ABSTRACT: The Problem of Time is due to conceptual gaps between General Relativity and the other observationally-confirmed theories of Physics; it is a major foundational issue in Quantum Gravity. The Problem of Time’s multiple facets were mostly pointed out over 50 years ago by Wheeler, DeWitt and Dirac. These facets were subsequently classified by Kuchar and Isham. They argued that the lion’s share of the problem consists of interferences between facets. They also posed the question of in which order the facets should be approached. By further considering the nature of each facet at the local classical level, the Author showed the facets to be two copies of Lie Theory — spacetime and canonical — with a Wheelerian 2-way route therebetween. This solves the facet ordering question. The resulting mathematical framework turns out moreover to be consistent enough to smooth out all local classical facet interferences as well.
It would furthermore be preferable if all of the Background Independence aspects, resultant Problem of Time facets, and strategies to resolve these, were treated in a globally well-defined manner. The current article begins to address this by classifying what is meant by `global’. Be this at the level of mathematical structure (Topology, Differential Geometry, Lie Theory, PDEs, Functional Analysis). At the level of which spaces the modelling actually requires (space, spacetime, configuration space, phase space, space of spacetimes…). Or at the level of each aspect of time, space or Background Independence more generally. We also include globalization strategies and justification of A Local Resolution of the Problem of Time being possible in the first place. This is based on Hausdorff paracompact spaces, which for now support a Shrinking Lemma.
[50 pages including 14 figures. First Article in a series of six on fundamental Physics and Applied Topology.]
Concepts of Shape 