Specific PDEs for Preserved Quantities in Geometry. III. 1-d Projective Transformations and Subgroups

Abstract

We extend finding geometrically-significant preserved quantities by solving specific PDEs to 1-$d$ projective transformations and subgroups. This can be viewed not only as a purely geometrical problem but also as a subcase of finding physical observables, and furthermore as part of extending the comparative study of Background Independence level-by-level in mathematical structure to include projective structure. Full 1-$d$ projective invariants are well-known to be cross-ratios. We moreover rederive this fact as the unique solution of 1-$d$ projective geometry's preserved equation PDE system. We also provide the preserved quantities for the 1-$d$ geometries whose only transformations are 1) special-projective transformations $Q$, giving differences of reciprocals. 2) $Q$ alongside dilations $D$, now giving ratios of difference of reciprocals. This analysis moreover firstly points to a new interpretation of cross-ratio: those ratios of differences that are concurrently differences of reciprocals, and secondly motivates 1) and 2) as corresponding to bona fide and distinctive Geometries.

11 pages including 1 figure. References updated

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Cited by in corpus (7)

• A Local Resolution of the Problem of Time. III. The other classical facets piecemeal
• Specific PDEs for Preserved Quantities in Geometry. I. Similarities and Subgroups
• Spaces of Observables from Solving PDEs. I. Translation-Invariant Theory
• Geometry from Brackets Consistency
• A Local Resolution of the Problem of Time. XIV. Grounding on Lie's Mathematics
• Specific PDEs for Preserved Quantities in Geometry. II. Affine Transformations and Subgroups
• A Local Resolution of the Problem of Time. VIII. Assignment of Observables