Black holes, hidden symmetries, and complete integrability

Abstract

The study of higher-dimensional black holes is a subject which has recently attracted a vast interest. Perhaps one of the most surprising discoveries is a realization that the properties of higher-dimensional black holes with the spherical horizon topology and described by the Kerr-NUT-(A)dS metrics are very similar to the properties of the well known four-dimensional Kerr metric. This remarkable result stems from the existence of a single object called the principal tensor. In our review we discuss explicit and hidden symmetries of higher-dimensional black holes. We start with the overview of the Liouville theory of completely integrable systems and introduce Killing and Killing-Yano objects representing explicit and hidden symmetries. We demonstrate that the principal tensor can be used as a `seed object' which generates all these symmetries. It determines the form of the black hole geometry, as well as guarantees its remarkable properties, such as special algebraic type of the spacetime, complete integrability of geodesic motion, and separability of the Hamilton-Jacobi, Klein-Gordon, and Dirac equations. The review also contains a discussion of different applications of the developed formalism and its possible generalizations.

The review paper, 217 pages, 7 figures. [v3: chapters 1-3 and appendices rearranged and extensively modified, references added]

References in corpus (98)

• Black Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume
• Black holes and the double copy
• Shadow of five-dimensional rotating Myers-Perry black hole
• General Kerr-NUT-AdS Metrics in All Dimensions
• Black Strings, Low Viscosity Fluids, and Violation of Cosmic Censorship
• Gravitational Perturbations of Higher Dimensional Rotating Black Holes: Tensor Perturbations
• The Kerr Metric
• Interpreting the C-metric
• An instability of higher-dimensional rotating black holes
• Complete Integrability of Geodesic Motion in General Kerr-NUT-AdS Spacetimes
• Classification of the Weyl Tensor in Higher Dimensions and Applications
• `Hidden' Symmetries of Higher Dimensional Rotating Black Holes
• Higher-Dimensional Black Holes: Hidden Symmetries and Separation of Variables
• Hidden Symmetry of Higher Dimensional Kerr-NUT-AdS Spacetimes
• Hidden Symmetries of Dynamics in Classical and Quantum Physics
• Rotating Black Holes in Higher Dimensional Einstein-Maxwell Gravity
• Algebraic classification of higher dimensional spacetimes based on null alignment
• Black Hole Thermodynamics with Conical Defects
• Generalization of the Geroch-Held-Penrose formalism to higher dimensions
• A Topology-Changing Phase Transition and the Dynamics of Flavour
• Type D Einstein spacetimes in higher dimensions
• Thermodynamics of Regular Accelerating Black Holes
• Closed conformal Killing-Yano tensor and Kerr-NUT-de Sitter spacetime uniqueness
• Electromagnetic Properties of Kerr-Anti-de Sitter Black Holes
• Ultraspinning instability of rotating black holes
• Separability of Dirac equation in higher dimensional Kerr-NUT-de Sitter spacetime
• Maxwell's Equations in the Myers-Perry Geometry
• End Point of the Ultraspinning Instability and Violation of Cosmic Censorship
• On conformal Killing-Yano tensors for Plebanski-Demianski family of solutions
• Separability of the Hamilton-Jacobi and Klein-Gordon Equations in Kerr-de Sitter Metrics
• Free motion around black holes with discs or rings: between integrability and chaos - II
• Gravitational instability of simply rotating AdS black holes in higher dimensions
• A supersymmetric black lens
• Stability of Five-dimensional Myers-Perry Black Holes with Equal Angular Momenta
• Constants of Geodesic Motion in Higher-Dimensional Black-Hole Spacetimes
• Perturbations of near-horizon geometries and instabilities of Myers-Perry black holes
• Free motion around black holes with discs or rings: between integrability and chaos - I
• Higher dimensional Kerr-Schild spacetimes
• Kerr-NUT-de Sitter Curvature in All Dimensions
• On (non)integrability of classical strings in p-brane backgrounds
