Some results on the integrability of Einstein's field equations for axistationary perfect fluids

TL;DR

This paper investigates the integrability of Einstein's field equations for axistationary perfect fluids using an orthonormal Lorentz frame, deriving conditions and relations that clarify the structure and classification of such spacetimes.

Contribution

It formulates the integrability conditions as a first order system and explores their implications, including relations between fluid shear and vorticity and Petrov type classifications.

Findings

Incompressible axistationary perfect fluids are of Petrov type I.

Integrability conditions help verify assumptions like Petrov types.

A relation between shear and vorticity for barotropic fluids is established.

Abstract

Using an orthonormal Lorentz frame approach to axistationary perfect fluid spacetimes, we have formulated the necessary and sufficient equations as a first order system, and investigated the integrability conditions of this set of equations. The integrability conditions are helpful tools when it comes to check the consequences and/or compatibility of certain simplifying assumptions, e.g. Petrov types. Furthermore, using this method, a relation between the fluid shear and vorticity is found for barotropic fluids. We collect some results concerning Petrov types, and it is found that an incompressible axistationary perfect fluid must be of Petrov type I.