Classification of the Weyl Tensor in Higher Dimensions and Applications
arXiv:0710.1598 · doi:10.1088/0264-9381/25/3/033001
Abstract
We review the theory of alignment in Lorentzian geometry and apply it to the algebraic classification of the Weyl tensor in higher dimensions. This classification reduces to the the well-known Petrov classification of the Weyl tensor in four dimensions. We discuss the algebraic classification of a number of known higher dimensional spacetimes. There are many applications of the Weyl classification scheme, especially in conjunction with the higher dimensional frame formalism that has been developed in order to generalize the four dimensional Newman--Penrose formalism. For example, we discuss higher dimensional generalizations of the Goldberg-Sachs theorem and the Peeling theorem. We also discuss the higher dimensional Lorentzian spacetimes with vanishing scalar curvature invariants and constant scalar curvature invariants, which are of interest since they are solutions of supergravity theory.
Topical Review for Classical and Quantum Gravity. Final published version
References in corpus (68)
- A Large Mass Hierarchy from a Small Extra Dimension
- An Alternative to Compactification
- The Hierarchy Problem and New Dimensions at a Millimeter
- A rotating black ring in five dimensions
- Type IIB Green-Schwarz superstring in plane wave Ramond-Ramond background
- Rotation and the AdS/CFT correspondence
- D--branes and Spinning Black Holes
- Penrose limits and maximal supersymmetry
- Exactly solvable model of superstring in Ramond-Ramond plane wave background
- A new maximally supersymmetric background of IIB superstring theory
- Brane-World Black Holes
- The General Kerr-de Sitter Metrics in All Dimensions
- Rotating Black Holes in Higher Dimensions with a Cosmological Constant
- Dirac Gaugino Masses and Supersoft Supersymmetry Breaking
- Classification of the Weyl Tensor in Higher Dimensions
- General Kerr-NUT-AdS Metrics in All Dimensions
- Higher dimensional black holes and supersymmetry
- Particle and light motion in a space-time of a five-dimensional rotating black hole
- A New Class of Exact Solutions in String Theory
- On solvable models of type IIB superstring in NS-NS and R-R plane wave backgrounds
- Black Rings, Supertubes, and a Stringy Resolution of Black Hole Non-Uniqueness
- Separability of Hamilton-Jacobi and Klein-Gordon Equations in General Kerr-NUT-AdS Spacetimes
- `Hidden' Symmetries of Higher Dimensional Rotating Black Holes
- All spacetimes with vanishing curvature invariants
- Vanishing Scalar Invariant Spacetimes in Higher Dimensions
- Alignment and algebraically special tensors in Lorentzian geometry
- Killing-Yano Tensors, Rank-2 Killing Tensors, and Conserved Quantities in Higher Dimensions
- Hidden Symmetry of Higher Dimensional Kerr-NUT-AdS Spacetimes
- On Spacetimes with Constant Scalar Invariants
- Asymptotic Flatness and Bondi Energy in Higher Dimensional Gravity
- Quest for Localized 4-D Black Holes in Brane Worlds
- Non-supersymmetric black rings as thermally excited supertubes
- Nariai, Bertotti-Robinson and anti-Nariai solutions in higher dimensions
- Robinson-Trautman spacetimes in higher dimensions
- Bianchi identities in higher dimensions
- Type D Einstein spacetimes in higher dimensions
- Ricci identities in higher dimensions
- Higher dimensional VSI spacetimes
- Strings on pp-waves and massive two dimensional field theories
- Gravitational field of a spinning radiation beam-pulse in higher dimensions
- Black holes on cylinders are not algebraically special
- Supermembranes with fewer supersymmetries
- Kerr-NUT-de Sitter Curvature in All Dimensions
- Quasi-local rotating black holes in higher dimension: geometry
- Lorentzian spacetimes with constant curvature invariants in three dimensions
- A class of exact pp-wave string models with interacting light-cone gauge actions
- Gravitational field of relativistic gyratons
- New supersymmetric AdS_3 solutions
- A Small Note on PP-Wave Vacua in 6 and 5 Dimensions
- Fermi Coordinates and Penrose Limits
- Holonomy groups and spacetimes
- Stationary Einstein-Maxwell fields in arbitrary dimensions
- WANDs of the Black Ring
- A Class of Exact Classical Solutions to String Theory
- Properties of Naked Black Holes
- Relativistic gyratons in asymptotically AdS spacetime
- On d=4,5,6 Vacua with 8 Supercharges
- Classification of Higher Dimensional Spacetimes
- Zero-Branes, Quantum Mechanics and the Cosmological Constant
- Supersymmetric gyratons in five dimensions
- String Gyratons in Supergravity
- Black rings with a small electric charge: gyromagnetic ratios and algebraic alignment
- The Petrov type of the five-dimensional Myers-Perry metric
- Vanishing Scalar Invariant Spacetimes in Supergravity
- The Petrov type of the BMPV metric
- Gravitational wave solutions in string and M-theory AdS Backgrounds
- A note on the peeling theorem in higher dimensions
- On the algebraic classification of spacetimes
Cited by in corpus (91)
- Black Holes in Higher Dimensions
- Black holes, hidden symmetries, and complete integrability
- `Hidden' Symmetries of Higher Dimensional Rotating Black Holes
- Higher-Dimensional Black Holes: Hidden Symmetries and Separation of Variables
- Spacetimes characterized by their scalar curvature invariants
- Hidden Symmetries of Higher Dimensional Black Holes and Uniqueness of the Kerr-NUT-(A)dS spacetime
- Kundt Spacetimes
- Algebraic classification of higher dimensional spacetimes based on null alignment
