Algebraic classification of higher dimensional spacetimes based on null alignment

Abstract

We review recent developments and applications of the classification of the Weyl tensor in higher dimensional Lorentzian geometries. First, we discuss the general setup, i.e. main definitions and methods for the classification, some refinements and the generalized Newman-Penrose and Geroch-Held-Penrose formalisms. Next, we summarize general results, such as a partial extension of the Goldberg-Sachs theorem, characterization of spacetimes with vanishing (or constant) curvature invariants and the peeling behaviour in asymptotically flat spacetimes. Finally, we discuss certain invariantly defined families of metrics and their relation with the Weyl tensor classification, including: Kundt and Robinson-Trautman spacetimes; the Kerr-Schild ansatz in a constant-curvature background; purely electric and purely magnetic spacetimes; direct and (some) warped products; and geometries with certain symmetries. To conclude, some applications to quadratic gravity are also overviewed.

59 pages, to appear as Topical Review in Classical and Quantum Gravity

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Cited by in corpus (72)

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• Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension
• Weyl doubling
• Type III and N universal spacetimes
• Separation of variables in Maxwell equations in Plebanski-Demianski spacetime
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• Embedding nonrelativistic physics inside a gravitational wave
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• Scalar Polynomial Curvature Invariant Vanishing on the Event Horizon of Any Black Hole Metric Conformal to a Static Spherical Metric
• Aligned Fields Double Copy to Kerr-NUT-(A)dS
• Spinor-helicity and the algebraic classification of higher-dimensional spacetimes
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• The Cartan Algorithm in Five Dimensions
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• All solutions of Einstein-Maxwell equations with a cosmological constant in 2+1 dimensions
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• Gyratons in the Robinson-Trautman and Kundt classes
• Asymptotic behaviour of Maxwell fields in higher dimensions
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• Universal Black Holes
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• Asymptotic behaviour of the Weyl tensor in higher dimensions
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• Siklos waves with torsion in 3D
• Absence of gyratons in the Robinson-Trautman class
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• On the Weyl Tensor Classification in All Dimensions and its Relation with Integrability Properties
• Twisting non-shearing congruences of null geodesics, almost CR structures, and Einstein metrics in even dimensions
• A novel classification method of 2+1 spacetimes based on the Cotton scalars
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• The Newman-Penrose formalism in quadratic gravity
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