Events and observables in generally invariant spacetime theories

TL;DR

This paper explores the nature of observables in generally invariant spacetime theories, proposing a new approach to defining events as point-coincidences of scalar fields, which helps identify physical spacetime.

Contribution

It introduces a generalized relational observable framework that does not depend on four invertible scalar fields, enhancing the understanding of spacetime events.

Findings

Defines events as point-coincidences of scalar fields

Shows the collection of point-coincidences forms a four-dimensional manifold

Identifies this manifold with physical spacetime in generic cases

Abstract

We address the problem of observables in generally invariant spacetime theories such as Einstein's general relativity. Using the refined notion of an event as a ``point-coincidence'' between scalar fields that completely characterise a spacetime model, we propose a 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.