From Mollusk to Swarms of Observers: A Fully General, Observer-Based Operational Framework for General Relativity

We present a thought experiment generalizing Einstein's "reference mollusk" - a congruence of observers, each carrying a clock and fixed spatial labels - to operationally realize any valid coordinate chart in general relativity. Each observer in a seed congruence carries a programmable device that computes and displays four coordinate values as functions of the four seed coordinates. Whenever the programmed transformation is a valid coordinate change on the working region, the displayed values implement a valid chart. A single congruence can directly display arbitrary alternative or additional charts - including light-cone coordinates with two null and two spacelike basis directions, Painlevé-Gullstrand coordinates inside the Schwarzschild horizon with four spacelike basis directions, and Kruskal-Szekeres coordinates with two null directions, none of which any congruence-adapted frame can produce directly. The same physical swarm can concurrently display multiple charts of different gauge-adaptation during a single deployment, operationally mirroring an atlas of compatible charts. Coordinate directions need not mirror observer motion: observers travel on timelike worldlines while the displayed coordinates may have null or spacelike basis directions. The metric is then reconstructed in the displayed coordinates by standard local-inertial measurements, keeping three layers manifestly distinct: the material reference apparatus, the coordinate labels it displays, and the metric subsequently reconstructed in those labels.