A Local Resolution of the Problem of Time. II. Configurational Relationalism via a generalization of Group Averaging (1905.06206v4)

Abstract: In this article, we consider a second Problem of Time Facet. This started life as Wheeler's Thin Sandwich Problem, within the narrow context of 1) GR-as-Geometrodynamics, in particular its momentum constraint. 2) A Lagrangian variables level treatment. Conceiving in terms of Barbour's Best Matching is a freeing from 1), now in the context of first-class linear constraints. Conceiving in terms of the underlying Background Independence aspect - the titular Configurational Relationalism - serves moreover to remove 2) as well. This is implemented by the $G$-act, $G$-all method. I.e. given an object $O$ that is not $G$-invariant, we act on it with $G$ and then apply an operation involving the whole group. This has the effect of double-cancelling the introduction of our group action, thus yielding a $G$-independent version of $O$. A first example of whole-group operation is group averaging. This is a valuable prototype by its familiarity to a large proportion of Mathematics and Physics majors through featuring in elementary Group Theory and Representation Theory courses (and which can be traced back to Cauchy). In particular, this is much more familiar than Thin Sandwiches or Best Matching! Secondly, extremization over the group, of which Best Matching, and taking infs or sups over the group, are examples. Some further significant examples of this method in modern geometry and topology include those of Kendall, Younes, Hausdorff and Gromov. In this way, we populate this approach with examples well beyond the usual GR literature's by DeWitt, Barbour and Fischer (which we also outline).