On the Meaning of the Principle of General Covariance

TL;DR

This paper formalizes the Principle of General Covariance as a physically testable principle using quasi-Minkowskian coordinates, clarifying the role of the metric field in defining spacetime points and addressing the Hole Argument.

Contribution

It introduces quasi-Minkowskian coordinates as an operational tool to give physical meaning to the principle of general covariance and clarifies the metric's role in individuating spacetime points.

Findings

Quasi-Minkowskian coordinates operationally define spacetime points.

The metric field provides the physical meaning of coordinates.

Addresses the Hole Argument using QMC and metric field.

Abstract

We present a definite formulation of the Principle of General Covariance (GCP) as a Principle of General Relativity with physical content and thus susceptible of verification or contradiction. To that end it is useful to introduce a kind of coordinates, that we call quasi-Minkowskian coordinates (QMC), as an empirical extension of the Minkowskian coordinates employed by the inertial observers in flat space-time to general observers in the curved situations in presence of gravitation. The QMC are operationally defined by some of the operational protocols through which the inertial observers determine their Minkowskian coordinates and may be mathematically characterized in a neighbourhood of the world-line of the corresponding observer. It is taken care of the fact that the set of all the operational protocols which are equivalent to measure a quantity in flat space-time split into inequivalent subsets of operational prescriptions under the presence of a gravitational field or when the observer is not inertial. We deal with the Hole Argument by resorting to de idea of the QMC and show how it is the metric field that supplies the physical meaning of coordinates and individuates point-events in regions of space-time where no other fields exist. Because of that the GCP has also value as a guiding principle supporting Einstein's appreciation of its heuristic worth in his reply to Kretschmann in 1918.