Gauge structure of the Einstein field equations in Bondi-like coordinates

Abstract

The characteristic initial (boundary) value problem has numerous applications in general relativity (GR) involving numerical studies, and is often formulated using Bondi-like coordinates. Recently it was shown that several prototype formulations of this type are only weakly hyperbolic. Presently we examine the root cause of this result. In a linear analysis we identify the gauge, constraint and physical blocks in the principal part of the Einstein field equations in such a gauge, and show that the subsystem related to the gauge variables is only weakly hyperbolic. Weak hyperbolicity of the full system follows as a consequence in many cases. We demonstrate this explicitly in specific examples, and thus argue that Bondi-like gauges result in weakly hyperbolic free evolution systems under quite general conditions. Consequently the characteristic initial (boundary) value problem of GR in these gauges is rendered ill-posed in the simplest norms one would like to employ. The possibility of finding good alternative norms, in which well-posedness is achieved, is discussed. So motivated, we present numerical convergence tests with an implementation of full GR which demonstrate the effect of weak hyperbolicity in practice.

arXiv comment23 pages, 3 figures, ancillary files, data and more supplemental material at 10.5281/zenodo.5618007, updated to match published version

References in corpus (68)

• The Large N Limit of Superconformal Field Theories and Supergravity
• Holography and colliding gravitational shock waves in asymptotically AdS_5 spacetime
• Numerical solution of gravitational dynamics in asymptotically anti-de Sitter spacetimes
• High-powered Gravitational News
• Unambiguous determination of gravitational waveforms from binary black hole mergers
• Toward standard testbeds for numerical relativity
• Choptuik scaling in null coordinates
• Characteristic extraction in numerical relativity: binary black hole merger waveforms at null infinity
• Computation of Displacement and Spin Gravitational Memory in Numerical Relativity
• The eth formalism in numerical relativity
• Mode coupling in the nonlinear response of black holes
• Adding Gravitational Memory to Waveform Catalogs using BMS Balance Laws
• Hyperboloidal layers for hyperbolic equations on unbounded domains
• Improved Cauchy-characteristic evolution system for high-precision numerical relativity waveforms
• A Characteristic Extraction Tool for Gravitational Waveforms
• Relativistic hydrodynamics on spacelike and null surfaces: Formalism and computations of spherically symmetric spacetimes
• An Introduction to Well-posedness and Free-evolution
• Holographic Collisions across a Phase Transition
• Bubble Wall Velocity from Holography
• Axisymmetric core collapse simulations using characteristic numerical relativity
• Implementation of standard testbeds for numerical relativity
• Paths to equilibrium in non-conformal collisions
• Regularity of the Einstein Equations at Future Null Infinity
• Hyperbolicity of second-order in space systems of evolution equations
• Strategies for the Characteristic Extraction of Gravitational Waveforms
• Simulating the dynamics of relativistic stars via a light-cone approach
• Real-Time dynamics and phase separation in a holographic first order phase transition
• Linearized solutions of the Einstein equations within a Bondi-Sachs framework, and implications for boundary conditions in numerical simulations
• Numerical stability of the Z4c formulation of general relativity
• Holographic Collisions in Non-conformal Theories
• Spherical symmetry as a test case for unconstrained hyperboloidal evolution
• Tetrad formalism for numerical relativity on conformally compactified constant mean curvature hypersurfaces
• Numerical relativity with characteristic evolution, using six angular patches
• Gravitational wave extraction based on Cauchy-characteristic extraction and characteristic evolution
• Numerical stability for finite difference approximations of Einstein's equations
• Extending Gravitational Wave Extraction Using Weyl Characteristic Fields
• Dynamics of Phase Separation from Holography
• A framework for large-scale relativistic simulations in the characteristic approach
• Crossing a large-$N$ phase transition at finite volume
• New Features of Gravitational Collapse in Anti-de Sitter Spacetimes
• Spectral Characteristic Evolution: A New Algorithm for Gravitational Wave Propagation
• Comparing Remnant Properties from Horizon Data and Asymptotic Data in Numerical Relativity
• Dynamics near a first order phase transition
• Spectral Cauchy-characteristic extraction of the gravitational wave news function
• Spherical symmetry as a test case for unconstrained hyperboloidal evolution II: gauge conditions
• General relativistic null-cone evolutions with a high-order scheme
• Binary Black Hole Waveform Extraction at Null Infinity
• Hyperbolicity of General Relativity in Bondi-like gauges
• The affine-null metric formulation of Einstein's equations
• High-accuracy waveforms for black hole-neutron star systems with spinning black holes
• The affine-null formulation of the gravitational equations: spherical case
• Global simulations of Minkowski space-time including space-like infinity
• First-order quasilinear canonical representation of the characteristic formulation of the Einstein equations
• Hyperbolicity of Physical Theories with Application to General Relativity
• Asymmetric shockwave collisions in ${\text{AdS}}_5$
• Critical phenomena in gravitational collapse with two competing massless matter fields
• Stationary Black Holes as Holographs II
• Gauge stability of 3+1 formulations of General Relativity
• Cauchy-Characteristic Matching In General Relativity
• Revisiting the characteristic initial value problem for the vacuum Einstein field equations
• Observational cosmology using characteristic numerical relativity: Characteristic formalism on null geodesics
• Hyperbolicity of High Order Systems of Evolution Equations
• Holographic systems far from equilibrium: a review
• The SpECTRE Cauchy-characteristic evolution system for rapid, precise waveform extraction
• The Bondi problem revisited: a spectral domain decomposition code
• On necessary and sufficient conditions for strong hyperbolicity
• Estimates for first-order homogeneous linear characteristic problems
• A symmetric hyperbolic formulation of the vacuum Einstein equations in affine-null coordinates

Cited by in corpus (7)

• Holographic Bubbles with Jecco: Expanding, Collapsing and Critical
• Fully relativistic three-dimensional Cauchy-characteristic matching for physical degrees of freedom
• Merging black holes with Cauchy-characteristic matching: Computation of late-time tails
• Gravitational-wave parameter inference with the Newman-Penrose scalar ${\psi }_4$
• Numerical convergence of model Cauchy-characteristic extraction and matching
• Simulations of gravitational collapse in null coordinates: III. Hyperbolicity
• Gravitational Radiation from an Accelerating Massive Particle in General Relativity