Coordinates, observables and symmetry in relativity

TL;DR

This paper explores how symmetry, coordinate interpretation, and observables interrelate in relativity, proposing a generalized relational approach to defining physical observables without relying on invertible scalar fields.

Contribution

It introduces a generalized relational framework for observables in spacetime theories, extending the concept of point-coincidences without needing four invertible scalar fields.

Findings

Point-coincidences form a four-dimensional manifold representing physical spacetime.

The generalized relational observables do not require four invertible scalar fields.

The work clarifies the role of symmetry and coordinates in the interpretation of spacetime.

Abstract

We investigate the interplay and connections between symmetry properties of equations, the interpretation of coordinates, the construction of observables, and the existence of physical relativity principles in spacetime theories. Using the refined notion of an event as a ``point-coincidence'' between scalar fields that completely characterise a spacetime model, we also propose a natural generalisation of the relational local observables that does not require the existence of four everywhere invertible scalar fields. The collection of all point-coincidences forms in generic situations a four-dimensional manifold, that is naturally identified with the physical spacetime.