Hartle's model within the general theory of perturbative matchings: the change in mass

TL;DR

This paper analyzes Hartle's model for slowly rotating stars in General Relativity, revealing that certain assumptions about metric function continuity at the boundary affect the calculation of the star's mass change.

Contribution

It applies modern perturbed matching theory to Hartle's model, showing that non-vanishing boundary energy density leads to discontinuities in second order metric functions.

Findings

Discontinuity in second order metric function when boundary energy density is non-zero.

Implication for accurate mass change calculations in rotating stars.

Highlights the importance of boundary conditions in perturbative models.

Abstract

Hartle's model provides the most widely used analytic framework to describe isolated compact bodies rotating slowly in equilibrium up to second order in perturbations in the context of General Relativity. Apart from some explicit assumptions, there are some implicit, like the "continuity" of the functions in the perturbed metric across the surface of the body. In this work we sketch the basics for the analysis of the second order problem using the modern theory of perturbed matchings. In particular, the result we present is that when the energy density of the fluid in the static configuration does not vanish at the boundary, one of the functions of the second order perturbation in the setting of the original work by Hartle is not continuous. This discrepancy affects the calculation of the change in mass of the rotating star with respect to the static configuration needed to keep the central energy density unchanged.