The Constraints in Spherically Symmetric General Relativity I --- Optical Scalars, Foliations, Bounds on the Configuration Space Variables and the Positivity of the Quasi-Local Mass

Abstract

We examine the constraints of spherically symmetric general relativity with one asymptotically flat region, exploiting both the traditional metric variables and variables constructed from the optical scalars. With respect to the latter variables, there exist two linear combinations of the Hamiltonian and momentum constraints which are related by time reversal. We introduce a one-parameter family of linear extrinsic time foliations of spacetime. The values of the parameter yielding globally valid gauges correspond to the vanishing of a timelike vector in the superspace of spherically symmetric geometries. We define a quasi-local mass on spheres of fixed proper radius which we prove is positive when the constraints are satisfied. Underpinning the proof are various local bounds on the configuration variables. We prove that a reasonable definition of the gravitational binding energy is always negative. Finally, we provide a tentative characterization of the configuration space of the theory in terms of closed bounded trajectories on the parameter space of the optical scalars.

45 pages, Plain Tex, 1 figure available from the authors.

References in corpus (7)

• Quasilocal Energy and Conserved Charges Derived from the Gravitational Action
• Geometrodynamics of Schwarzschild Black Holes
• Quasi-Local Gravitational Energy
• Spherically Symmetric Gravity as a Completely Integrable System
• On the Symmetries of Hamiltonian Systems
• Recollapse of the closed Tolman spacetimes
• The Constraints in Spherically Symmetric General Relativity II --- Identifying the Configuration Space: A Moment of Time Symmetry

Cited by in corpus (17)

• Late time behaviour of the maximal slicing of the Schwarzschild black hole
• Quasi-local variables in spherical symmetry: numerical applications to dark matter and dark energy sources
• Constant mean curvature slices in the extended Schwarzschild solution and collapse of the lapse: Part I
• Bounds on 2m/R for static spherical objects
• Flat foliations of spherically symmetric geometries
• Global Prescribed Mean Curvature foliations in cosmological spacetimes with matter, Part I
• Local Prescribed Mean Curvature foliations in cosmological spacetimes
• Boundary Conditions and Quasilocal Energy in the Canonical Formulation of All 1+1 Models of Gravity
• A 3+1 Computational Scheme for Dynamic Spherically Symmetric Black Hole Spacetimes -- I: Initial Data
• Sufficient Conditions for Apparent Horizons in Spherically Symmetric Initial Data
• Necessary Conditions for Apparent Horizons and Singularities in Spherically Symmetric Initial Data
• The Constraints in Spherically Symmetric General Relativity II --- Identifying the Configuration Space: A Moment of Time Symmetry
• A 3+1 Computational Scheme for Dynamic Spherically Symmetric Black Hole Spacetimes -- II: Time Evolution
• The inequality between size and charge in spherical symmetry
• Embedding spherical spacelike slices in a Schwarzschild solution
• Geometric Bounds in Spherically Symmetric General Relativity
• Trapped surfaces in cosmological spacetimes