Compact body in a tidal environment: New types of relativistic Love numbers, and a post-Newtonian operational definition for tidally induced multipole moments
Eric Poisson
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 first is the radius of curvature of the external spacetime in which the body is inserted, the second is the scale of spatial inhomogeneity of the curvature, and the third is 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. The first is ${k}_{\ensuremath{\ell}}$, the familiar Love number that measures the linear response to a static tidal field. A second is ${p}_{\ensuremath{\ell}}$, which measures the quadratic response to the tidal field. A third is ${\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{k}}_{\ensuremath{\ell}}$, which is associated with the first time derivative of the tidal field. And the fourth is ${\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{k}}_{\ensuremath{\ell}}$, associated with the second time derivative of the tidal field. All Love numbers are gauge-invariant constants that appear in the body's exterior metric. Their computation (not carried out here, except for black holes) requires extending the metric to the body's interior. 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; such subtraction prescriptions are artificial and subjective. 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, moving on a given world line. The tidally induced multipole moments provide a description of this multipole 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. The computation of the multipole moments is carried out to the first post-Newtonian order, but the general method can in principle be extended to higher orders.