The Emergence of Measured Geometry in Self-Gravitating Systems

TL;DR

The study investigates whether geometry is a fixed Euclidean backdrop or an emergent, relational construct in gravity by analyzing central configurations of the Newtonian $N$-body problem. It introduces a scale-invariant measure $V$ built from inter-particle distances and shows a robust radial inhomogeneity in nearest-neighbor separations, implying a position-dependent measuring geometry within an otherwise Euclidean setting. Interpreting these results through Poincaré's operational geometry and Einstein's rod–clock perspective, the work argues that measured geometry arises from the relational structure and interactions of matter, with Newton–Cartan and emergent-gravity frameworks offering a continuum bridge. The findings advocate a unified, relational framework for emergent geometry in gravitational systems and highlight potential implications for cosmology and quantum gravity, where geometry is conceived as arising from underlying relational data rather than being fundamental.

Abstract

This work investigates the geometrical properties of self-gravitating $N$-body systems from the perspective established by Henri Poincaré and Albert Einstein concerning the operational nature of measured geometry. Utilizing recent numerical analyses of central configurations--special equilibrium solutions to the Newtonian $N$-body problem--we uncover systematic spatial variations in nearest-neighbor particle separations correlated with the radial distance from the system's center of mass. We argue that these variations reflect a context-dependent, emergent effective geometry shaped by gravitational interactions, in accordance with Poincaré's assertion that measured geometry depends on the forces influencing measuring devices, and Einstein's view that rods and clocks define physical geometry through their local dynamics. By revisiting these foundational insights within a modern computational framework, we provide evidence that geometry in self-gravitating Newtonian systems is not a fixed background, but an emergent construct arising from internal physical interactions.

Figures (5)

• Figure 1: Three-dimensional CC of 1000 equal-mass particles with variety very close to the absolute minimum of $V$.
• Figure 2: Three-dimensional CC of 1000 equal-mass particles with variety about $1.5\%$ above the absolute minimum of $V$.
• Figure 3: The previous 1000-particle CC roughly $1.5\%$ above the absolute minimum of $V$. We applied a rainbow color scale, where particles shift from red to purple as their distance to the nearest-neighbor decreases.
• Figure 4: A plot of the nearest-neighbor distance as a function of the distance to the center of mass for the 1000-particle CC roughly $1.5\%$ above the absolute minimum of $V$.
• Figure 5: The previous 1000-particle CC roughly $1.5\%$ above the absolute minimum of $V$. A rainbow color scale is used to encode filament density, defined as the inverse of the mean nearest-neighbor distance, with purple indicating higher densities and red lower densities. Noise particles are shown in gray.