Constraints in spherically symmetric classical general relativity. I. Optical scalars, foliations, bounds on the configuration space variables, and the positivity of the quasilocal mass

This is the first of a series of papers in which we examine the constraints of spherically symmetric general relativity with one asymptotically flat region. Our approach is manifestly invariant under spatial diffeomorphisms, 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 from the other by time reversal. Boundary conditions on the spherically symmetric three-geometries and extrinsic curvature tensors are discussed. We introduce a one-parameter family of foliations of spacetime involving a linear combination of the two scalars characterizing a spherically symmetric extrinsic curvature tensor. We can exploit this gauge to express one of these scalars in terms of the other and thereby solve the radial momentum constraint uniquely in terms of the radial current. The values of the parameter yielding potentially globally regular gauges corresponding to the vanishing of a timelike vector in the superspace 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 expressed as a volume integral over the sources and is positive. We provide two proofs of the positivity of the QLM. If the dominant energy condition (DEC) and the constraints are satisfied positivity can be established in a manifestly gauge-invariant way.

This is most easily achieved exploiting the optical scalars. In the second proof we specify the foliation. The payoff is that the weak energy condition replaces the DEC and the Hamiltonian constraint replaces the full constraints. Underpinning this proof is a bound on the derivative of the circumferential radius of the geometry with respect to its proper radius. We show that, when the DEC is satisfied, analogous bounds exist on the optical scalar variables and, following on from this, on the extrinsic curvature tensor. We compare the difference between the values of the QLM and the corresponding material energy to prove that a reasonable definition of the gravitational binding energy is always negative. Finally, we summarize our understanding of the constraints in a tentative characterization of the configuration space of the theory in terms of closed bounded trajectories on the parameter space of the optical scalars.