On a Five-Dimensional Version of the Goldberg-Sachs Theorem

Abstract

Previous work has found a higher-dimensional generalization of the "geodesic part" of the Goldberg-Sachs theorem. We investigate the generalization of the "shear-free part" of the theorem. A spacetime is defined to be algebraically special if it admits a multiple Weyl Aligned Null Direction (WAND). The algebraically special property restricts the form of the "optical matrix" that defines the expansion, rotation and shear of the multiple WAND. After working out some general constraints that hold in arbitrary dimensions, we determine necessary algebraic conditions on the optical matrix of a multiple WAND in a five-dimensional Einstein spacetime. We prove that one can choose an orthonormal basis to bring the 3 × 3 optical matrix to one of three canonical forms, each involving two parameters, and we discuss the existence of an "optical structure" within these classes. Examples of solutions corresponding to each form are given. We give an example which demonstrates that our necessary algebraic conditions are not sufficient for a null vector field to be a multiple WAND, in contrast with the 4d result.

On the uniqueness of the Myers-Perry spacetime as a type II(D) solution in six dimensions

“…fall off as r 1−n or faster [25,27]. Using the Bianchi identities (B7) and (B8) (with (5), (9), (13) and (15)), at the leading order one finds (n − 3)…”

“…For our purposes, terms of higher order are not needed. It is just important to observe that these can be determined recursively to any desired order once the leading terms in (15), (16) are known [37], and do not involve any integration functions other that Φ 0 and b ij . This implies (cf.…”

“…More generally, it appears of interest to study and classify all vacuum solutions of type II or more special, i.e., those satisfying (1). In five dimensions, a first step in this direction was an extension of the Goldberg-Sachs theorem [11,14,15]. 1 This defines three branches of non-Kundt solutions [15], according to the possible rank (3, 2, or 1) of the optical matrix (defined below in (2)).…”

“…In five dimensions, a first step in this direction was an extension of the Goldberg-Sachs theorem [11,14,15]. 1 This defines three branches of non-Kundt solutions [15], according to the possible rank (3, 2, or 1) of the optical matrix (defined below in (2)). All such solutions (including a cosmological constant) have been fully classified in [18][19][20] (see also [21,22] for earlier results in special cases).…”

On higher dimensional Einstein spacetimes with a non-degenerate double Weyl aligned null direction

“…Various results partially extending the Goldberg-Sachs theorem to higher dimensions have been obtained in recent years [5][6][7]10,13,15,17,21]. Here we point out some additional restrictions which apply to the spacetimes of section 2.2 (in section 3.1) as well as to general type II Einstein spacetimes (in section 3.2), at least when n = 6.…”