Type I vacuum solutions with aligned Papapetrou fields: An intrinsic characterization

Abstract

We show that Petrov type I vacuum solutions admitting a Killing vector whose Papapetrou field is aligned with a principal bivector of the Weyl tensor are the Kasner and Taub metrics, their counterpart with timelike orbits and their associated windmill-like solutions, as well as the Petrov homogeneous vacuum solution. We recover all these metrics by using an integration method based on an invariant classification which allows us to characterize every solution. In this way we obtain an intrinsic and explicit algorithm to identify them.

Intrinsic, deductive, explicit, and algorithmic characterization of the Szekeres-Szafron solutions

“…A suitable procedure is to analyze every particular case in order to understand the minimal set of elements of the curvature tensor that are necessary to label these geometries, an approach adapted to each particular geometry we want to characterize. This is the method we have achieved here in labeling the Szekeres-Szafron metrics, and it is also the one used in previous articles when characterizing different families of solutions [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34].…”

“…He partially performed this type of analysis for the spherically symmetric spacetimes [16,17], a result we attained in two recent papers [18,19]. This kind of IDEAL characterization has also been achieved for other geometrically significant families of metrics and for physically relevant solutions of the Einstein equations [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. The use of the appellation IDEAL (as an acronym) seems to be adequate because the conditions obtained are Intrinsic (depending only of the metric tensor), Deductive (not involving inductive or inferential methods or arguments), Explicit (expressing the solution non implicitly) and ALgorithmic (giving the solution as a flow chart with a finite number of steps).…”

On Collinear Closed One-Forms

“…Collinear one-forms appear in many problems of theoretical physics, for example in general relativity: type I vacuum solutions with aligned Papapetrou fields [6] or triplet ansatz [2,11]. Ranks (of group of the periods) of collinear Morse forms (closed oneforms with nondegenerate singularities) have been studied in [10].…”

Ranks of collinear Morse forms

“…Collinear 1-forms which are the Weyl tensor invariants arise in the problem of classification of type I vacuum solutions with aligned Papapetrou fields [2]. The triplet The corresponding foliation F ω is a characteristic of the class (Lemma 3.2).…”