The complex Goldberg–Sachs theorem in higher dimensions

Abstract

We study the geometric properties of holomorphic distributions of totally null m-planes on a (2m+ǫ)dimensional complex Riemannian manifold (M, g), where ǫ ∈ {0, 1} and m ≥ 2. In particular, given such a distribution N , say, we obtain algebraic conditions on the Weyl tensor and the Cotton-York tensor which guarrantee the integrability of N , and in odd dimensions, of its orthogonal complement. These results generalise the Petrov classification of the (anti-)self-dual part of the complex Weyl tensor, and the complex Goldberg-Sachs theorem from four to higher dimensions.Higher-dimensional analogues of the Petrov type D condition are defined, and we show that these lead to the integrability of up to 2 m holomorphic distributions of totally null m-planes. Finally, we adapt these findings to the category of real smooth pseudo-Riemannian manifolds, commenting notably on the applications to Hermitian geometry and Robinson (or optical) geometry. complex sphere. More recently, it was noted by Mason and the present author [MT10] that the higherdimensional Kerr-NUT-AdS metric [CLP06] is characterised by a discrete set of Hermitian structures, and its Weyl tensor satisfies an algebraic condition generalising the four-dimensional Petrov type D condition. As in four dimensions [WP70], these results were shown to arise from the existence of a conformal Killing-Yano 2-form.Such findings suggest that a higher-dimensional Goldberg-Sachs theorem should be formulated in the context of null structures, and to this end, an invariant classication of the curvature tensors with respect to an almost null structure appears to be the most natural framework. Such a classification already exists in almost Hermitian geometry [FFS94,TV81], but curvature prescriptions that are sufficient for the integrability of an almost Hermitian structure do not appear to have been investigated. In Lorentzian geometry, the Weyl tensor has also been subject to a classification [CMPP04a, CMPP04b, MCPP05, PPCM04, PPO07, OPP07] which has mostly focused on the properties of null geodesics. In fact, according to this approach, the geodesic part of the Goldberg-Sachs theorem admits a generalisation to higher dimensions [DR09], but its shearfree part does not. In fact, shearfree congruences of null geodesics in more than four dimensions, which, as remarked in [Tra02b], are no longer equivalent to Robinson structures, have not featured so prominently in the solutions to Einstein's field equations [FS03, PPCM04].On the other hand, the present author [TC11] put forward a higher-dimensional generalisation of the Petrov type II condition, which, together with a genericity assumption on the Weyl tensor and a degeneracy condition on the Cotton-York tensor, guarantees the existence of a Robinson structure on a five-dimensional Lorentzian manifold. A counterexample to the converse is given: the black ring solution [ER02] admits pairs of null structures, but the Weyl tensor fails to be 'algebraically special relative to it' in the sense of Theorem 1.1 below. In the same refere...

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

“…This is the only part of the theorem relevant to this paper, and hereafter this will be understood. It should also be noted that a different formulation of the higher-dimensional Goldberg-Sachs theorem has been studied in [16,17].2 As a consequence, for example, for n ≥ 6 there exist static vacuum black holes with horizons of non-constant curvature [28].…”

“…This is the only part of the theorem relevant to this paper, and hereafter this will be understood. It should also be noted that a different formulation of the higher-dimensional Goldberg-Sachs theorem has been studied in [16,17].…”

Spinor-helicity and the algebraic classification of higher-dimensional spacetimes

“…been the subject of much previous work concerning, for instance, problems of existence and stability of solutions [68][69][70][71][72][73][74][75][76][77][78]. Another interesting problem to investigate with our formalism is the use of curvature (and Cartan) invariants to characterise spacetimes; see [79] for a brief introduction and [80][81][82][83] for recent work on this topic.…”

Twistor Geometry of Null Foliations in Complex Euclidean Space

“…For a proof, see for instance [33,34]. The argument leading up to Theorem 3.5 equally applies to the even-dimensional case -simply substitute γ-plane for α-plane.…”