ABSTRACT: We consider global issues with Relationalism: a first piece of Background Independence that resolves various Problem of Time facets and is locally implemented by Lie derivatives. Spacetime Relationalism and Configurational Relationalism are quite similar, though the second of these requires supplementing with a heterogeneous Temporal Relationalism. Avoiding both zeroes and Killing vectors is involved, as are some fibre bundle effects, including bibundles, monopoles on configuration space and the Gribov effect. Most of Relationalism’s globality follows from quotient configuration spaces being stratified manifolds. Nine distinct strategies for dealing with stratified manifolds are compared. Compactness and metric-space guarantees for nice (in particular Hausdorff) stratified manifolds are provided. Fibre bundles do not suffice for stratified manifolds, so general bundles, differential spaces, presheaves and sheaves are brought in instead. Hausdorff paracompact (HP) spaces continue to afford simplifications, with some support when local compactness and second-coutability apply. Relational Particle Mechanics, Gauge Theory and GR examples of these various global issues are provided.
[79 pages, including 25 figures. Second of a series of six Fundamental Physics and Applied Topology Articles.]
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.]
This rests the previous preprint – on Packing Theory applied to Social Distancing – on solid Graph Theory foundations that describe its topological-level structure. Further theoretical description of its metric and real-plane-embedded flat-geometric levels of structure are left for another occasion.