Six new mechanics corresponding to further shape theories

Abstract

In this paper, suite of relational notions of shape are presented at the level of configuration space geometry, with corresponding new theories of shape mechanics and shape statistics. These further generalize two quite well known examples: (i) Kendall’s (metric) shape space with his shape statistics and Barbour’s mechanics thereupon. (ii) Leibnizian relational space alias metric scale-and-shape space to which corresponds Barbour–Bertotti mechanics. This paper’s new theories include, using the invariant and group namings, (iii) Angle alias conformal shape mechanics. (iv) Area ratio alias e shape mechanics. (v) Area alias e scale-and-shape mechanics. (iii)–(v) rest respectively on angle space, area-ratio space, and area space configuration spaces. Probability and statistics applications are also pointed to in outline.(vi) Various supersymmetric counterparts of (i)–(v) are considered. Since supergravity differs considerably from GR-based conceptions of background independence, some of the new supersymmetric shape mechanics are compared with both. These reveal compatibility between supersymmetry and GR-based conceptions of background independence, at least within these simpler model arenas.