Complete classification of purely magnetic, non-rotating and non-accelerating perfect fluids

TL;DR

This paper fully classifies purely magnetic, non-rotating perfect fluids, revealing new solutions and conditions under which these fluids can exist, especially highlighting the unique spatially homogeneous case with specific Petrov type and shear properties.

Contribution

It provides a complete classification of purely magnetic, non-rotating perfect fluids, including a new explicit solution with specific geometric and physical properties.

Findings

Purely magnetic, non-rotating perfect fluids are non-accelerating if and only if the density gradient vanishes.

A new spatially homogeneous solution of Bianchi type VI_0 is found, with degenerate shear and Petrov type I(M^∞).

Rotating dust models cannot be purely magnetic under certain Petrov type conditions.

Abstract

Recently the class of purely magnetic non-rotating dust spacetimes has been shown to be empty (Wylleman, Class. Quant. Grav. 23, 2727). It turns out that purely magnetic rotating dust models are subject to severe integrability conditions as well. One of the consequences of the present paper is that also rotating dust cannot be purely magnetic when it is of Petrov type D or when it has a vanishing spatial gradient of the energy density. For purely magnetic and non-rotating perfect fluids on the other hand, which have been fully classified earlier for Petrov type D (Lozanovski, Class. Quant. Grav. 19, 6377), the fluid is shown to be non-accelerating if and only if the spatial density gradient vanishes. Under these conditions, a new and algebraically general solution is found, which is unique up to a constant rescaling, which is spatially homogeneous of Bianchi type $VI_0$, has degenerate shear and is of Petrov type I($M^\infty)$ in the extended Arianrhod-McIntosh classification. The metric and the equation of state are explicitly constructed and properties of the model are briefly discussed. We finally situate it within the class of normal geodesic flows with degenerate shear tensor.