Gravitational field of compact objects in general relativity

TL;DR

This paper develops an analytic approximate solution for the gravitational field of slowly rotating, slightly deformed compact objects in general relativity, linking interior and exterior solutions.

Contribution

It derives a generalized interior solution and an exterior solution equivalent to the Hartle-Thorne approximation, addressing limitations of previous models.

Findings

Standard models cannot serve as interior sources for the Kerr solution.

Derived a unified interior and exterior solution for slowly rotating objects.

Connected approximate solutions to exact solutions like Quevedo-Mashhoon.

Abstract

We study some exact and approximate solutions of Einstein's equations that can be used to describe the gravitational field of astrophysical compact objects in the limiting case of slow rotation and slight deformation. First, we show that none of the standard models obtained by using Fock's method can be used as an interior source for the approximate exterior Kerr solution. We then use Fock's method to derive a generalized interior solution, and also an exterior solution that turns out to be equivalent to the exterior Hartle-Thorne approximate solution that, in turn, is equivalent to an approximate limiting case of the exact Quevedo-Mashhoon solution. As a result we obtain an analytic approximate solution that describes the interior and exterior gravitational field of a slowly rotating and slightly deformed astrophysical object.