The Erez–Rosen Solution Versus the Hartle–Thorne Solution

Kuantay Boshkayev,Ainur Urazalina,Algis Malybayev,Gulmira Nurbakyt,Hernando Quevedo

doi:10.3390/sym11101324

Abstract

In this work, we investigate the correspondence between the Erez–Rosen and Hartle–Thorne solutions. We explicitly show how to establish the relationship and find the coordinate transformations between the two metrics. For this purpose the two metrics must have the same approximation and describe the gravitational field of static objects. Since both the Erez–Rosen and the Hartle–Thorne solutions are particular solutions of a more general solution, the Zipoy–Voorhees transformation is applied to the exact Erez–Rosen metric in order to obtain a generalized solution in terms of the Zipoy–Voorhees parameter δ = 1 + s q . The Geroch–Hansen multipole moments of the generalized Erez–Rosen metric are calculated to find the definition of the total mass and quadrupole moment in terms of the mass m, quadrupole q and Zipoy–Voorhees δ parameters. The coordinate transformations between the metrics are found in the approximation of ∼q. It is shown that the Zipoy–Voorhees parameter is equal to δ = 1 - q with s = - 1 . This result is in agreement with previous results in the literature.

Highlights

There exists quite a large number of exact and approximate solutions of the Einstein field equations (EFE) in the literature [1,2]
We investigated the Erez–Rosen and Hartle–Thorne metrics for small quadrupole moment and found their relationship in the absence of rotation by using the perturbation method
We showed that the approximate Erez–Rosen line element coincides with the Hartle–Thorne solution, in the limit of ∼ q, when the former is considered with a Zipoy–Voorhees transformation

Summary

Introduction

There exists quite a large number of exact and approximate solutions of the Einstein field equations (EFE) in the literature [1,2]. Though the ER metric is exact and describes only the exterior part of the static deformed object, in turn, the HT metric is approximate and can be used to investigate both interior and exterior fields of slowly rotating and slightly deformed astrophysical objects in the strong field regime In this regard, it is interesting to show the relationship between these solutions in the limiting static case with a small deformation. It has been shown that the general form of the QM solution with the Zipoy–Voorhees parameter in the limiting case is equivalent to the exterior HT solution up to the first order in the quadrupole parameter q and to the second order terms in the rotation parameter a [2,19] Hartle developed his formalism in order to investigate the physical properties of slowly rotating relativistic stars in his pioneering paper in 1967 [4].

The Erez–Rosen Metric

The Linearized Erez–Rosen Solution in Terms of the Zipoy–Voorhees Parameter

The Exterior Hartle–Thorne Solution

Coordinate Transformations

Conclusions