Tidal deformation of a slowly rotating material body. External metric

TL;DR

This work develops a relativistic external metric for a slowly rotating, tidally deformed body in general relativity, introducing four rotational-tidal Love numbers in addition to the standard gravitoelectric and gravitomagnetic Love numbers $K^{\rm el}_2$ and $K^{\rm mag}_2$. By constructing tidal potentials from the spin and tidal tensors and solving perturbatively in two gauges (light-cone and Regge-Wheeler), the authors show that all six Love numbers are gauge-invariant and vanish for a black hole, while remaining undetermined without an internal matter model (to be addressed in Paper II). A post-Newtonian analysis provides scaling insights for the new rotational-tidal Love numbers, suggesting how they depend on the body's radius $R$ and compactness, with specific results such as $\bigl( GM/c^2 \bigr)^5 \mathfrak{F}^{\sf o} = -\frac{75}{1792} R^5$. Overall, the paper connects external tidal response to internal structure and lays groundwork for interpreting gravitational-wave signals from tidally deformed, slowly rotating bodies such as neutron stars.

Abstract

We construct the external metric of a slowly rotating, tidally deformed material body in general relativity. The tidal forces acting on the body are assumed to be weak and to vary slowly with time, and the metric is obtained as a perturbation of a background metric that describes the external geometry of an isolated, slowly rotating body. The tidal environment is generic and characterized by two symmetric-tracefree tidal moments E_{ab} and B_{ab}, and the body is characterized by its mass M, its radius R, and a dimensionless angular-momentum vector χ^a << 1. The perturbation accounts for all couplings between χ^a and the tidal moments. The body's gravitational response to the applied tidal field is measured in part by the familiar gravitational Love numbers K^{el}_2 and K^{mag}_2, but we find that the coupling between the body's rotation and the tidal environment requires the introduction of four new quantities, which we designate as rotational-tidal Love numbers. All these Love numbers are gauge invariant in the usual sense of perturbation theory, and all vanish when the body is a black hole.