Petrov types of slowly rotating fluid balls

TL;DR

This paper classifies the Petrov types of slowly rotating perfect fluid space-times, establishing conditions for various types and analyzing specific models like incompressible fluids and Tolman IV solutions.

Contribution

It provides new theorems on Petrov types of rotating fluid models and demonstrates limitations on Petrov types for certain physically realistic fluids.

Findings

Petrov type II solutions reduce to de Sitter in static limit

Incompressible rotating fluids cannot be Petrov type D

Rotation function for Tolman IV can be solved in quadratures

Abstract

Circularly rotating axisymmetric perfect fluid space-times are investigated to second order in the small angular velocity. The conditions of various special Petrov types are solved in a comoving tetrad formalism. A number of theorems are stated on the possible Petrov types of various fluid models. It is shown that Petrov type II solutions must reduce to the de Sitter spacetime in the static limit. Two space-times with a physically satisfactory energy-momentum tensor are investigated in detail. For the rotating incompressible fluid, it is proven that the Petrov type cannot be D. The equation of the rotation function $\omega $ can be solved for the Tolman type IV fluid in terms of quadratures. It is also shown that the rotating version of the Tolman IV space-time cannot be Petrov type D.