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