Spherically Symmetric Gravity as a Completely Integrable System

Abstract

It is shown - in Ashtekar's canonical framework of General Relativity - that spherically symmetric (Schwarzschild) gravity in 4 dimensional space-time constitutes a finite dimensional completely integrable system. Canonically conjugate observables for asymptotically flat space-times are masses as action variables and - surprisingly - time variables as angle variables, each of which is associated with an asymptotic "end" of the Cauchy surfaces. The emergence of the time observable is a consequence of the Hamiltonian formulation and its subtleties concerning the slicing of space and time and is not in contradiction to Birkhoff's theorem. The results are of interest as to the concept of time in General Relativity. They can be formulated within the ADM formalism, too. Quantization of the system and the associated Schr\"odinger equation depend on the allowed spectrum of the masses.