dependencies on orbital separation and equation ofstate found by Newtonian calculations. For some equations of state, the MTCS isvery sensitive to the neutron star crust region, demonstrating the importanceof treating this region accurately.

Nonlinear mode-tide coupling in coalescing binary neutron stars with relativistic corrections