The space-time line element for static ellipsoidal objects

TL;DR

This paper derives a new static ellipsoidal spacetime metric characterized by linear eccentricity, which generalizes Schwarzschild and is potentially more applicable for observational studies of ellipsoidal objects.

Contribution

The paper introduces a novel line element for static ellipsoidal objects based on linear eccentricity, simplifying analysis and observations compared to previous models.

Findings

Reduces to Schwarzschild metric when eccentricity is zero.

Approximates Schwarzschild at large distances by neglecting higher-order terms.

Provides a measurable parameter for modeling ellipsoidal astrophysical objects.

Abstract

In this paper, we solved the Einstein's field equation and obtained a line element for static, ellipsoidal objects characterized by the linear eccentricity ($\eta$) instead of quadrupole parameter ($q$). This line element recovers the Schwarzschild line element when $\eta$ is zero. In addition to that it also reduces to the Schwarzschild line element, if we neglect terms of the order of $r^{-2}$ or higher which are present within the expressions for metric elements for large distances. Furthermore, as the ellipsoidal character of the derived line element is maintained by the linear eccentricity ($\eta$), which is an easily measurable parameter, this line element could be more suitable for various analytical as well as observational studies.