Axial symmetry and conformal Killing vectors

Abstract

Axisymmetric spacetimes with a conformal symmetry are studied and it is shown that, if there is no further conformal symmetry, the axial Killing vector and the conformal Killing vector must commute. As a direct consequence, in conformally stationary and axisymmetric spacetimes, no restriction is made by assuming that the axial symmetry and the conformal timelike symmetry commute. Furthermore, we prove that in axisymmetric spacetimes with another symmetry (such as stationary and axisymmetric or cylindrically symmetric spacetimes) and a conformal symmetry, the commutator of the axial Killing vector with the two others mush vanish or else the symmetry is larger than that originally considered. The results are completely general and do not depend on Einstein's equations or any particular matter content.

15 pages, Latex, no figures

Cited by in corpus (54)

• Singularity Theorems and Their Consequences
• Critical Collapse of Cylindrically Symmetric Scalar Field in Four-Dimensional Einstein's Theory of Gravity
• Cylindrical Systems in General Relativity
• Two diverse models of embedding class one
• Geometric inequalities for axially symmetric black holes
• Gravitational Collapse of Self-Similar Perfect Fluid in 2+1 Gravity
• Symmetry-preserving matchings
• Axially symmetric Einstein-Straus models
• Asymptotic Structure with a positive cosmological constant
• Geometrical inequalities bounding angular momentum and charges in General Relativity
• Wormholes with a barotropic equation of state admitting a one-parameter group of conformal motions
• Review on exact and perturbative deformations of the Einstein-Straus model: uniqueness and rigidity results
• Generalisation of the Einstein-Straus model to anisotropic settings
• Axially symmetric, asymptotically flat vacuum metric with a naked singularity and closed timelike curves
• Stationary and Axisymmetric Perfect-Fluid Solutions with Conformal Motion
• Lie symmetries for equations in conformal geometries
• Mining metrics for buried treasure
• Wormholes admitting conformal Killing vectors and supported by generalized Chaplygin gas
• Interpretation of the Weyl tensor
• Stationary axisymmetric exteriors for perturbations of isolated bodies in general relativity, to second order
• Achronal ANEC, Weak Cosmic Censorship, and AdS/CFT duality
• An approximate global solution of Einstein's equation for a rotating compact source with linear equation of state
• A cylindrically symmetric, static anisotropic fluid spacetime and the naked singularity
• Cylindrically symmetric, Asymptotically flat, Petrov Type D spacetime with a Naked Curvature Singularity and Matter Collapse
• Myers-Perry Black Hole in an External Gravitational Field
• Some Restrictions on Symmetry Groups of Axially Symmetric Spacetimes
• Conformal symmetry wormholes and the null energy condition
• Cylindrically symmetric inhomogeneous dust collapse with a zero expansion component
• Stable phantom-energy wormholes admitting conformal motions
• Stable wormholes on a noncommutative-geometry background admitting a one-parameter group of conformal motions
• Study of stationary rigidly rotating anisotropic cylindrical fluids with new exact interior solutions of GR. 5. Dust limit and discussion
• Cylindrical symmetric, non-rotating and non-static or static black hole solutions and the naked singularities
• Comment on "Conformally flat stationary axisymmetric metrics"
• On the Stability of Pressure Isotropy Condition in Palatini $f(R)$ Gravity
• Axially symmetric equilibrium regions of Friedmann-Lemaitre-Robertson-Walker universes
• Gauge fixing and regularity of axially symmetric and axistationary second order perturbations around spherical backgrounds
• Unwrapping Closed Timelike Curves
• The averaged null energy condition on holographic evaporating black holes
• Microstate solutions from black hole deconstruction
• On global models for isolated rotating axisymmetric charged bodies; uniqueness of the exterior field
• The limit of Kerr-de Sitter spacetime with infinite angular-momentum parameter $a$
• Stationary and Axisymmetric Perfect Fluids with one Conformal Killing Vector
• Conical singularity in spacetimes with NUT is observer-dependent
• Chronology protection in stationary three-dimensional spacetimes
• Avenues for Analytic exploration in Axisymmetric Spacetimes. Foundations and the Triad Formalism
• A Structure Theorem for Stationary Perfect Fluids
• Gravitational Collapse of an Imperfect Non Adiabatic Fluid
• Axial symmetry cosmological constant vacuum solution of field equations with a curvature singularity, closed time-like curves and deviation of geodesics
• Gravitational Collapse of a Massless Scalar Field and a Perfect Fluid with Self-Similarity of the First Kind in (2+1) Dimensions
• No-hair and almost-no-hair results for static axisymmetric black holes and ultracompact objects in astrophysical environments
• Interior spacetimes sourced by stationary differentially rotating irrotational cylindrical fluids: anisotropic pressure
• Chronology protection in stationary 3D spacetimes
• Conformal-symmetry wormholes supported by a perfect fluid
• Ultra-massive spacetimes in 2+1 dimensions with positive Λ