Abstract
Parallel transport of line elements, surface elements etc. along geodesics and more general curves in a projectively connected manifold is investigated analytically and in terms of geometrical constructions. Projective curvature is characterized geometrically by a projective analogue of the geodesic deviation equation and by a geometrical construction. The results are interpreted physically as statements about free fall world lines in space-time.
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Ehlers, J., Pirani, F. A. E., Schild, A.: General relativity, papers in honour of J. L. Synge (ed. L. O'Raifeartaigh) 63–84. Oxford: Clarendon Press 1972. See also N. M. J. Woodhouse, J. Math. Phys.14, 495–501 (1973).
Weyl, H.: Raum, Zeit, Materie, Fifth edition, Berlin: Springer 1923, Nachr. Ges. Wiss. Göttingen (1921), 99–112; K. L. Stellmacher, Math. Ann.123, 34–52 (1951).
Synge, J. L.: Relativity: the general theory Chap. III, § 8. Amsterdam: North-Holland 1960.
Pirani, F. A. E.: Bull. Acad. Polonica, Math.-Astr.-Phys. Series,13, 239–242 (1965).
Pirani, F. A. E., Schild, A.: Bull. Acad. Polonica, Math.-Astr.-Phys. Series,9, 543–547 (1961); Perspectives in geometry and relativity, essays in honor of Václav Hlavatý (ed. B. Hoffmann) 291–309. Bloomington: Indiana University Press 1966.
Pirani, F. A. E.: Symposia Mathematica, Rome (to be published).
Penrose, R.: Battelle Rencontres 1967, Lectures in Mathematics and Physics (ed. C. M. DeWitt and J. A. Wheeler), 121–235. New York: Benjamin 1968.
Geroch, R.: Ph. D. thesis, Princeton University (1967); Ann. Physics48, 526–540 (1968); J. Math. Phys.9, 450–465 (1968).
Eardly, D., Sachs, R. K.: J. Math. Phys.14, 209–212 (1973). See also D. Eardley, Ph. D. Thesis, University of California at Berkeley (1972).
Levi-Civita, T.: The absolute differential calculus, Chap. VII, 208–220; London: Blackie 1927, J. L. Synge and A. Schild, Tensor Calculus, Chap. 3, Sec. 3.3. University of Toronto Press 1949.
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This paper is dedicated to our friend John Archibald Wheeler, geometer and physicist, who celebrated his sixtieth birthday on July 9, 1971.
This work was supported in part by the National Science Foundation (Grant No. GP-34639X). One of the authors (A.S.) did much of this work while visiting the Université Libre de Bruxelles (summer, 1968), Cambridge University (summer, 1970), and the Nordic Institute for Theoretical Atomic Physics (1970-71); he wishes to thank these institutions and Drs. I. Prigogine, D. Sciama, and C. Møller for their kind hospitality.
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Ehlers, J., Schild, A. Geometry in a manifold with projective structure. Commun.Math. Phys. 32, 119–146 (1973). https://doi.org/10.1007/BF01645651
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DOI: https://doi.org/10.1007/BF01645651