Relativistic theory of tidal Love numbers
Taylor Binnington, Eric Poisson
TL;DR
The paper presents a fully relativistic formulation of tidal Love numbers, defining electric-type $k_{ m el}$ and magnetic-type $k_{ m mag}$ for compact bodies and deriving their gauge-invariant meanings in a linear perturbation framework. By solving the external vacuum problem in Eddington–Finkelstein coordinates and the internal stellar problem for polytropes using a light-cone gauge, the authors obtain precise matching conditions at the stellar surface that yield the Love numbers. A key finding is that nonrotating black holes have vanishing tidal Love numbers, while neutron-star–like bodies exhibit strong, EOS-dependent tidal deformabilities that decrease with increasing compactness and stiffness. The numerical results provide comprehensive data for various multipoles ($l=2$–$5$) and polytropic indices, informing gravitational-wave modeling and EOS inference from tidal imprints in inspiral signals.
Abstract
In Newtonian gravitational theory, a tidal Love number relates the mass multipole moment created by tidal forces on a spherical body to the applied tidal field. The Love number is dimensionless, and it encodes information about the body's internal structure. We present a relativistic theory of Love numbers, which applies to compact bodies with strong internal gravities; the theory extends and completes a recent work by Flanagan and Hinderer, which revealed that the tidal Love number of a neutron star can be measured by Earth-based gravitational-wave detectors. We consider a spherical body deformed by an external tidal field, and provide precise and meaningful definitions for electric-type and magnetic-type Love numbers; and these are computed for polytropic equations of state. The theory applies to black holes as well, and we find that the relativistic Love numbers of a nonrotating black hole are all zero.