Affine-null metric formulation of General Relativity at two intersecting null hypersurfaces

Abstract

We revisit Winicour's affine-null metric initial value formulation of General Relativity, where the characteristic initial value formulation is set up with a null metric having two affine parameters. In comparison to past work, where the application of the formulation was aimed for the timelike-null initial value problem, we consider here a boundary surface that is a null hypersurface. All of the initial data are either metric functions or first derivatives of the metric. Given such a set of initial data, Einstein equations can be integrated in a hierarchical manner, where first a set of equations is solved hierarchically on the null hypersurface serving as a boundary. Second, with the obtained boundary values, a set of differential equations, similar to the equations of the Bondi-Sachs formalism, comprising of hypersurface and evolution equations is solved hierarchically to find the entire space-time metric. An example is shown how the double null Israel black hole solution arises after specification to spherical symmetry and vacuum. This black hole solution is then discussed to with respect to Penrose conformal compactification of spacetime.

arXiv comment28 pages, matches (up to reference listing) published version

References in corpus (43)

• Soft Hair on Black Holes
• Aspects of the BMS/CFT correspondence
• Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited
• Horizon formation and far-from-equilibrium isotropization in supersymmetric Yang-Mills plasma
• Superrotation Charge and Supertranslation Hair on Black Holes
• Supertranslations and Superrotations at the Black Hole Horizon
• Conserved charges of the extended Bondi-Metzner-Sachs algebra
• Extended Symmetries at the Black Hole Horizon
• High-powered Gravitational News
• Bondi-Sachs Formalism
• A perturbative and gauge invariant treatment of gravitational wave memory
• Symmetries and charges of general relativity at null boundaries
• Choptuik scaling in null coordinates
• Stable characteristic evolution of generic 3-dimensional single-black-hole spacetimes
• Center-of-mass angular momentum and memory effect in asymptotically flat spacetimes
• Relativistic hydrodynamics on spacelike and null surfaces: Formalism and computations of spherically symmetric spacetimes
• Simulating the dynamics of relativistic stars via a light-cone approach
• Moving Black Holes in 3D
• Linearized solutions of the Einstein equations within a Bondi-Sachs framework, and implications for boundary conditions in numerical simulations
• Horizon Shells and BMS-like Soldering Transformations
• News from Critical Collapse: Bondi Mass, Tails and Quasinormal Modes
• The Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions
• Covariant double-null dynamics: $(2+2)$-splitting of the Einstein equations
• The sky pattern of the linearized gravitational memory effect
• Observer dependence of angular momentum in general relativity and its relationship to the gravitational-wave memory effect
• Spectral Cauchy Characteristic Extraction of strain, news and gravitational radiation flux
• Retarded radiation from colliding black holes in the close limit
• Complete null data for a black hole collision
• Gauge Invariant Spectral Cauchy Characteristic Extraction
• The affine-null metric formulation of Einstein's equations
• Explicit Global Coordinates for Schwarzschild and Reissner-Nordstroem
• Maximally extended, explicit and regular coverings of the Schwarzschild - de Sitter vacua in arbitrary dimension
• Radiation Memory, Boosted Schwarzschild Spacetimes and Supertranslations
• Characteristic Formulation for Metric $f(R)$ Gravity
• Prescriptions for measuring and transporting local angular momenta in general relativity
• The Bondi-Sachs metric at the vertex of a null cone: axially symmetric vacuum solutions
• Worldtube conservation laws for the null-timelike evolution problem
• Kerr Black Holes and Nonlinear Radiation Memory
• Simple, explicitly time-dependent and regular solutions of the linearized vacuum Einstein equations on a null cone
• Boosted Schwarzschild Metrics from a Kerr-Schild Perspective
• Master equation solutions in the linear regime of characteristic formulation of general relativity
• Some notes on the Kruskal - Szekeres completion
• An alternative approach to the static spherically symmetric vacuum global solution to the Einstein's equations

Cited by in corpus (13)

• Hyperbolically symmetric static fluids: A general study
• Geodesics of the hyperbolically symmetric black hole
• Symmetries, charges and conservation laws at causal diamonds in general relativity
• The affine-null formulation of the gravitational equations: spherical case
• Hyperbolically symmetric versions of Lemaitre-Tolman-Bondi spacetimes
• Influence of Electromagnetic Field on Hyperbolically Symmetric Source
• Non-static hyperbolicallly symmetric fluids
• Simulations of gravitational collapse in null coordinates: I. Formulation and weak-field tests in generalised Bondi gauges
• Slowly rotating Kerr metric derived from the Einstein equations in affine-null coordinates
• Characteristic initial value problems for the Einstein-Maxwell-scalar field equations in spherical symmetry
• Conformal compactification and affine-null metric formulation of the Einstein equations
• Spherically symmetric black holes and affine-null metric formulation of Einstein's equations
• Hierarchical formulation of the self-gravitating, n-dimensional, charged scalar field in spherical symmetry in affine null formalism