Bel-Debever criteria for the classification of the Weyl tensors in higher dimensions

Abstract

An extension to higher dimensions of the Bel-Debever characterization of the Weyl tensor is considered. This provides algebraic conditions that uniquely determine the multiplicity of a Weyl aligned null direction (WAND), and thus the principal Weyl type, in a frame independent way. The specification of several "subtypes" is also encompassed by the criteria. We further comment on a Cartan-like geometrical interpretation of WANDs in terms of their invariance properties under parallel transport around infinitesimal loops. As a result, restrictions on the algebraic types permitted in spacetimes that admit a recurrent/covariantly constant vector field are outlined.

arXiv comment8 pages

References in corpus (9)

• Classification of the Weyl Tensor in Higher Dimensions
• Classification of the Weyl Tensor in Higher Dimensions and Applications
• Alignment and algebraically special tensors in Lorentzian geometry
• Robinson-Trautman spacetimes in higher dimensions
• Type D Einstein spacetimes in higher dimensions
• Higher dimensional Kerr-Schild spacetimes
• WANDs of the Black Ring
• Time-Dependent Multi-Centre Solutions from New Metrics with Holonomy Sim(n-2)
• Black rings with a small electric charge: gyromagnetic ratios and algebraic alignment

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