Abstract
In his famous paper of 1916 on the foundation of general relativity Einstein has formulated the following principle that he himself calls “the postulate of general covariance”1:
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Notes
A. Einstein: Die Grundlage der allgemeinen Relätivitatstheorie. Ann. d. Phys. 49 (1916) 769–822. Quoted from § 3.
ibid. § 2.
ibid. § 3.
E. Kretschmann: Über den physikalischen Sinn der Relativitätspostulate. A. Einsteins neue und seine ursprüngliche Relativitatstheorie. Ann. d. Phys. 53 (1917) 575–614. Quoted from p. 576.
A. Einstein: Prinzipielles zur allgemeinen Relativitätstheorie. Ann. d. Phys. 55 (1918) 241–244. Quoted from p. 242.
A. Einstein and L. Infeld: The Evolution of Physics. Cambridge 1938. Quoted from p. 212. As I learned from Don Howard the book was written entirely by Infeld,and Einstein only gave his name to fasten the sale.
A. Einstein: Die Grundlage der allgemeinen Relätivitatstheorie. Ann. d. Phys, 49 (1916) 769–822. Quoted from § 2.
ibid. § 3.
E. Cartan: Sur les variétés a connexion affine et la théorie de la relativité géneralisée. Ann. sci. Ecole Normale Supér. 40 (1923) 326–412 and 41 (1924) 1-25.
N. Bourbaki: Elements of Mathematics. Theory of Sets. Reading Mass., 1968. CH. IV. For physical application see G. Ludwig: Die Grundstrukturen einer physikalischen Theorie. Berlin 1978.
For the following see E. Scheibe: Invariance and Covariance. In: Scientific Philosophy Today, Essays in Honor of Mario Bunge. Ed. by J. Agassi and R. S. Cohen. Dordrecht 1982. 311–31.
For details see the Encyclopedic Dictionary of Mathematics. Ed. by S. Iyanaga and Y. Kawada. Cambridge, Mass., 1977. 92 D and (a narrower concept) 108 Z.
A recent exception is W. G. Dixon: Special Relativity. CUP 1978. pp. 42 ff.
F. Klein: Elementarmathematik vom höheren Standpunkt aus. Vol. II: Geometrie. Berlin 1925.
The Standard monograph is J. A. Schouten: Ricci-Calculus. Berlin 1954.
See Misner, Ch. W., Thorne, K.S., and J. A. Wheeler: Gravitation. San Francisco 1973. What I am emphasizing is that, although the definition of, say, the concept of a vector field need not refer to coordinate systems, the definition is based on the concept of a differentiable manifold and this concept usually is defined by using coordinate systems.
A different analysis of the principle of general covariance can be found in S. Weinberg: Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York 1972. pp. 91 ff.
For some further thoughts on the matter see the paper mentioned in n. 11.
There are, of course, explications different from (C+). One possibility is to restrict the whole question to field theories in the sense of (4). Yet the problem of proving (C+) thus modified again is a matter not too easily settled.
J. L. Anderson: Principles of Relativity Physics. New York 1967. J. L. Anderson: Covariance, Invariance, and Equivalence: a Viewpoint. Gen. Rel. Grav. 2 (1971) 161-72.
M. Friedman: Relativity Principles, Absolute Objects, and Symmetry Groups. In: Space, Time, and Geometry. Ed. by P. Suppes. Dordrecht 1973. 296–320.
This assumption simplifies the concept formation and the argument. But it seems not essential for the matter.
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Scheibe, E. (1991). Covariance and the Non-Preference of Coordinate Systems. In: Brittan, G.G. (eds) Causality, Method, and Modality. The University of Western Ontario Series in Philosophy of Science, vol 48. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3348-7_3
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