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

We present a thought experiment extending Einstein's reference mollusk to explore operational realizations of coordinate systems in general relativity. Observers carrying four independently programmable clock-like devices assign numerical labels to events, constructing coordinate charts without the structural constraints of mollusk-type frames. This approach decouples coordinate assignment from both motion and geometry, enabling the representation of arbitrary charts. The framework offers pedagogical clarity: coordinate freedom appears as the capacity to freely program clocks, subject only to the (pre-deployment) programming condition that their collective readings fulfill the minimum mathematical properties required of all charts (in particular smoothness). Following the logical structure of differential geometry, charts precede metric determination, which arises from comparing displayed values with measurements in local inertial frames. We emphasize two key aspects. First, any smooth observer congruence suffices to construct every valid coordinate chart, generalizing Einstein's construction. Second, and more strongly, a single congruence suffices to simultaneously also represent any other chart: by corresponding programming of displays, observers following fixed timelike worldlines can display multiple charts (if more than four numbers are displayed), and emulate also any smooth coordinate transformation, including displaying charts with null coordinate directions in the spacetime in which the swarm is deployed. The timelike motion of the observers is only required to fulfill the task of covering a spacetime region, but does not restrict the freedom of programming.