Scale Invariance, Variety and Central Configurations

TL;DR

This paper explores scale invariance in physics through the N-body problem, revealing that critical points of a scale-invariant function called variety lead to pattern formation similar to cosmic structures, offering new insights into relational dynamics.

Contribution

It formalizes the concept of scale invariance using the variety function in the N-body problem and links critical points to pattern formation in shape space, providing a novel perspective on cosmic structure emergence.

Findings

Critical points of the variety function correspond to stable configurations.

Deviations from minimal variety lead to filament and pattern formation.

Shape space structure influences the dynamics and attractors.

Abstract

Scale invariance has received very little attention in physics. Nevertheless, it provides a natural conceptual foundation for a relational understanding of the universe, where absolute size loses meaning and only dimensionless ratios retain physical significance. We formalize this idea through the $N$-body problem, introducing a scale-invariant function--the variety, $V$--built from the square root of the center-of-mass moment of inertia and the Newtonian potential. Critical points of $V$, known as central configurations, correspond to special particle arrangements that preserve their shape under homothetic collapse or expansion. Numerical exploration of these critical points reveals that even slight deviations from the absolute minimum of $V$, which corresponds to a remarkably uniform configuration, lead to the spontaneous formation of filaments, loops, voids and other patterns reminiscent of the cosmic web. This behavior is a consequence of the intrinsic structure of shape space--the space of configurations modulo translations, rotations and dilatations--in which regions of higher variety act as attractors. Our results suggest that scale-invariant dynamics not only captures the relational nature of physical laws but also naturally generates organized patterns, offering a novel perspective on the formation of cosmic structures and on the emergence of a gravitational arrow of time from scale-invariant, relational dynamics.