• Merger Transitions in Brane--Black-Hole Systems: Criticality, Scaling, and Self-Similarity
• Charged particle in higher dimensional weakly charged rotating black hole spacetime
• Generalized hidden symmetries and the Kerr-Sen black hole
• The Black Ring is Unstable
• Complete Set of Commuting Symmetry Operators for the Klein-Gordon Equation in Generalized Higher-Dimensional Kerr-NUT-(A)dS Spacetimes
• Closed conformal Killing-Yano tensor and geodesic integrability
• Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets
• Particle Motion and Scalar Field Propagation in Myers-Perry Black Hole Spacetimes in All Dimensions
• Symmetries of the Dirac operator with skew-symmetric torsion
• Closed conformal Killing-Yano tensor and uniqueness of generalized Kerr-NUT-de Sitter spacetime
• Integrability and the Kerr-(A)dS black hole in five dimensions
• Quantum integrability of quadratic Killing tensors
• Stationary strings and branes in the higher-dimensional Kerr-NUT-(A)dS spacetimes
• A Note on Separability of Field Equations in Myers-Perry Spacetimes
• New Black Holes in Five Dimensions
• Gyromagnetic Ratio of Charged Kerr-Anti-de Sitter Black Holes
• Kerr-Schild Structure and Harmonic 2-forms on (A)dS-Kerr-NUT Metrics
• Generalized Kerr-NUT-de Sitter metrics in all dimensions
• Dynamics of test particles in the general five-dimensional Myers-Perry spacetime
• Particle dynamics near extreme Kerr throat and supersymmetry
• Electromagnetic Field in Higher-Dimensional Black-Hole Spacetimes
• Bumpy black holes
• Perturbations of higher-dimensional spacetimes
• Perturbations and Stability of Higher-Dimensional Black Holes
• 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
• Newman-Penrose formalism in higher dimensions: vacuum spacetimes with a non-twisting geodetic multiple Weyl aligned null direction
• The Black Hole Meissner Effect and Blandford-Znajek Jets
• On the parameters of the Kerr-NUT-(anti-)de Sitter space-time
• Hidden symmetry in the presence of fluxes
• Parallel-propagated frame along null geodesics in higher-dimensional black hole spacetimes
• Search for microscopic black holes and string balls in final states with leptons and jets with the ATLAS detector at sqrt(s) = 8 TeV
• Hidden symmetries of Eisenhart-Duval lift metrics and the Dirac equation with flux
• Conformal Killing Tensors and covariant Hamiltonian Dynamics
• Hidden Symmetries of Higher-Dimensional Black Hole Spacetimes
• C-metrics in Gauged STU Supergravity and Beyond
• Ricci-flat spacetimes admitting higher rank Killing tensors
• Solving parallel transport equations in the higher-dimensional Kerr-NUT-(A)dS spacetimes
• Black rings with a small electric charge: gyromagnetic ratios and algebraic alignment
• Tidal effects around higher-dimensional black holes
• Quasi-normal modes for doubly rotating black holes
• Do Killing-Yano tensors form a Lie Algebra?
• Symmetry operators and separability of massive Klein-Gordon and Dirac equations in the general 5-dimensional Kerr-(anti-)de Sitter black hole background
• Geometry of Lax pairs: particle motion and Killing-Yano tensors
• Stationary Strings in the Spacetime of Rotating Black Holes in Five-Dimensional Minimal Gauged Supergravity
• Particle Dynamics in Weakly Charged Extreme Kerr Throat
• A Killing tensor for higher dimensional Kerr-AdS black holes with NUT charge
• A simple test for spacetime symmetry
• Weak electromagnetic field admitting integrability in Kerr-NUT-(A)dS spacetimes
• Five-Dimensional Black Hole Capture Cross-Sections
• Integrability Conditions for Killing-Yano Tensors and Conformal Killing-Yano Tensors
• On hidden symmetries of extremal Kerr-NUT-AdS-dS black holes
• Separability in Cohomogeneity-2 Kerr-NUT-AdS Metrics
• Weakly charged generalized Kerr-NUT-(A)dS spacetimes
• SUSY in the Spacetime of Higher-Dimensional Rotating Black Holes
• NUT-like and near-horizon limits of Kerr-NUT-(A)dS spacetimes
• Electron in higher-dimensional weakly charged rotating black hole spacetimes
• Membranes with a symmetry of cohomogeneity one
• Self-similar motion of a Nambu-Goto string

Cited by in corpus (149)

