CONSTRAINTS IN SPHERICALLY SYMMETRICAL CLASSICAL GENERAL-RELATIVITY .1. OPTICAL SCALARS, FOLIATIONS, BOUNDS ON THE CONFIGURATION-SPACE VARIABLES, AND THE POSITIVITY OF THE QUASI-LOCAL MASS

This is the first of a series of papers in which we examine the constr aints of spherically symmetric general relativity with one asymptotica lly hat region. Our approach is manifestly invariant under spatial dif feomorphisms, exploiting both traditional metric variables as well as the optical scalar variables introduced recently in this context. With respect to the latter variables, there exist two linear combinations of the Hamiltonian and momentum constraints one of which is obtained f rom the other by time reversal. Boundary conditions on the spherically symmetric three-geometries and extrinsic curvature tensors are discus sed. We introduce a one-parameter family of foliations of spacetime in volving a linear combination of the two scalars characterizing a spher ically symmetric extrinsic curvature tenser. We can exploit this gauge to express one of these scalars in terms of the other and thereby sol ve the radial momentum constraint uniquely in terms of the radial curr ent. The values of the parameter yielding potentially globally regular gauges correspond to the vanishing of a timelike vector in the supers pace of spherically symmetric geometries. We define a quasilocal mass (QLM) on spheres of fixed proper radius which provides observables of the theory. When the constraints are satisfied the QLM can be expresse d as a volume integral over the sources and is positive. We provide tw o proofs of the positivity of the QLM. If the dominant energy conditio n (DEC) and the constraints are satisfied positivity can be establishe d in a manifestly gauge-invariant way. This is most easily achieved ex ploiting the optical scalars. In the second proof we specify the folia tion. The payoff is that the weak energy condition replaces the DEC an d the Hamiltonian constraint replaces the full constraints. Underpinni ng this proof is a bound on the derivative of the circumferential radi us of the geometry with respect to its proper radius. We show that, wh en the DEC is satisfied, analogous bounds exist on the optical scalar variables and, following on from this, on the extrinsic curvature tens er. We compare the difference between the values of the QLM and the co rresponding material energy to prove that a reasonable definition of t he gravitational binding energy is always negative. Finally, we summar ize our understanding of the constraints in a tentative characterizati on of the configuration space of the theory in terms of closed bounded trajectories on the parameter space of the optical scalars.