Axially symmetric static sources of gravitational field

TL;DR

This paper introduces a general method for constructing static, axially symmetric interior solutions to Einstein's equations that match exterior Weyl solutions, ensuring physical plausibility and regularity for non-spherical gravitational sources.

Contribution

It presents a novel ansatz-based approach to generate interior solutions that smoothly connect to exterior Weyl metrics, extending known spherical models to non-spherical cases.

Findings

Solutions satisfy energy conditions across parameter ranges

Models converge to known spherical solutions in the limit

Total mass matches the exterior monopole moment

Abstract

A general procedure to find static and axially symmetric, interior solutions to the Einstein equations is presented. All the so obtained solutions, verify the energy conditions for a wide range of values of the parameters, and match smoothly to some exterior solution of the Weyl family, thereby representing globally regular models describing non spherical sources of gravitational field. In the spherically symmetric limit, all our models converge to the well known incompressible perfect fluid solution.The key stone of our approach is based on an ansatz allowing to define the interior metric in terms of the exterior metric functions evaluated at the boundary source. Some particular sources are obtained, and the physical variables of the energy-momentum tensor are calculated explicitly, as well as the geometry of the source in terms of the relativistic multipole moments. The total mass of different configurations is also calculated, it is shown to be equal to the monopole of the exterior solution.