On the Goldberg-Sachs theorem in higher dimensions in the non-twisting case

Abstract

We study a generalization of the "shear-free part" of the Goldberg-Sachs theorem for Einstein spacetimes admitting a non-twisting multiple Weyl Aligned Null Direction (WAND) l in n>=6 spacetime dimensions. The form of the corresponding optical matrix $\rho$ is restricted by the algebraically special property in terms of the degeneracy of its eigenvalues. In particular, there necessarily exists at least one multiple eigenvalue and further constraints arise in various special cases. For example, when $\rho$ is non-degenerate and the Weyl components ${\Phi }_{ij}$ are non-zero, all eigenvalues of $\rho$ coincide and such spacetimes thus correspond to the Robinson-Trautman (RT) class. On the other hand, in certain degenerate cases all non-zero eigenvalues can be distinct. We also present explicit examples of Einstein spacetimes admitting some of the permitted forms of $\rho$, including examples violating the "optical constraint". The obtained restrictions on $\rho$ are, however, in general not sufficient for l to be a multiple WAND, as demonstrated by a few "counterexamples". We also discuss the geometrical meaning of these restrictions in terms of integrability properties of certain null distributions. Finally, we specialize our analysis to the six-dimensional case, where all the permitted forms of $\rho$ are given in terms of just two parameters. In the appendices some examples are given and certain results pertaining to (possibly) twisting mWANDs of Einstein spacetimes are presented.

arXiv comment38 pages

References in corpus (25)

• A rotating black ring in five dimensions
• Classification of the Weyl Tensor in Higher Dimensions
• Particle and light motion in a space-time of a five-dimensional rotating black hole
• Vanishing Scalar Invariant Spacetimes in Higher Dimensions
• On Spacetimes with Constant Scalar Invariants
• Robinson-Trautman spacetimes in higher dimensions
• Bianchi identities in higher dimensions
• Generalization of the Geroch-Held-Penrose formalism to higher dimensions
• Type D Einstein spacetimes in higher dimensions
• Ricci identities in higher dimensions
• Generalizations of pp-wave spacetimes in higher dimensions
• A Bestiary of Higher Dimensional Taub-NUT-AdS Spacetimes
• Nuttier (A)dS Black Holes in Higher Dimensions
• Kerr-Schild spacetimes with (A)dS background
• WANDs of the Black Ring
• Killing-Yano tensors and multi-hermitian structures
• On a five-dimensional version of the Goldberg-Sachs theorem
• Newman-Penrose formalism in higher dimensions: vacuum spacetimes with a non-twisting geodetic multiple Weyl aligned null direction
• Algebraically special axisymmetric solutions of the higher-dimensional vacuum Einstein equation
• Type III and N Einstein spacetimes in higher dimensions: general properties
• The complex Goldberg-Sachs theorem in higher dimensions
• On asymptotically flat algebraically special spacetimes in higher dimensions
• On higher dimensional Einstein spacetimes with a warped extra dimension
• Optical structures, algebraically special spacetimes, and the Goldberg-Sachs theorem in five dimensions
• On algebraically special vacuum spacetimes in five dimensions

Cited by in corpus (8)

• Static and radiating p-form black holes in the higher dimensional Robinson-Trautman class
• Charging Kerr-Schild spacetimes in higher dimensions
• Asymptotic behaviour of Maxwell fields in higher dimensions
• Spinors and the Weyl Tensor Classification in Six Dimensions
• On the uniqueness of the Myers-Perry spacetime as a type II(D) solution in six dimensions
• On the Weyl Tensor Classification in All Dimensions and its Relation with Integrability Properties
• On the uniqueness of the Kerr-(A)dS metric as a type II(D) solution in six dimensions
• Exact solutions and their interpretation - session A1