- Lorentzian spacetimes with constant curvature invariants in four dimensions
- Metrics With Vanishing Quantum Corrections
- General Kundt spacetimes in higher dimensions
- Weyl compatible tensors
- Generalized Killing-Yano equations in D=5 gauged supergravity
- New Exact Solutions of Quadratic Curvature Gravity
- Hidden Symmetry and Exact Solutions in Einstein Gravity
- Higher dimensional bivectors and classification of the Weyl operator
- Symmetry and Equivalence in Teleparallel Gravity
- Curvature operators and scalar curvature invariants
- From Petrov-Einstein to Navier-Stokes
- Interpreting spacetimes of any dimension using geodesic deviation
- Embedding nonrelativistic physics inside a gravitational wave
- Bel-Debever criteria for the classification of the Weyl tensors in higher dimensions
- Newman-Penrose formalism in higher dimensions: vacuum spacetimes with a non-twisting geodetic multiple Weyl aligned null direction
- Separability of the massive Dirac's equation in 5-dimensional Myers-Perry black hole geometry and its relation to a rank-three Killing-Yano tensor
- Supersymmetry, holonomy and Kundt spacetimes
- Aligned Fields Double Copy to Kerr-NUT-(A)dS
- Parallel-propagated frame along null geodesics in higher-dimensional black hole spacetimes
- Spinor classification of the Weyl tensor in five dimensions
- First-order symmetries of Dirac equation in curved background: a unified dynamical symmetry condition
- Covariant gravitational dynamics in 3+1+1 dimensions
- A higher-dimensional generalization of the geodesic part of the Goldberg-Sachs theorem
- Lorentzian manifolds and scalar curvature invariants
- Separability of a modified Dirac equation in a five-dimensional rotating, charged black hole in string theory
- Slice-reducible conformal Killing tensors, photon surfaces and shadows
- Averaging in cosmological models using scalars
- Symmetry operators and separability of massive Klein-Gordon and Dirac equations in the general 5-dimensional Kerr-(anti-)de Sitter black hole background
- Disformal map and Petrov classification in modified gravity
- Petrov type I Condition and Dual Fluid Dynamics
- The Cartan Algorithm in Five Dimensions
- Pseudo-Riemannian VSI spaces II
- Mathematical General Relativity
- Petrov type I Spacetime and Dual Relativistic Fluids
- Optical structures, algebraically special spacetimes, and the Goldberg-Sachs theorem in five dimensions
- Petrov type I Condition and Rindler Fluid in Vacuum Einstein-Gauss-Bonnet Gravity
- Refinements of the Weyl tensor classification in five dimensions
- Near Horizon Extremal Geometry Perturbations: Dynamical Field Perturbations vs. Parametric Variations
- Geometric Horizons in the Kastor-Traschen Multi Black Hole Solutions
- Generalized Formalism in Gauge-Invariant Gravitational Perturbations
- Pseudo-Riemannian VSI spaces
- Algebraic classification of five-dimensional spacetimes using scalar invariants
- 4D neutral signature VSI and CSI spaces
- What is General Relativity?
- Classification of Kerr-de Sitter-like spacetimes with conformally flat in all dimensions
- Algebraic classification of 2+1 geometries: a new approach
- Spinors and the Weyl Tensor Classification in Six Dimensions
- Spatially Homogeneous Teleparallel Gravity: Bianchi I
- Algebraic classification of the Weyl tensor in higher dimensions based on its "superenergy" tensor
- On the Weyl Tensor Classification in All Dimensions and its Relation with Integrability Properties
- Second-order symmetric Lorentzian manifolds II: structure and global properties
- Teleparallel Geometry with a Single Affine Symmetry
- Dual Fluid for the Kerr Black Hole
- Nonvanishing Local Scalar Invariants even in VSI Spacetimes with all Polynomial Curvature Scalar Invariants Vanishing
- Type II Einstein spacetimes in higher dimensions
- A novel classification method of 2+1 spacetimes based on the Cotton scalars
- On the algebraic classification of pseudo-Riemannian spaces
- Exact Solutions of General Relativity and Quadratic Gravity in Arbitrary Dimension
- Spinor calculus on 5-dimensional spacetimes
- SO(n + 1) Symmetric Solutions of the Einstein Equations in Higher Dimensions
- Algebraic classification of spacetimes using discriminating scalar curvature invariants
- Flat space compressible fluid as holographic dual of black hole with curved horizon
- Averaging in cosmological models
- -Preserving Diffeomorphisms of Lorentzian Manifolds
- Generalization Of The Gross-Perry Metrics
- On scalar curvature invariants in three dimensional spacetimes
- Hidden Symmetries and Black Holes
- Persistence in black hole lattice cosmological models
- Mathematical Properties of a Class of Four-dimensional Neutral Signature Metrics
- On the Equivalence of Spacetimes, the Cartan-Karlhede Algorithm
- Note on the invariant classification of vacuum type D spacetimes
- All non-expanding gravitational waves in -dimensional (anti-)de Sitter space
- Universality and constant scalar curvature invariants
- On the uniqueness of the Kerr-(A)dS metric as a type II(D) solution in six dimensions
- Killing Spinors and Related Symmetries in Six Dimensions
- Double-extended Kerr-Schild form for electrovacuum solutions
- Locally Boost Isotropic Spacetimes and the Type Condition
- 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
- Interpreting gravitational fields of Topologically Massive Gravity using geodesic deviation
- An invariant characterization of the quasi-spherical Szekeres dust models
- Geometric Horizons: A Frame Approach