Higher dimensional bivectors and classification of the Weyl operator

Abstract

We develop the bivector formalism in higher dimensional Lorentzian spacetimes. We define the Weyl bivector operator in a manner consistent with its boost-weight decomposition. We then algebraically classify the Weyl tensor, which gives rise to a refinement in dimensions higher than four of the usual alignment (boost-weight) classification, in terms of the irreducible representations of the spins. We are consequently able to define a number of new algebraically special cases. In particular, the classification in five dimensions is discussed in some detail. In addition, utilizing the (refined) algebraic classification, we are able to prove some interesting results when the Weyl tensor has (additional) symmetries.

25 pages; v2: some notation changed, typos fixed; CQG accepted

References in corpus (11)

• Classification of the Weyl Tensor in Higher Dimensions and Applications
• Spacetimes characterized by their scalar curvature invariants
• Kundt Spacetimes
• Robinson-Trautman spacetimes in higher dimensions
• Lorentzian spacetimes with constant curvature invariants in four dimensions
• General Kundt spacetimes in higher dimensions
• Lorentzian spacetimes with constant curvature invariants in three dimensions
• Gauduchon-Tod structures, Sim holonomy and De Sitter supergravity
• Bel-Debever criteria for the classification of the Weyl tensors in higher dimensions
• Algebraically special axisymmetric solutions of the higher-dimensional vacuum Einstein equation
• Type II Einstein spacetimes in higher dimensions

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• Geometric Horizons
• Spinor-helicity and the algebraic classification of higher-dimensional spacetimes
• Type II universal spacetimes
• Weyl Tensor Classification in Four-dimensional Manifolds of All Signatures
• Lorentzian manifolds and scalar curvature invariants
• Pseudo-Riemannian VSI spaces II
• The Cartan Algorithm in Five Dimensions
• Optical structures, algebraically special spacetimes, and the Goldberg-Sachs theorem in five dimensions
• Refinements of the Weyl tensor classification in five dimensions
• Algebraic classification of five-dimensional spacetimes using scalar invariants
• Pseudo-Riemannian VSI spaces
• $\mathcal{I}$-degenerate pseudo-Riemannian metrics
• Cartan Invariants and Event Horizon Detection, Extended Version
• A Generalization of the Goldberg-Sachs Theorem and its Consequences
• On the Weyl Tensor Classification in All Dimensions and its Relation with Integrability Properties
• Spinors and the Weyl Tensor Classification in Six Dimensions
• On the algebraic classification of pseudo-Riemannian spaces
• Averaging in cosmology based on Cartan scalars
• Wick rotations and real GIT
• Geometric horizons in binary black hole mergers
• Real GIT with applications to compatible representations and Wick-rotations
• Electric and magnetic Weyl tensors in higher dimensions
• Curvature Invariants and the Geometric Horizon Conjecture in a Binary Black Hole Merger
• A peeling theorem for the Weyl tensor in higher dimensions
• On the Pursuit of Generalizations for the Petrov Classification and the Goldberg-Sachs Theorem