Revisiting Hartle's model using perturbed matching theory to second order: amending the change in mass

TL;DR

This paper revisits Hartle's model of rotating compact stars using a modern, gauge-independent perturbative matching theory, revealing a necessary correction to the mass change due to a jump in the second-order perturbation function.

Contribution

It introduces a coordinate- and gauge-independent second-order matching framework and corrects the mass change calculation in Hartle's model.

Findings

The second-order perturbation function $m_0$ has a jump at the star's surface.

The mass change must be amended to account for the jump in $m_0$.

The correction impacts the modeling of rotating compact stars.

Abstract

Hartle's model describes the equilibrium configuration of a rotating isolated compact body in perturbation theory up to second order in General Relativity. The interior of the body is a perfect fluid with a barotropic equation of state, no convective motions and rigid rotation. That interior is matched across its surface to an asymptotically flat vacuum exterior. Perturbations are taken to second order around a static and spherically symmetric background configuration. Apart from the explicit assumptions, the perturbed configuration is constructed upon some implicit premises, in particular the continuity of the functions describing the perturbation in terms of some background radial coordinate. In this work we revisit the model within a modern general and consistent theory of perturbative matchings to second order, which is independent of the coordinates and gauges used to describe the two regions to be joined. We explore the matching conditions up to second order in full. The main particular result we present is that the radial function $m_0$ (in the setting of the original work) of the second order perturbation tensor, contrary to the original assumption, presents a jump at the surface of the star, which is proportional to the value of the energy density of the background configuration there. As a consequence, the change in mass needed by the perturbed configuration to keep the value of the central energy density unchanged must be amended. We also discuss some subtleties that arise when studying the deformation of the star.