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 the tool 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.
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Notes
For instance, if E is the electric field, Q[E(P,x)] would be the set of all possible measurement methods to determine the components of E at P in the coordinates x λ.
Wald refers to an RRP as a”principle of special covariance” [11].
The improper elements of the full Poincaré group should be excluded on account of the existence of phenomena that violate parity and or time reversal symmetry.
τ e and τ r are invariants once the origin for the proper time has been set which amounts to setting the origin of the time \(\tilde{t}\equiv\frac{1}{c}\tilde{x}^{0}\).
I am indebted to Lluís Bel for bringing to my attention this reference.
This means that the space-time must contain regions free from essential singularities where there may be observers of the class \(\mathcal{K}\) who may use QMC(\(C\mathbf{\omega}\))’s to label events in a neighborhood of their world-lines.
Of course here we are considering two observers: O and an inertial one, each of them in a different space-time where the same physical entity is present, for instance an electric field.
These are well-behaved coordinate systems that do not introduce coordinate singularities in the regions where the form of the equations describing the behaviour of the physical quantities is being considered.
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Acknowledgements
I would like to thank Professor Jesús Martín for useful and detailed conversations and Professor Lluís Bel for his always interesting comments and suggestions. I also thank one of the referees for his recommendations that helped to improve the clarity of the paper. This work was started long ago while I was on sabbatical leave at the Department of Atomic, Molecular and Nuclear Physics of the University of Seville by the kind invitation of Professor Luís Rull. It has been partially supported by the Spanish Ministry of Economy and Competitiveness through research project FIS2010-15492, the Basque government through Grant GICO7/51-IT-221-07 and the UFI 11/55 program of the University of the Basque country UPV/EHU.
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Chamorro, A. On the Meaning of the Principle of General Covariance. Int J Theor Phys 52, 117–129 (2013). https://doi.org/10.1007/s10773-012-1309-1
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DOI: https://doi.org/10.1007/s10773-012-1309-1