Compact body in a tidal environment: New types of relativistic Love numbers, and a post-Newtonian operational definition for tidally induced multipole moments

TL;DR

This paper develops a comprehensive relativistic framework for understanding tidal deformation of compact bodies, introducing new types of Love numbers and a robust operational definition for tidally induced multipole moments within a post-Newtonian context.

Contribution

It introduces new relativistic Love numbers, defines a robust operational method for multipole moments, and connects these to observable tidal effects in general relativity.

Findings

Defined four types of relativistic Love numbers including new quadratic and time-derivative types.

Proposed a post-Newtonian operational definition for tidally induced multipole moments.

Linked Love numbers to observable tidal accelerations in a relativistic setting.

Abstract

We examine the tidal deformation of a nonrotating compact body (material body or black hole) in general relativity. The body's exterior metric is calculated in a simultaneous expansion in powers of the ratio between the distance to the body and three distinct length scales: the radius of curvature of the external spacetime in which the body is inserted, the scale of spatial inhomogeneity of the curvature, and the scale of temporal variation. The metric is valid in the body's immediate neighborhood, which excludes the external matter responsible for the tidal environment. The body's tidal response is encapsulated in four types of relativistic Love numbers: $k_\ell$, the familiar Love number that measures the linear response to a static tidal field, $p_\ell$, which measures the quadratic response to the tidal field, $\dot{k}_\ell$ and $\ddot{k}_\ell$, associated with first and second time derivatives of the tidal field, respectively. The Love numbers acquire an operational meaning through the definition of tidally induced multipole moments. Previously proposed definitions for the moments suffer from ambiguities associated with the subtraction of a "pure tidal field" from the full metric. A robust operational definition is proposed here. It relies on inserting the body's local metric within a global metric constructed in post-Newtonian theory; the global metric includes the external matter responsible for the tidal environment. When viewed in the post-Newtonian spacetime, the compact body appears as a skeletonized object with a specific multipole structure. The tidally induced multipole moments provide a description of this structure. They manifest themselves, for example, in the body's tidal acceleration, which is nonlinear in the tidal field. At leading order in the tidal interaction, the acceleration is proportional to the $k_2$ Love number as calculated in full general relativity.