• The Spectra of Gravitational Atoms
• Type D Spacetimes and the Weyl Double Copy
• Charged black-bounce spacetimes
• Massive Vector Fields in Rotating Black-Hole Spacetimes: Separability and Quasinormal Modes
• Instability of the Proca field on Kerr spacetime
• S-duality and the Double Copy
• Maxwell's Equations in the Myers-Perry Geometry
• The universal Teukolsky equations and black hole perturbations in higher-derivative gravity
• Axisymmetric black holes allowing for separation of variables in the Klein-Gordon and Hamilton-Jacobi equation
• Separation of Maxwell equations in Kerr-NUT-(A)dS spacetimes
• Weyl doubling
• Misner Gravitational Charges and Variable String Strengths
• Shadows of charged rotating black holes: Kerr-Newman versus Kerr-Sen
• NUTs and bolts beyond Lovelock
• Quasinormal modes of massive vector fields on the Kerr spacetime
• Subleading Eikonal, AdS/CFT and Double Stress Tensors
• The modification of photon trapping orbits as a diagnostic of non-Kerr spacetimes
• Separation of variables in Maxwell equations in Plebanski-Demianski spacetime
• Black holes with synchronised Proca hair: linear clouds and fundamental non-linear solutions
• Kerr-de Sitter Quasinormal Modes via Accessory Parameter Expansion
• The dark side of fuzzball geometries
• Love Numbers for Rotating Black Holes in Higher Dimensions
• Gravitational perturbations of rotating black holes in Lorenz gauge
• Light propagation in a plasma on an axially symmetric and stationary spacetime: Separability of the Hamilton-Jacobi equation and shadow
• General Smarr relation and first law of Nutty dyonic black hole
• Quasinormal modes of Proca fields in a Schwarzschild-AdS spacetime
• Homoclinic orbit and the violation of the chaos bound around a black hole with anisotropic matter fields
• Killing tensor and Carter constant for Painleve-Gullstrand form of Lense-Thirring spacetime
• Aligned Fields Double Copy to Kerr-NUT-(A)dS
• Killing tensors and photon surfaces in foliated spacetimes
• Quasinormal Modes of C-metric from SCFTs
• Generalized Carter constant for quadrupolar test bodies in Kerr spacetime
• Black Hole Horizons from the Double Copy
• Massive vector fields in Kerr--Newman and Kerr--Sen black hole spacetimes
• Metric perturbations of Kerr spacetime in Lorenz gauge: Circular equatorial orbits
• Boson and Dirac stars in $D\ge 4$ dimensions
• Free motion around black holes with discs or rings: between integrability and chaos -- VI. The Melnikov method
• Analytic Solution for the Motion of Spinning Particles in Kerr Space-Time
• Slice-reducible conformal Killing tensors, photon surfaces and shadows
• Killing horizons and surface gravities for a well-behaved three-function generalisation of the Kerr spacetime
• Electromagnetic fields on Kerr spacetime, Hertz potentials and Lorenz gauge
• Dynamics of null particles and shadow for general rotating black hole
• Excitations of the Myers-Perry Black Holes
• Dynamics, symmetries, anomaly and vortices in a rotating cosmic string background
• Non-linear effects in EMRI dynamics and their imprints on gravitational waves
• Plebański-Demiański solutions with dynamical torsion and nonmetricity fields
• Photon Emission Near Myers-Perry Black Holes in the Large Dimension Limit
• Principal Tensor Strikes Again: Separability of Vector Equations with Torsion
• Free motion around black holes with discs or rings: between integrability and chaos -- V
• Throat effects on strong gravitational lensing in Kerr-like wormholes
• Eisenhart Lift of $2$--Dimensional Mechanics
• On integrability of geodesics in near-horizon extremal geometries: Case of Myers-Perry black holes in arbitrary dimensions
• Scalar fields on λ-deformed cosets
• On the integrability of extended test body dynamics around black holes
• Circular string in a Black p-brane leading to chaos
• Duality and $\mu$-separability of Maxwell equations in Kerr-NUT-(A)dS spacetime
• Gravitational Waves on Kerr Black Holes I: Reconstruction of Linearized Metric Perturbations
• The spinning self-force EFT: 1SF waveform recursion relation and Compton scattering
• Slowly rotating black holes with exact Killing tensor symmetries
• Generalized Lense--Thirring metrics: higher-curvature corrections and solutions with matter
• Hidden symmetry and (super)conformal mechanics in a monopole background
• Black hole photon ring beyond General Relativity: an integrable parametrization
• Love symmetry in higher-dimensional rotating black hole spacetimes
• Positive Energy Functional for Massless Scalars in Rotating Black Hole Backgrounds of Maximal Ungauged Supergravity
• On Integrability of the Geodesic Deviation Equation
• IDEAL characterization of higher dimensional spherically symmetric black holes
• On symmetry operators for the Maxwell equation on the Kerr-NUT-(A)dS spacetime
• Killing-Yano Cotton Currents
• Tensionless Strings and Killing(-Yano) Tensors
• Symplectic mechanics of relativistic spinning compact bodies II.: Canonical formalism in the Schwarzschild spacetime
• Symmetry axes of Kerr-NUT-(A)dS spacetimes
• Geometrical origin of the Kodama vector
