The Emergence of Measured Geometry in Self-Gravitating Systems

TL;DR

This paper explores how the geometry of self-gravitating systems emerges from gravitational interactions, challenging the notion of fixed background geometry by analyzing spatial variations in particle configurations.

Contribution

It introduces a modern computational approach to demonstrate that measured geometry in Newtonian systems is emergent and context-dependent, aligning with foundational ideas from Poincaré and Einstein.

Findings

Spatial variations in particle separations correlate with radial distance from the center of mass.

Evidence suggests geometry arises from internal gravitational interactions rather than being fixed.

The approach links numerical analysis of equilibrium configurations to foundational concepts of physical geometry.

Abstract

This work investigates the geometrical properties of self-gravitating $N$-body systems from the perspective established by Henri Poincar\'e 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 Poincar\'e'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.