Spherical Relationalism

TL;DR

This paper explores relational particle models on spherical spaces, extending classical and quantum frameworks to closed geometries like spheres, with implications for cosmology and background independence in physics.

Contribution

It introduces spherical relational models, analyzing their classical and quantum properties, and connects them to cosmological models and geometrodynamics.

Findings

Shape space for N particles on S^1 is T^{N-1}.

S^2 and S^3 cases relate to skies and cosmologies.

Models bridge relational particle theories and quantum cosmology.

Abstract

This paper considers passing from the usual $\mathbb{R}^d$ model of absolute space to $\mathbb{S}^d$ at the level of relational particle models. Both approaches' $d = 1$ cases are rather simpler than their $d \geq 2$ cases, with $N$ particles in $\mathbb{S}^1$ admitting a straightforward reduction with shape space $\mathbb{T}^{N - 1}$. The $\mathbb{S}^2$ and $\mathbb{S}^3$ cases - observed skies and the simplest closed GR cosmologies respectively -- are also considered, the latter in the contexts of both static and dynamical radius of the model universe. The space of relational triangles on $\mathbb{S}^2$ is hyperbolic 3-space $\mathbb{H}^3$. Overall, by passing to a closed underlying absolute space, and then to dynamical notion of space, we close some of the modelling gaps between relational particle models and geometrodynamics or its inhomogeneous perturbative regime of interest in cosmology. Quantum counterparts are also outlined, for use as model arenas of quantum cosmology. These models are useful in further considerations of both classical and quantum background independence.