Spherically Symmetric Gravity as a Completely Integrable System

TL;DR

This paper demonstrates that spherically symmetric gravity in 4D can be formulated as a finite-dimensional integrable system with mass and time as conjugate variables, providing insights into the concept of time in General Relativity.

Contribution

It shows that Schwarzschild gravity can be described as a completely integrable system with novel conjugate variables, including time, within the Hamiltonian framework.

Findings

Masses serve as action variables for asymptotically flat spacetimes.

Time variables emerge as angle variables associated with spacetime ends.

Quantization depends on the spectrum of the mass variables.

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.