  We compute the internal modes of a nonspinning neutron star and its tidal
metric perturbation in general relativity, and determine the effect of
relativistic corrections to the modes on mode coupling. Claims have been made
that a new hydrodynamic instability can occur in a neutron star in a binary
neutron star system triggered by the nonlinear coupling of the companion's
tidal field to pairs of p-modes and g-modes in it as the binary inspirals
toward merger. This "PG" instability may be significant since it can influence
the binary's inspiral phase by extracting orbital energy, thereby potentially
causing large deviations in their gravitational waveforms from those predicted
by theoretical models that do not account for it. This can result in incorrect
parameter estimation, at best, or mergers going undetected, at worst, owing to
the use of deficient waveform models. On the other hand, better modeling of
this instability and its effect on binary orbits can unravel a new phenomenon
and shed light on stellar instabilities, via gravitational wave observations.
So far, all mode-tide coupling instability studies have been formulated in
Newtonian perturbation theory. Neutron stars are compact objects, so
relativistic corrections might be important. We present and test a new code to
calculate the relativistic eigenmodes of nonrotating relativistic stars. We use
these relativistic tide and neutron star eigenmodes to compute the mode-tide
coupling strength (MTCS) for a few selected equations of state. The MTCS thus
calculated can be at most tens of percent different from its purely Newtonian
value, but we confirm the dependencies on orbital separation and equation of
state found by Newtonian calculations. For some equations of state, the MTCS is
very sensitive to the neutron star crust region, demonstrating the importance
of treating this region accurately.
