Abstract
Space and time and the even more basic notion of spacetime, and the structures assigned to them, belong to the most fundamental concepts of science. So far, every physical theory of some generality and scope, whether it is a classical or a quantum theory, a particle or a field theory, presupposes for the formulation or its laws and for its interpretation some spacetime geometry, and the choice of this geometry predetermines to some extent the laws which are supposed to govern the behaviour of matter, the laws of primary concern to physics. Thus Galileo’s assertion1 still applies: ‘He who undertakes to deal with questions of natural sciences without the help of geometry is attempting the unfeasible.’
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References and Notes
I. G. Galilei, Dialogo. Opere. VIL, p 299, Edizione nazionale Florence (1890–1909).
E. P. Wigner, Annals Math. 40, 139 (1939).
A. Einstein, Annalen der Physik 17, 891 (1905).
H. Minkowski, Göttinger Nachr. (1908), p. 53; see also his Address delivered at the 80th Assembly of German Natural Scientists and Physicians, at Cologne, September 1908, reprinted in the well-known Dover paperback The Principle of Relativity. The quotation on page 73 is taken from this address.
P. G. Roll, R. Krotkov, and R. H. Dicke, Ann. Phys. (N.Y.) 26, 442 (1964); V. B. Braginsky and V. J. Panov, Zh. Eksp. Teor. Fiz. 61, 875 (1971).
A. Einstein, Jb. Radioakt. Electronik 4, 411 (1907); Z. Math. Phys. 62, 225 (1913); Sber. preuss. Akad. Wiss. 844 (1915); Annalen der Physik 49, 769 (1916). The statement in the text emphasizes what is now considered to be the essential content of Einstein’s analysis. Historically, the role of the connection as an important structure independently of the metric was recognized only in 1917 and the following years. Seeref. 27 and the remarks on page 78. For a detailed account of the history including more references see the paper by J. Mehra in this volume.
H. Weyl, Philosophy of Mathematics and Natural Science, Princeton (1949). The quotation is taken from section 18, part C.
See, e.g., H. Poincaré, Revue de Métaphysique et de Morale, vol. 3 (1895).
See, e.g., M. Jammer, Concepts of Space, Cambridge/U.S. A., (1954). This book contains many references to original sources as well as to historical accounts. Many interesting remarks about the development of spacetime concepts, from a modern point of view, are found in ref. 7.
For a new version of this attitude concerning the status of physical geometry see P. Lorenzen, Philosophia naturalis 6, 415 (1961).
W. K. Clifford, in The Common Sense of the Exact Sciences (ed. J. R. Newman), New York (1946), p. 202; J. A. Wheeler, Geometrodynamics, New York (1962).
R.Penrose, Structure of Spacetime, in Battelle Rencontres (ed. C. M. De Witt and J. A. Wheeler ), New York (1968), p. 121.
G. W. Leibniz, Math. Schriften 7, p. 18 fed. Gerhardt), Berlin (1849). The causal interpretation of time order and simultaneity is also used by Kant.
H. Weyl, Raum, Zeit, Materie, 5th ed., Berlin (1923). See in particular p. 142.
See, e.g., ref. 9, ch. IV.
Über die wissenschaftliche Fassung des Galileischen Beharrungsgesetzes, Ber. kgl. Ges. Wiss., Math.-Phys. Kl. (1885), p. 333.
See ref. 15, section 20.
F. Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen, Erlangen (1972).
V. Bargmann, Ann. Math. 59,1 (1954). See also J. M. Juch, Foundations of Quantum Mechanics, Reading/Mass. (1968), ch. 13.
E. Cartan, Ann. Ec. Norm. Sup. 40, 325 (1923); 41, 1 (1924).
K. Friedrichs, Math. Ann. 98, 566 (1927).
P. Havas, Rev. Mod. Phys. 36, 938 (1964).
A. Trautman, in Perspectives in Geometry and Relativity (ed. B. Hoffmann ), Bloomington (1966), p. 413.
The following reasoning follows largely A. Trautman; see ref. 24 and his Brandeis lectures (1964). It is worth noticing that the section in which Schrödinger introduces gravitational fields on pages 56–60 of his beautiful book Space-Time Structure, Cambridge (1950) - can be read verbatim as an introduction to Cartan’s geometrical version of Newton’s theory of gravity, although they are meant to apply to relativity (and do, of course).
O. H. L. Heclcmann and E. Schücking, ‘Newtonsche und Einsteinsche Kosmologie’, in Encycl. of Physics (ed. S. Flügge), 53, 489 (1959).
T. Levi-Civita, Rend. Palermo 42, 173 (1917); J. A. Schouten, Verh. Kon. Akad. Amsterdam 12, 95 (1918).
C. N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954); R. Hermann, Vector Bundles in Mathematical Physics, vol. 1, New York (1970), ch. 3.
G. Holland, Ph.D-thesis, University of Hamburg (1965).
E. Mach, Die Mechanik in ihrer Entwicklung, 7th ed., Leipzig (1912).
See G. Rosen, Am. Journ. Phys. 40, 683 (1972) and the references given there.
The distinction between a ‘weak’ and a ‘strong’ principle of equivalence is due to R. H. Dicke; see The Theoretical Significance of Experimental Relativity, New York (1964).
