Quadrupolar metrics

TL;DR

This paper reviews the modeling of gravitational fields of compact stars with quadrupolar deviations in general relativity, discussing exact solutions, solution-generating techniques, and methods for interior solutions.

Contribution

It provides a comprehensive review of static and stationary quadrupolar metrics, solution techniques, and approaches for interior solutions of compact stars in general relativity.

Findings

Reviewed main static and axisymmetric exact solutions satisfying physical conditions.

Discussed solution-generating techniques like Lie symmetries, Bäcklund transformations, and inverse scattering.

Highlighted the need for alternative methods for physically meaningful interior solutions.

Abstract

We review the problem of describing the gravitational field of compact stars in general relativity. We focus on the deviations from spherical symmetry which are expected to be due to rotation and to the natural deformations of mass distributions. We assume that the relativistic quadrupole moment takes into account these deviations, and consider the class of axisymmetric static and stationary quadrupolar metrics which satisfy Einstein's equations in empty space and in the presence of matter represented by a perfect fluid. We formulate the physical conditions that must be satisfied for a particular spacetime metric to describe the gravitational field of compact stars. We present a brief review of the main static and axisymmetric exact solutions of Einstein's vacuum equations, satisfying all the physical conditions. We discuss how to derive particular stationary and axisymmetric solutions with quadrupolar properties by using the solution generating techniques which correspond either to Lie symmetries and B\"ackund transformations of the Ernst equations or to the inverse scattering method applied to Einstein's equations. As for interior solutions, we argue that it is necessary to apply alternative methods to obtain physically meaningful solutions, and review a method which allows us to generate interior perfect-fluid solutions.