Optical geometries

Abstract

We study the notion of optical geometry, defined to be a Lorentzian manifold equipped with a null line distribution, from the perspective of intrinsic torsion. This is an instance of a non-integrable version of holonomy reduction in Lorentzian geometry. These generate congruences of null curves, which play an important rôle in general relativity. Conformal properties of these are investigated. We also extend this concept to generalised optical geometries as introduced by Robinson and Trautman.

Almost Robinson geometries

“…V be a p2m `2q-dimensional oriented complex vector space equipped with a non-degenerate symmetric bilinear form r g. We introduce abstract indices following the convention of [24]: minuscule Roman indices starting with the beginnning of the alphabet a, b, c, . .…”

“…It is clear that a Robinson structure pN, Kq on pV, gq determines in particular an optical structure, namely K, in the sense of [24]. We therefore have a filtration of vector subspaces…”

“…In a recent article [24], the authors give a comprehensive review of the notion of optical structure on a Lorentzian manifold pM, gq, simply understood as a null line distribution K on M. Many of the geometric properties of this distribution and its orthogonal complement are encoded in terms of its screen bundle H K " K K {K, which is naturally equipped with a bundle metric h inherited from g. One may naturally wish to endow H K with further bundle structures. In the present article, where we assume M to have dimension 2m `2, we equip H K with a bundle complex structure J compatible with h. Such a structure was introduced by Nurowski and Trautman in [64,105,106], where it is equivalently described in terms of a totally null complex pm `1q-plane distribution N .…”

“…In dimension four, as is well-known [71,64,105,106,24], an almost Robinson structure pN, Kq is essentially equivalent to an optical structure. The key point, here, is that the involutivity of the totally null complex 2-plane distribution N is equivalent to the congruence K of null curves tangent to K being geodesic and non-shearing, that is, the conformal class of the bundle metric h is preserved along the curves of K. What is more, the rank-one complex distribution N { C K descends to the leaf space M of K, thereby endowing it with a Cauchy-Riemann (CR) structure.…”

“…The real span of the intersection N X N then determines the line distribution K. Following their terminology, we shall refer to the pair pN, Kq as an (almost) Robinson structure. The structure group of the frame bundle is reduced to pR ą0 ˆUpmqq˙pR 2m q ˚, which is a subgroup of the group Simp2mq, which characterises optical structures, and as in [24], we shall describe the geometric properties of an almost Robinson structure in terms of its intrinsic torsion.…”