Problem of Time: Temporal Relationalism Compatibility of Other Local Classical Facets Completed

TL;DR

This paper explores how various classical facets of the Problem of Time in General Relativity can be made compatible with Temporal Relationalism, focusing on foliation and Refoliation Invariance within a TR framework.

Contribution

It provides a detailed analysis of how to adapt classical solutions to the Problem of Time to be consistent with Temporal Relationalism, especially regarding foliation and hypersurface deformations.

Findings

TR-compatible Refoliation Invariance formulated

TR version of hypersurface kinematics developed

Classical resolutions integrated within TR framework

Abstract

Temporal Relationalism (TR) is that there is no time for the universe as a whole at the primary level. Time emerges rather at a secondary level; one compelling idea for this is Mach's `time is to be abstracted from change'. TR leads to, and better explains, the well-known Frozen Formalism Problem encountered in GR at the quantum level. Indeed, abstraction from change is a type of emergent time resolution of this.Moreover, the Frozen Formalism Problem is but one of the many Problem of Time facets, which are furthermore notoriously interconnected. The other local and already classically present facets are as follows. 2) The GR Thin Sandwich involves a subcase of Best Matching, which is one particular implementation of Configurational Relationalism. 3) The Constraint Closure Problem, 4) the Problem of Observables or Beables, 5) Spacetime Relationalism, 6) the Spacetime Construction Problem, and 7) the Foliation Dependence Problem as resolved in classical GR by Refoliation Invariance. In this Article, I bring together the individual classical resolutions of these, and how these can be rendered compatible with TR. Having covered that in detail for 2) to 6) elsewhere, the rest of the current Article is dedicated to the detailed form that 7) and its TR compatible modification takes. I.e. I consider TR implementing foliations, the TR versions of Refoliation Invariance and the associated TR version of hypersurface kinematics and hypersurface deformations. These require a TR counterpart of the Arnowitt--Deser--Misner split.