Gravitational Metric of a Star

TL;DR

This paper develops a recursive method to compute the gravitational metric of a star with arbitrary multipole moments in general relativity, extending to higher orders and connecting to black hole solutions.

Contribution

It introduces a recursive approach in de Donder gauge to derive the metric outside a star with complex multipole structure to any perturbative order.

Findings

Expressed the metric in terms of tensor bubble integrals in momentum space.

Reproduced the Kerr black hole solution at a specific multipole configuration.

Showed how slight deviations from Kerr multipoles can model Kerr-like stars.

Abstract

Solving the classical equations of motion in general relativity recursively, we consider the metric of a spatially localized and stationary source of matter. Having in mind a star of general composition, we characterize it by means of its infinite set of mass and current multipoles. Specializing to de Donder gauge we set up the recursive equations that produce the metric outside the star to any desired order in perturbation theory, expanded both in Newton's constant and in the order of multipoles. Up to second post-Minkowskian order we express the result to any order in the multipole expansion in terms of generalized (tensor) bubble integrals in momentum space and a corresponding simple expansion in inverse distances. In a special corner of the space of multipoles we recover the Kerr black hole solution to the given order. By tweaking just slightly the multipoles away from the Kerr limit the metric will describe stars that are Kerr-like and yet are not black holes. A subtlety with respect to the gauge ambiguity of de Donder gauge is also pointed out.