Complete classification of purely magnetic, nonrotating, nonaccelerating perfect fluids

Abstract

Recently the class of purely magnetic nonrotating dust spacetimes has been shown to be empty [L. Wylleman, Classical Quantum Gravity 23, 2727 (2006).]. 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 nonrotating perfect fluids on the other hand, which have been fully classified earlier for Petrov type $D$ [C. Lozanovski, Classical Quantum Gravity 19, 6377 (2002).], the fluid is shown to be nonaccelerating 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 ${\mathrm{VI}}_{0}$, has degenerate shear, and is of Petrov type $\mathrm{I}({M}^{\ensuremath{\infty}})$ 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.