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Abstract

Gravitomagnetic quasi-normal modes of neutron stars are resonantly excited by
tidal effects during a binary inspiral, leading to a potentially measurable
effect in the gravitational wave signal. We take an important step towards
incorporating these effects in waveform models by developing a relativistic
effective action for the gravitomagnetic dynamics that clarifies a number of
subtleties. Working in the slow-rotation limit, we first consider the
post-Newtonian approximation and explicitly derive the effective action from
the equations of motion. We demonstrate that this formulation opens a novel way
to compute mode frequencies, yields insights into the relevant matter
variables, and elucidates the role of a shift symmetry of the fluid properties
under a displacement of the gravitomagnetic mode amplitudes. We then construct
a fully relativistic action based on the symmetries and a power counting
scheme. This action involves four coupling coefficients that depend on the
internal structure of the neutron star and characterize the key matter
parameters imprinted in the gravitational waves. We show that, after fixing one
of the coefficients by normalization, the other three directly involve the two
kinds of gravitomagnetic Love numbers (static and irrotational), and the mode
frequencies. We discuss several interesting features and dynamical consequences
of this action. Our results provide the foundation for deriving precision
predictions of gravitomagnetic effects, and the nuclear physics they encode,
for gravitational-wave astronomy.