Scale Invariance, Variety and Central Configurations

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.

💡 Research Summary

The paper revisits the Newtonian N‑body problem from a relational, scale‑invariant perspective, arguing that absolute size is physically meaningless and only dimensionless ratios should enter the fundamental description. By quotienting out global translations, rotations, and dilatations the authors define a reduced configuration space – “shape space” – in which each point represents the pure geometric shape of the particle system. Within this space they introduce a single scale‑invariant scalar, the “variety” V, defined as the ratio of the mass‑weighted root‑mean‑square inter‑particle distance ℓ_rms to the mean‑harmonic distance ℓ_mhl. Because ℓ_rms ∝ √I_cm (the centre‑of‑mass moment of inertia) and ℓ_mhl^{-1} ∝ –V_New (the Newtonian potential), V can be written compactly as V = –(1/M^{5/2}) √I_cm V_New. V therefore measures clustering: when a few particles approach each other ℓ_mhl drops sharply while ℓ_rms changes little, causing V to increase.

Critical points of V, where ∇V = 0, are precisely the well‑known central configurations (CCs) of the N‑body problem. In a CC each particle’s acceleration is proportional to its position relative to the centre of mass, i.e. ∑_{j≠i} G m_j (r_i – r_j)/|r_i – r_j|^3 = –λ (r_i – r_cm). Such configurations support homothetic collapse/expansion solutions as well as static equilibria, and they have been extensively studied because they underpin many analytical results in celestial mechanics.

The novelty of the present work lies in exploring CCs that are not at the absolute minimum of V but only slightly above it. Using numerical optimisation for systems of 5 000 equal‑mass particles in two dimensions and 1 000 particles in three dimensions, the authors locate two families of CCs:

• Near‑minimum V: The particles form an almost perfectly uniform sphere (or disc) with a sharp outer boundary. Radial profiles show concentric shells of high particle density, reflecting a highly ordered, radially quantised arrangement. This reproduces earlier findings that the global minimum of V corresponds to an extremely uniform configuration.

• Slightly higher V (≈1 %–1.5 % above the minimum): Even this modest increase triggers a dramatic reorganisation. Particles self‑assemble into filamentary strands, closed loops, and knot‑like nodes, producing a network that closely resembles the cosmic web observed in large‑scale structure surveys. Nearest‑neighbour distance distributions shift toward shorter separations and become broader, indicating local clustering along one‑dimensional filaments while the overall volume remains comparable to the uniform case.

The authors interpret these results through the geometry of shape space. Regions of low V correspond to deep potential wells where the uniform configuration is stable. Small perturbations push the system toward neighboring regions of slightly higher V, which act as attractors because the curvature of shape space there favours configurations with higher “variety”. Thus, scale‑invariant dynamics alone generate structure without invoking additional forces, stochastic noise, or cosmological assumptions.

A further conceptual contribution is the discussion of an emergent arrow of time. In a closed N‑body system with zero total energy and angular momentum the underlying equations are time‑reversal symmetric, yet the evolution towards higher variety (greater clustering) is statistically one‑way. This provides a relational definition of temporal direction based on the monotonic increase of a complexity‑like quantity, echoing ideas of “gravitational arrow of time”.

In summary, the paper (1) introduces a clean, dimensionless scalar V that encapsulates both the inertial size and the Newtonian potential, (2) shows that central configurations are the stationary points of V, (3) demonstrates numerically that configurations only marginally above the absolute minimum of V spontaneously develop filamentary, web‑like structures reminiscent of the cosmic web, and (4) argues that the intrinsic geometry of shape space naturally drives this behaviour and supplies a relational basis for both structure formation and the emergence of a gravitational arrow of time. The work suggests that adopting scale invariance as a foundational principle could reshape our understanding of cosmology, structure formation, and the very notion of time in classical physics.