Review on the matching conditions for the tidal problem: towards the application to more general contexts

TL;DR

This paper reviews the matching conditions used in the tidal problem for stars, providing a geometric framework that can be applied to more general matter configurations and stellar models.

Contribution

It offers a geometric derivation of the matching conditions for the tidal problem, enabling their application to diverse matter fields and complex stellar structures.

Findings

Recovered known results for perfect fluid stars

Extended matching conditions to superfluid neutron stars

Provided a geometric framework for general matter configurations

Abstract

The tidal problem is used to obtain the tidal deformability (or Love number) of stars. The semi-analytical study is usually treated in perturbation theory as a first order perturbation problem over a spherically symmetric background configuration consisting of a stellar interior region matched across a boundary to a vacuum exterior region that models the tidal field. The field equations for the metric and matter perturbations at the interior and exterior regions are complemented with corresponding boundary conditions. The data of the two problems at the common boundary are related by the so called matching conditions. These conditions for the tidal problem are known in the contexts of perfect fluid stars and superfluid stars modelled by a two-fluid. Here we review the obtaining of the matching conditions for the tidal problem starting from a purely geometrical setting, and present them so that they can be readily applied to more general contexts, such as other types of matter fields, different multiple layers or phase transitions. As a guide on how to use the matching conditions, we recover the known results for perfect fluid and superfluid neutron stars.