Vacuum type I spacetimes and aligned Papapetrou fields: symmetries

Abstract

Abstract. We analyze type I vacuum solutions admitting an isometry whose Killing 2-form is aligned with a principal bivector of the Weyl tensor, and we show that these solutions belong to a family of type I metrics which admit a group G 3 of isometries. We give a classification of this family and we study the Bianchi type for each class. The classes compatible with an aligned Killing 2-form are also determined. The SzekeresBrans theorem is extended to non vacuum spacetimes with vanishing Cotton tensor.

On the super-energy radiative gravitational fields

“…We can quote some examples of non radiative fields: (i) All the type D vacuum solutions are known [20,21]. (ii) A result inferred by Szekeres [22] and confirmed by Brans [23] states that no vacuum solutions of type IM ∞ exist (see [25] for a generalization).…”

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

“…In the second case φ = φ(t) and then Eq. (25) implies that b(ρ) = 0 if, and only if, b(ω) = 0, and then proposition 12 applies and the spacetime admits a Killing vector which is parallel to b. In the first case, for metrics of class I, the spacetime is a FLRW universe as a consequence of lemma 12 in Appendix B.…”

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

“…12 Some of the conditions on the Killing 2-form imposed in the literature quoted above could be very restrictive. Thus, in a previous paper 13 we have shown that the Petrov type I vacuum spacetimes admitting an isometry whose Killing 2-form is aligned with a Weyl principal bivector belong to two classes of metrics which admit a 3-dimensional group of isometries of Bianchi type I or II. In the present work we show that a close relation between the Weyl principal directions and the isometry group exists in these classes.…”

“…So, we can take ν 2 = ν 3 = 0 in the second structure equations (13). Taking into account the definition of ν i from (14) we obtain…”

“…Now we shall go on the integration of (13). As in the previous section it will be useful to distinguish the cases of d τ to be the timelike principal direction θ 0 or a spacelike principal direction θ i .…”