Spherically symmetric gravity and the notion of time in general relativity

Summary

Abstract

It is shown - in Ashtekar's canonical framework of General Relativity - that the constraints of spherically symmetric (Schwarzschild) gravity in 4 dimensional space-time can be solved completely yielding two canonically conjugate observables for asymptotically flat spacetimes, namely mass and - surprisingly - time. 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. Our results can be expressed within the ADM formalism, too, and their relation to the equivalent ones Kuchař obtained recently are briefly discussed. Quantization of the system and the associated Schrödinger equation depend on the allowed spectrum of the masses.

Introduction

The issue ‘time’ is perhaps the most crucial one in canonical - especially quantum - gravity and a number of different approaches have been pursued in recent years (see the excellent reviews[1, 2]). Whereas the discussions of general aspects are certainly essential, one might possibly learn a lot by the analysis of a single - even very simple - system for which the corresponding quantisation can be carried through completely and in which the quantity time appears as a classical and quantum gauge invariant ‘observable’.

Such a system is spherically symmetric pure gravity. In view of Birkhoff's theorem which seems to eliminate completely the notion of time for such systems this assertion may appear to be quite surprising.