2010/11/09 by A. A. Coley, Sigbjørn Hervik, Coley, Alan +1
Physics and Astronomy · #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories
paper · pdf · doi:10.48550/arxiv.1011.2175
openalex publication_date 2010/11/09 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
The Weyl and Ricci tensors can be algebraically classified in a Lorentzian\nspacetime of arbitrary dimensions using alignment theory. Used in tandem with\nthe boost weight decomposition and curvature operators, the algebraic\nclassification of the Weyl tensor and the Ricci tensor in higher dimensions can\nthen be refined utilizing their eigenbivector and eigenvalue structure,\nrespectively. In particular, for a tensor of a particular algebraic type, the\nassociated operator will have a restricted eigenvector structure, and this can\nthen be used to determine necessary conditions for a particular algebraic type.\nWe shall present an analysis of the discriminants of the associated\ncharacteristic equation for the eigenvalues of an operator to determine the\nconditions on (the associated) curvature tensor for a given algebraic type. We\nwill describe an algorithm which enables us to completely determine the\neigenvalue structure of the curvature operator, up to degeneracies, in terms of\na set of discriminants. We then express these conditions (discriminants) in\nterms of these polynomial curvature invariants. In particular, we can use the\ntechniques described to study the necessary conditions in arbitrary dimensions\nfor the Weyl and Ricci curvature operators (and hence the higher dimensional\nWeyl and Ricci tensors) to be of algebraic type II or D, and create syzygies\nwhich are necessary for the special algebraic type to be fulfilled. We are\nconsequently able to determine the necessary conditions in terms of simple\nscalar polynomial curvature invariants in order for the higher dimensional Weyl\nand Ricci tensors to be of type II or D. We explicitly determine the scalar\npolynomial curvature invariants for a Weyl or Ricci tensor to be of type II or\nD in 5D. A number of simple examples are presented and, in particular, we\npresent a detailed analysis of the important example of a 5D rotating black\nring.\n