Centrifugal Force and Ellipticity behaviour of a slowly rotating ultra compact object

TL;DR

This paper derives a general expression for the ellipticity of slowly rotating compact objects using general relativity and analyzes centrifugal force behavior, revealing reversal and maximum ellipticity points.

Contribution

It introduces a new approach to calculate ellipticity in relativistic rotating fluids using optical reference geometry and compares results with previous full spacetime formalisms.

Findings

Centrifugal reversal occurs at R/Rs ≈ 1.45

Maximum ellipticity occurs at R/Rs ≈ 2.75

Results agree with Chandrasekhar and Miller's formalism

Abstract

Using the optical reference geometry approach, we have derived in the following, a general expression for the ellipticity of a slowly rotating fluid configuration using Newtonian force balance equation in the conformally projected absolute 3-space, in the realm of general relativity. Further with the help of Hartle-Thorne (H-T) metric for a slowly rotating compact object, we have evaluated the centrifugal force acting on a fluid element and also evaluated the ellipticity and found that the centrifugal reversal occurs at around $R/R_s \approx 1.45$, and the ellipticity maximum at around $R/R_s \approx 2.75$. The result has been compared with that of Chandrasekhar and Miller which was obtained in the full 4-spacetime formalism.