For a review and detailed references, see ref. 26.
J. L. Anderson, Principles of Relativity Physics, New York (1967), section 4. 3.
K. S. Thorne, C. M. Will and W. T. Ni, in Proceedings of the Conference on Experimental Tests of Gravitational Theories (ed. R. W. Davies), NASA-JPL Technical Memorandum 33–499. See also Will’s lectures presented at the 56th Fermi summer school, Varenna 1972.
E. Breitenberger, Nuovo Cimento ser. 11, IB, 1 (1971).
H. Weyl, Mathematische Analyse des Raumproblems, Berlin (1923), especially lecture 3.
V. A. Fock, Theory of Space, Time and Gravitation, Moscow 1955, Section 8 and Appendix A.
E. C. Zeeman, J. Math. Phys. 5, 490 (1964).
A. D. Alexandrov, Commun. of the State University of Leningrad, No. 8, 103 (1953).
Ref. 7, section 17.
F. J. M. Farley, J. Bailey, R. C. A. Brown, M. Giesch, H. Jöstlein, S. van der Meer, E. Picasso and M. Tannenbaum, Nuovo Cimento 45, 281 (1966).
More precisely: elementary-particle experiments do not determine the variability of the metric field. This fact has been emphasized by Thorne and Will, see ref. 35.
Our formulation of the principle differs only slightly from that of Dicke given in ref. 32. The present formulation avoids inconsistencies which arise from the noncommutativity of covariant derivatives if one speaks of ‘all’ rather than of ‘some’ local laws.
A. Schild, ch. 1 in Relativity Theory and Astrophysics (ed. J. Ehlers); Lectures in Applied Mathematics 8, Providence (1967).
S. Deser, General Relativity and Gravitation 1, 9 (1970).
J. Ehlers, F. A. E. Pirani, A. Schild, ch. 4 in General Relativity (ed. L. O’Raifeartaigh), Oxford (1972), p. 63. See also the forthcoming lectures by J. Ehlers given at the Banff Summer School on Relativity and Gravitation, August 1972, to be published by the Canadian Physical Society.
M. Castagnino, J. Math. Phys. 12, 2203 (1971); E. Kronheimer and R. Penrose, Proc. Camb. Phil. Soc. 63, 481 (1967); H. Reichenbach, Axiomatik der relativistischen Raum-Zeit-Lehre, Braunschweig (1924); J. L. Synge, Relativity: the special theory, Amsterdam (1956), Relativity: the general theory, Amsterdam (1964); A. Trautman, ‘The general theory of relativity’, Soviet Physics Usp$19, 319 (1966), see also ref$124; H. Weyl, see refs$115 and 37; N. M. J. Woodhouse, J. Math. Phys. 14, 495 (1973).
A projective structure can be defined as an equivalence class of symmetric linear connections all having the same geodesies except for parametrization. For a direct definition (avoiding reference to connections) see ref. 47.
R. F. Marzke and J. A. Wheeler, in Gravitation and Relativity (ed. H. Y. Chiu and W. F. Hoffman ), New York (1964).
B. Hoffmann and W. Kundt, in Recent Developments in General Relativity, p. 303, Warsaw (1962).
See, in particular, H. Weyl, Physik. Z. 22, 473 (1921).
Instead of obtaining a metric by adding to a conformai structure a particular kind of projective structure, as indicated in the text and elaborated in ref. 47, one can also get a metric by supplementing the conformai (or causal) structure by a measure. Whereas this last procedure may lend itself more readily to generalizations at the microscopic level, as advocated by D. Finkelstein (ref. 62), there appears to be no phenomenological motivation for introducing a spacetime measure (4-volume) as a primitive concept.
See ref. 13, sections 4 and 5 for an excellent exposition.
These notions have been defined precisely by L. Markus, Ann. Math. 62, 411 (1955).
The connection between spacetime orientations and elementary symmetry violations has been investigated by R. Geroch, Ph. D. Thesis, Princeton 1967; and by Ya. B. Zeldovich and I. D. Novikov, Pisma V Red. Zh. E.T.F. 6, 772 (1967).
This theory is outlined in A. Trautman’s contribution to this volume, where relevant references will be found.
See, e.g., Schrôdinger’s book mentioned in note 25, and M. A. Tonnelat, Les théories unitaires Vélectromagnétisme et de la gravitation, Paris 1965.
R. Penrose, An Analysis of the Structure of Spacetime, Adams Prize essay, 1966, and ref. 13.
G. F. Chew, Sci. Progr. 51, 529 (1963).
See S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge 1973, and the references therein.
See K. Menger’s contribution to the well-known book Albert Einstein, (ed. A. Schilpp), Evanston, 111., U.S.A. 1949; also Topology without points, Rice Institute Pamphlet, 1932.
D. Finkelstein, Phys. Rev. 184, 1261 (1969); also Coral Gables Conference on Fundamental Interactions at High Energy, New York 1969, p. 338. The quotation is taken from the last reference.
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Ehlers, J. (1973). The Nature and Structure of Spacetime. In: Mehra, J. (eds) The Physicist’s Conception of Nature. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2602-4_6
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