Initial boundary value problems for Einstein's field equations and geometric uniqueness

Abstract

While there exist now formulations of initial boundary value problems for Einstein's field equations which are well posed and preserve constraints and gauge conditions, the question of geometric uniqueness remains unresolved. For two different approaches we discuss how this difficulty arises under general assumptions. So far it is not known whether it can be overcome without imposing conditions on the geometry of the boundary. We point out a natural and important class of initial boundary value problems which may offer possibilities to arrive at a fully covariant formulation.

19 pages

References in corpus (5)

• Far-from-constant mean curvature solutions of Einstein's constraint equations with positive Yamabe metrics
• Boundary conditions for coupled quasilinear wave equations with application to isolated systems
• On the Asymptotics for the Vacuum Einstein Constraint Equations
• The Constraint Equations
• Is general relativity `essentially understood' ?

Cited by in corpus (22)

• Continuum and Discrete Initial-Boundary-Value Problems and Einstein's Field Equations
• Spacelike matching to null infinity
• Peeling or not peeling -- is that the question ?
• Asymptotics of Schwarzschild black hole perturbations
• On the AdS stability problem
• Self-gravitating Klein-Gordon fields in asymptotically Anti-de-Sitter spacetimes
• Hyperboloidal evolution of test fields in three spatial dimensions
• On the Use of Multipole Expansion in Time Evolution of Non-linear Dynamical Systems and Some Surprises Related to Superradiance
• A review of total energy-momenta in GR with a positive cosmological constant
• On quasi-local Hamiltonians in General Relativity
• The Initial-Boundary Value Problem in General Relativity
• The initial boundary value problem for the Einstein equations with totally geodesic timelike boundary
• Boundary Conditions for the Gravitational Field
• Disembodied boundary data for Einstein's equations
• Conformal structures of static vacuum data
• Constraint preserving boundary conditions for the Baumgarte-Shapiro-Shibata-Nakamura formulation in spherical symmetry
• Geometric Boundary Data for the Gravitational Field
• Killing boundary data for anti-de Sitter-like spacetimes
• Geometric Asymptotics and Beyond
• On the initial boundary value problem for the vacuum Einstein equations and geometric uniqueness
• Time-like hypersurfaces of prescribed mean extrinsic curvature
• Stability of the Toroidal AdS Schwarzschild Solution in the Einstein--Klein-Gordon System