Spherically Symmetric Scalar Field Collapse: An Example of the Spacetime Problem of Time

TL;DR

This paper extends a canonical formalism for spherical symmetry to include a scalar field, revealing challenges in defining a spacetime scalar time variable due to the spacetime problem of time.

Contribution

It introduces a formalism incorporating a scalar field into spherical symmetry and discusses the difficulties in establishing a spacetime scalar time variable.

Findings

Super-Hamiltonian and supermomentum constraints simplify with scalar field inclusion.

The natural time variable loses its spacetime scalar property when matter is coupled.

A candidate spacetime scalar time variable is proposed, but integrating it into phase space remains problematic.

Abstract

A canonical formalism for spherical symmetry, originally developed by Kucha\v{r} to describe vacuum Schwarzschild black holes, is extended to include a spherically symmetric, massless, scalar field source. By introducing the ADM mass as a canonical coordinate on phase space, one finds that the super-Hamiltonian and supermomentum constraints for the coupled system simplify considerably. Yet, despite this simplification, it is difficult to find a functional time formalism for the theory. First, the configuration variable that played the role of time for the vacuum theory is no longer a spacetime scalar once spherically symmetric matter is coupled to gravity. Second, although it is possible to perform a canonical transformation to a new set of variables in terms of which the super-Hamiltonian and supermomentum constraints can be solved, the new time variable also fails to be a spacetime scalar. As such, our solutions suffer from the so-called {\it spacetime problem of time}. A candidate for a time variable that {\it is} a spacetime scalar is presented. Problems with turning this variable into a canonical coordinate on phase space are discussed.