RELATIONAL QUADRILATERALLAND. I CONFIGURATION SPACE COORDINATES

Relational particle models (RPM’s) are toy models of many aspects of GR in geometrodynamical form. They are suitable as toy models for studying 1) strategies for the problem of time in quantum gravity, in particular timeless, semiclassical, histories and observables approaches and combinations of these. 2) Various other quantum-cosmological issues: structure formation/inhomogeneity, the significance of uniform states... They are relational in that only relative ratios of separations, relative angles and relative times are significant; more widely, this is a ‘Leibniz–Mach–Barbour’ brand of relationalism. My recent review of RPM’s mostly used the relational triangle as an example; this series of papers extends that to the relational quadrilateral. The relational quadrilateral is more useful in these respects than previously investigated simpler RPM’s via simultaneously possessing linear constraints and nontrivial subsystems; also its configuration space is now a nontrivial complex-projective space. This paper studies quadrilateralland’s configura-tion space. In particular, what the relational quadrilateral counterparts of triangleland’s A) Dragt-type coordinates (ellipticity, anisoscelesness, and triangle area, which is also a democracy invariant), B) subsystem-split parabolic coordi-nates and C) the most blockwise-simple coordinates (spherical polars). These were key to unlocking the dynamics, QM and problem of time calculations for the triangle, and their counterparts turn out to be likewise for the quadrilateral in Papers II, III and IV respectively. I show these are now A) a hexuplet of shape coordinates (which now exclude the democracy invariant square root of the sum of squares of areas), B) a linear combination of these termed Kuiper coordinates, and C) the Gibbons–Pope-type coordinates. Each of these is given a lucid new interpretation in terms of quadrilaterals. I furthermore investigate qualitatively-significant regions of the configuration space of quadrilaterals, in anticipation of timeless and Halliwell-type combined Problem of Time strategies and of uniformity in Quantum Cosmology. PACS: 04.60Kz