# Tidal deformation of a slowly rotating material body: Interior metric and Love numbers

## TL;DR

This paper derives and calculates the interior metric and Love numbers for slowly rotating bodies under tidal deformation, including new rotational-tidal Love numbers, with implications for neutron star responses in binary systems.

## Contribution

It introduces two new rotational-tidal Love numbers for barotropic fluids and provides explicit calculations for polytropes, expanding understanding of tidal responses in rotating bodies.

## Key findings

- Two rotational-tidal Love numbers depend on internal structure.
- Universal fixed value for two Love numbers across all barotropes.
- Estimated internal currents in neutron stars reach kilometers per second.

## Abstract

The metric outside a compact body deformed by a quadrupolar tidal field is universal up to its Love numbers, constants which encode the tidal response's dependence on the body's internal structure. For a non-rotating body, the deformed external geometry is characterized by the familiar gravitational Love numbers $K_2^{\text{el}}$ and $K_2^{\text{mag}}$. For a slowly rotating body, these must be supplemented by rotational-tidal Love numbers, which measure the response to couplings between the body's spin and the external tidal field. By integrating the interior field equations, I find that the response of a barotropic perfect fluid to spin-coupled tidal perturbations is described by two rotational-tidal Love numbers, which I calculate explicitly for polytropes. Two other rotational-tidal Love numbers identified in prior work are found to have a fixed, universal value for all barotropes. Equipped with the complete interior solution, I calculate the amplitude of the time-varying internal currents induced by the gravitomagnetic part of the tidal field. For a typical neutron star in an equal-mass binary system, the size of the equatorial velocity perturbation is on the order of kilometers per second.