Purely magnetic locally rotationally symmetric spacetimes

Abstract

We consider all purely magnetic, locally rotationally symmetric (LRS) spacetimes. It is shown that such spacetimes belong to either LRS class I or III by the Ellis classification. For each class the most general solution is found exhibiting a disposable function and three parameters. A Segré classification of purely magnetic LRS spacetimes is given together with the compatibility requirements of two general energy–momentum tensors. Finally, implicit solutions are obtained, in each class, when the energy–momentum tensor is a perfect fluid.

Complete classification of purely magnetic, nonrotating, nonaccelerating perfect fluids

“…Taking an orthonormal eigenframe B of H ab , we may assume that H 22 = H 33 . In this case the (22), (33) and (23) components of (25), (26) immediately give q 1 = −r 1 , σ 23 = 0, n 33 = n 22 and h 1 = 0 (such that h 3 = −h 2 ), while the (12), and (13) components of (25), (26) together with the (2,3) components of (24) lead to…”

“…More precisely, we drop the purely magnetic condition and look at the class A of non-vacuum, non-conformally flat, non-rotating perfect fluids (M, g ab , u a ) which have a degenerate shear tensor σ ab = 0 and vanishing spatial gradients of energy density and pressure. As ω a =u a = D a p = D a µ = 0, one derives D a θ = 0 from (22) and (12) with f = µ, after which D a (σ bc σ bc ) = 0 follows from (19) and (12) with f = θ. The shear being degenerate, σ bc σ bc is the only independent scalar which may be built from σ ab .…”

Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

“…Adding the expressions in (24) and (26) for C + and C − , one obtains the well-known formula for the Weyl tensor in four-dimensional General Relativity in terms of E ab and H ab [9], which are usually referred to as the electric and magnetic parts of the Weyl tensor. Since they are respectively equivalent with C + and C − this justifies the above definition of Weyl electric and magnetic parts, for general n.…”

“…hold, where c = +1 in the PE and c = −1 in the PM case (see, e.g., [26]). For a Petrov type I Weyl tensor one can always take a Weyl canonical transversal (Ψ 0 = Ψ 4 = 0, Ψ 1 = Ψ 3 = 0) or longitudinal (Ψ 0 = Ψ 4 = 0, Ψ 1 = Ψ 3 = 0) frame and add these to the PE/PM conditions (42).…”

An exhaustive classification of aligned Petrov type D purely magnetic perfect fluids

“…Solutions are then locally rotationally symmetric and belong to one of the classes discussed in detail in [9,10]: they either have ρ − ρ = µ − µ and are non-rotating with non-vanishing shear, or they have ρ + ρ = µ + µ and are rotating and shearfree.…”