The Erez-Rosen solution versus the Hartle-Thorne solution

TL;DR

This paper explores the relationship between the Erez-Rosen and Hartle-Thorne solutions by deriving coordinate transformations and relating their parameters, enhancing understanding of static gravitational fields with quadrupole moments.

Contribution

It explicitly establishes the coordinate transformations and parameter relations between the Erez-Rosen and Hartle-Thorne metrics for static objects.

Findings

Derived the coordinate transformations between the two metrics.

Expressed the Zipoy-Voorhees parameter in terms of quadrupole moment.

Confirmed the relation $ ext{delta} = 1 - q$ with previous literature.

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 $\delta=1+sq$. 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 $\delta$ parameters. The coordinate transformations between the metrics are found in the approximation of $\sim$q. It is shown that the Zipoy-Voorhees parameter is equal to $\delta=1-q$ with $s=-1$. This result is in agreement with previous results in the literature.