On the possibility of emergent light cones from relational shape dynamics

TL;DR

This paper demonstrates that a universal propagation speed, akin to the speed of light, can emerge from purely relational, scale-invariant N-body dynamics on shape space, offering a new perspective on the origin of relativistic causality.

Contribution

It shows how an effective Lorentzian structure and a maximal signal speed can arise dynamically from shape space spectral properties in relational dynamics.

Findings

Emergent light-cone velocity $c_{rel}$ from shape space spectrum.

Relational dynamics can produce Lorentzian causal structure.

Spectral universality underpins the emergence of relativistic kinematics.

Abstract

We show that a universal propagation scale can emerge from purely relational, scale-invariant $N$-body dynamics formulated on shape space, i.e. the space of configurations modulo translations, rotations, and dilatations. Although pure shape dynamics treats only unparametrized curves as fundamental, we adopt an affine parametrization as a gauge choice to perform a perturbative analysis, with all physically meaningful results expressed in parametrization-independent terms. Linear perturbations around central configurations satisfy second-order equations on shape space whose high-frequency spectrum defines a dimensionless constant $c_{\mathrm{rel}}$. Under general conditions of reparametrization invariance, spectral universality, and strict hyperbolicity, $c_{\mathrm{rel}}$ functions as an emergent light-cone velocity, endowing the product space $\mathbb{R} \times \mathcal{S}$ with an effective Lorentzian structure. These results suggest that causal structure and a maximal signal speed may arise dynamically from the geometry and spectral properties of relational configuration space, providing a novel perspective on the origin of relativistic kinematics.