• Conformal Killing Initial Data
• Twisted form hierarchies, Killing-Yano equations and supersymmetric backgrounds
• Scale invariance and constants of motion
• Conformally Coupled Scalar in Rotating Black Hole Spacetimes
• Stationary black holes with stringy hair
• Geometry, conformal Killing-Yano tensors and conserved "currents"
• Hidden symmetries from distortions of the conformal structure
• New Metrics Admitting the Principal Killing-Yano Tensor
• Quasinormal modes of Dirac spinors in the background of rotating black holes in four and five dimensions
• An exact model for evaporating primordial black holes in cosmological space-time
• Solutions of the massive Dirac equation in the near-horizon metric of the extremal five dimensional Myers-Perry black hole with equal angular momenta
• Nonlinear supersymmetry as a hidden symmetry
• Resonant Islands of Effective-One-Body Dynamics
• Nonlocal modification of the Kerr metric
• On the Interaction between Ultralight Bosons and Quantum-Corrected Black Holes
• Chiral anomalies in black hole spacetimes
• Two-dimensional twistor manifolds and Teukolsky operators
• Hidden Conformal Symmetries from Killing Towers with an Application to Large-D/CFT
• Invariant prolongation of the Killing tensor equation
• Some metrics admitting nonpolynomial first integrals of the geodesic equation
• Algebraic classification of the gravitational field in Weyl-Cartan space-times
• Surface gravity from tidal acceleration
• Light propagation in Kerr spacetime
• Symmetries, Spinning Particles and the TCFH of D=4,5 Minimal Supergravities
• Aschenbach effect for spinning particles in Kerr-(A)dS spacetime
• Off-shell duality invariance of Schwarzschild perturbation theory
• Conserved quantities and integrability for massless spinning particles in general relativity
• Black-hole hair from vector dark matter accretion
• Exact solutions of the Klein-Gordon equation in external electromagnetic fields on 3D de Sitter background
• Properties of general stationary axisymmetric spacetimes: circularity and beyond
• Linear Stability of Black Holes and Naked Singularities
• First Order Symmetry Operators for the Linearized Field Equation of Metric Perturbations
• Killing tensors of generalized Lense-Thirring space-time
• Evolution and detection of vector superradiant instabilities
• Tidal Forces in Kerr-AdS and Grey Galaxies
• TCFHs, hidden symmetries and type II theories
• Bilocal geodesic operators in static spherically-symmetric spacetimes
• Conformal bridge in a cosmic string background
• Separability, plane wave limits and black holes
• Principal Killing strings in higher-dimensional Kerr-NUT-(A)dS spacetimes
• Generalized wave operators, weighted Killing fields, and perturbations of higher dimensional spacetimes
• TCFHs, hidden symmetries and M-theory backgrounds
• On causality violation on a Kerr-de Sitter spacetime
• Kerr-AdS type higher dimensional black holes with non-spherical cross-sections of horizons
• CFT Duals for Black Rings and Black Strings
• First BGG operators on homogeneous conformal geometries
• Separation of variables in the WZW models
• Types of the geodesic motions in Kerr-Sen-AdS${}_4$ spacetime
• Homogeneous Symmetry Operators in Kerr--NUT--AdS Spacetimes
• Gravitational Faraday Holonomy
• Spinoptics in the Kerr Spacetime: Polarized Wave Scattering
• Universal Horizons without Hypersurface Orthogonality
• Hidden symmetries of near-horizon extremal Kerr-AdS-NUT geometries
• Dynamics of test particles in the five-dimensional Gödel spacetime
• Closed conformal Killing-Yano initial data
• Charged traversable wormholes: charge without charge
• Algebraic classification of the gravitational field in general metric-affine geometries
• Chaos in a generalized Euler's three-body problem
• On First Order Symmetry Operators for the Field Equations of Differential Forms
• The TCFHs of D=11 AdS backgrounds and hidden symmetries
• TCFHs, IIB warped AdS backgrounds and hidden symmetries
• The Radial Action from Probe Amplitudes to All Orders
• TCFHs and hidden symmetries of type IIA AdS backgrounds
• Light-ring pairs from $A$-discriminantal varieties
• Fast and accurate analytical formulas for light propagation in general static, spherically symmetric spacetimes
• Brane Symmetries Revisited: Symmetries of Tensile and Tensionless Branes in Possibly Degenerate Metrics and their Manifestations
• The Bohlin variant of the Eisenhart lift
• Spinning test particles in the spacetime of a global monopole
• Poincaré asymptotic expansion in black hole theory
• Conformal bridge transformation, $\mathcal{P}\mathcal{T}$- and super- symmetry
• Higher-dimensional black holes with multiple equal rotations
• The eye of the storm: A regular Kerr black hole
• Symmetry operators for the conformal wave equation in rotating black hole spacetimes
• The Kerr-de Sitter spacetime in Bondi coordinates
• Shadow cast by a rotating black hole with anisotropic matter
• On a lower-dimensional Killing vector origin of irreducible Killing tensors
• Null integrability and photon phenomenology in the accelerated Schwarzschild black hole