An approximate global solution of Einstein's equation for a rotating compact source with linear equation of state

TL;DR

This paper develops an approximate rotating fluid solution to Einstein's equations with a specific linear equation of state, analyzing its properties, matching conditions, and Petrov classification, with implications for models like Wahlquist and Kerr.

Contribution

It introduces a new approximate metric for rotating perfect fluids with a linear EOS, fully matched to an exterior, and analyzes its Petrov type and physical limitations.

Findings

The metric matches well with asymptotically flat exteriors.

Wahlquist's interior cannot be matched to flat exterior.

Interior solutions are limited to Petrov types I, D, or O.

Abstract

We use analytic perturbation theory to present a new approximate metric for a rigidly rotating perfect fluid source with equation of state (EOS) $\epsilon+(1-n)p=\epsilon_0$. This EOS includes the interesting cases of strange matter, constant density and the fluid of the Wahlquist metric. It is fully matched to its approximate asymptotically flat exterior using Lichnerowicz junction conditions and it is shown to be a totally general matching using Darmois-Israel conditions and properties of the harmonic coordinates. Then we analyse the Petrov type of the interior metric and show first that, in accordance with previous results, in the case corresponding to Wahlquist's metric it can not be matched to the asymptotically flat exterior. Next, that this kind of interior can only be of Petrov types I, D or (in the static case) O and also that the non-static constant density case can only be of type I. Finally, we check that it can not be a source of Kerr's metric.