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

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 δM 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.

Slowly rotating supercompact Schwarzschild stars

“…The complementary functions here satisfy the homogeneous forms of equations (3.74) and (3.75) 85) which have the following behaviors near the origin…”

“…Recently Reina & Vera [84,85] revisited Hartle's framework within the context of the modern theory of perturbed matchings [56]. They found that the perturbative functions at first and second order are continuous across the boundary of the configuration except when the energy density is discontinuous there.…”

Completion of the universal I–Love–Q relations in compact stars including the mass

“…Assumption (iii) has significant implications on the properties of the stellar models. As shown by Reina & Vera (2015) using the framework of Mars (2005), a relevant perturbative function presents a discontinuity proportional to the (background) energy density in the stellar surface. This function enters the computation of the mass of the star at second order δ M and, therefore, the original expression for the total mass of the rotating star given in Hartle (1967) has to be amended.…”

First order perturbations of hypersurfaces of arbitrary causal character

“…There is, however, one assumption taken explicitly in [7] regarding the causal character of the hypersurfaces, as they are assumed to be timelike or spacelike everywhere. This method has been applied to linear order to obtain uniqueness results for the Einstein-Strauss model [9] and to second order to revisit the problem of slowly rotating stars [14].…”