Ellipsoidal shapes in general relativity: general definitions and an application

TL;DR

This paper extends the concept of ellipsoids to curved spaces in general relativity, exploring their use in modeling rotating bodies and presenting new solutions for perfect-fluid spacetimes with specific symmetries.

Contribution

It introduces a generalized definition of ellipsoids in curved spaces and applies it to derive new solutions for rotating perfect-fluid bodies in general relativity.

Findings

Derived a class of perfect-fluid metrics with ellipsoidal symmetry

Obtained a vacuum solution with a non-zero cosmological constant

Presented interior NUT-spacetimes with specific symmetries

Abstract

A generalization of the notion of ellipsoids to curved Riemannian spaces is given and the possibility to use it in describing the shapes of rotating bodies in general relativity is examined. As an illustrative example, stationary, axisymmetric perfect-fluid spacetimes with a so-called confocal inside ellipsoidal symmetry are investigated in detail under the assumption that the 4-velocity of the fluid is parallel to a time-like Killing vector field. A class of perfect-fluid metrics representing interior NUT-spacetimes is obtained along with a vacuum solution with a non-zero cosmological constant.