On the Meaning of the Principle of General Covariance
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On the Meaning of the Principle of General Covariance Alberto Chamorro
Received: 30 March 2012 / Accepted: 14 August 2012 / Published online: 7 September 2012 © Springer Science+Business Media, LLC 2012
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. Keywords General relativity · General covariance · Coordinates · Hole argument
1 Introduction Since first formulated nine decades ago the question of the meaning of the GCP has been a subject of polemic and confusion. Thus Kretschmann [1] in 1917 claimed the GCP to be devoid of physical content and that given enough mathematical ingenuity any theory could be set in a general covariant form. Einstein [2] begrudgingly accepted the objection stating however the heuristic value the GCP had in searching for a good theory and that was a reason to prefer General Relativity to Newtonian gravitation which—in his opinion—would only A. Chamorro () Department of Theoretical Physics, University of the Basque Country, 48080 Bilbao, Spain e-mail: [email protected]
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Int J Theor Phys (2013) 52:117–129
be awkwardly cast into generally covariant form. Einstein was soon proved wrong as Cartan [3] in 1923 and Friedrichs [4] in 1927 found serviceable generally covariant formulations of Newtonian gravitation theory. See also Misner et al. (1973, Chap. 12) [5]. In his excellent book Fock [6] makes interesting and critical remarks about the term “general relativity” adopted by Einstein to name his theory of gravitation and the connection of the term with general covariance that, in his view, is merely a logical requirement that is always satisfiable. Fock rightly points out that though Einstein had agreed with Kretschman’s ob
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