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We examine the tidal deformation of a nonrotating compact body (material bodyor black hole) in general relativity. The body's exterior metric is calculatedin a simultaneous expansion in powers of the ratio between the distance to thebody and three distinct length scales: the radius of curvature of the externalspacetime in which the body is inserted, the scale of spatial inhomogeneity ofthe curvature, and the scale of temporal variation. The metric is valid in thebody's immediate neighborhood, which excludes the external matter responsiblefor the tidal environment. The body's tidal response is encapsulated in fourtypes of relativistic Love numbers: $k_\ell$, the familiar Love number thatmeasures the linear response to a static tidal field, $p_\ell$, which measuresthe 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 thedefinition of tidally induced multipole moments. Previously proposeddefinitions for the moments suffer from ambiguities associated with thesubtraction of a "pure tidal field" from the full metric. A robust operationaldefinition is proposed here. It relies on inserting the body's local metricwithin a global metric constructed in post-Newtonian theory; the global metricincludes the external matter responsible for the tidal environment. When viewedin the post-Newtonian spacetime, the compact body appears as a skeletonizedobject with a specific multipole structure. The tidally induced multipolemoments provide a description of this structure. They manifest themselves, forexample, in the body's tidal acceleration, which is nonlinear in the tidalfield. At leading order in the tidal interaction, the acceleration isproportional to the $k_2$ Love number as calculated in full general relativity.

physical review. d/physical review. d.

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