Complete classification of purely magnetic, nonrotating, nonaccelerating perfect fluids

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 V I0, has degenerate shear and is of Petrov type I(M ∞ ) 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.

Gravito-electromagnetic analogies

“…Note however that, by contrast with the latter, these are not the only independent scalar invariants one can construct from C αβµν ; there are also two cubic invariants, see e.g. [32,42,29,41,74,44].…”

“…6 we briefly review the formal analogies between the electric (E αβ ) and magnetic (H αβ ) parts of the Weyl tensor, and the electric (E α ) and magnetic (B α ) fields, e.g. [31,32,33,34,35,36,37,38,39,40,41,42,75]. One of its interesting results is that in the case of vacuum, in the linear regime, one obtains a set of equations formally similar to Maxwell's equations in a Lorentz frame, only with the gravitational tidal tensors (E ij = E ij , H ij = H ij ) in place of E α and B α .…”

Complex windmill transformation producing new purely magnetic fluids

“…As stated in the previous section, if ∂ w k = 0 ⇒ a x = 0, then, by a theorem due to Wylleman and Van den Bergh [34], a unique PMpf is obtained. The solution set for the WVL PMpf is…”

“…It is shown that there are two PMpf spacetimes that have related windmill PMaf spacetimes. One of these is the PMpf solution independently found by the present authors [34] (henceforth, referred to as the WVL spacetime), and the other is given by a general solution to the confluent hypergeometric equation by Kummer's functions. The spatial gradient of the energy-density function and the 4-acceleration vector are both 0 for the former spacetime while non-zero for the latter.…”

“…That is, for example, only vacuum spacetimes could be considered, although a complete integration of the resulting equations may be formidable. Figure 1 shows the classification tree for Petrov type I PMpf's, which is a refinement of the one presented in [34]. …”