Abstract
As a preparation for studying quantum models, we analyze unusual features of Einstein's theory of gravitation in a three-dimensional space-time. In three dimensions, matter curves space-time only locally and the gravitational field has no dynamical degrees of freedom. The standard correspondence of Einstein's theory with Newton's theory breaks down. A dust distribution moves without any geodesic deviation between the particles. The cosmological models and relativistic stars behave in a qualitatively different way from their Newtonian counterparts. These features are important for the correct understanding of mini-superspace models.
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References
Whitrow, G. J. (1955).Br. J. Philos. Sci.,6, 13–31; or M. Jammer (1969).Concepts of Space, 2nd ed. (Harvard University Press, Cambridge, Massachusetts).
t'Hooft, G., and Veltman, M. (1972).Nucl. Phys.,B44, 189.
Schwinger, J. (1962).Phys. Rev.,128, 2425.
Glimm, J., and Jaffe, A. (1981).Quantum Physics, A Functional Integral Point of View (Springer, Berlin).
Deser, S. and van Nieuwehuizen, P. (1974).Phys. Rev. D,10, 401.
Kuchař, K. (1981). InQuantum Gravity 2: A Second Oxford Symposium. C. J. Isham, R. Penrose, and D. W. Sciama, eds. (Clarendon Press, Oxford).
Leutwyler, H. (1966).Nuovo Cim.,42, 159.
Ryan, M. (1972).Hamiltonian Cosmology (Springer, Berlin); MacCallum, M. A. H. (1975). InQuantum Gravity: An Oxford Symposium, C. J. Isham, R. Penrose, and D. W. Sciama, eds. (Clarendon Press, Oxford).
Einstein, A. (1916). Reprinted in H. A. Lorentz et al. (1952).The Principle of Relativity, trans. by W. Perrett and G. B. Jeffery (Dover, New York), p. 111.
Cartan, E. (1922).J. Math. Pure Appl,1, 141.
Vermeil, H. (1917).Nachr. Ges. Wiss. Gottingen,334.
Weyl, H. (1921).Raum, Zeit, Materie (Springer, Berlin).
Lovelock, D. (1971).J. Math. Phys.,12, 498.
Infeld, L., and Schild, A. (1949).Revs. Mod. Phys.,21, 408; Taub, A. (1964).J. Math. Phys.,5, 112.
Kuchař, K. (1976).J. Math. Phys.,17, 792.
Kuchař, K. (1974).J. Math. Phys.,15, 708; Hojman, S., Kuchař, K., and Teitelboim, C. (1974).Ann. Phys. (N. Y.),96, 88.
Eisenhart, L. P. (1949).Riemannian Geometry (Princeton University Press, Princeton, New Jersey), p. 92.
Kuchař, K. (1978).J. Math. Phys.,19, 390.
Allen, M., and Kuchař, K. Work in progress.
Deser, S., Jackiw, R., and Templeton, S. (1982).Ann. Phys. (N. Y.),140, 372; Levin, J. (1964). Brandeis thesis (unpublished; available through Ann Arbor microprints).
Misner, C., Thorne, K., and Wheeler, J. (1973).Gravitation (W. H. Freeman, San Francisco), Chap. 18.
Misner, C., Thorne, K., and Wheeler, J. (1973). Gravitation (W. H. Freeman, San Francisco), pp. 728, 729.
Giddings, S., to appear inAm. J. Phys.
Bondi, H. (1960).Cosmology, 2nd ed. (Cambridge University Press, Cambridge), p. 73.
Israel, W. (1966),Il Nuovo Cim.,X44, 1; Israel, W. (1967).Il Nuovo Cim.,X48, 463; Israel, W. (1967).Phys. Rev.,153, 1388.
Staruszkiewicz, A. (1963).Acta. Phys. Polon.,24, 735.
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Giddings, S., Abbott, J. & Kuchař, K. Einstein's theory in a three-dimensional space-time. Gen Relat Gravit 16, 751–775 (1984). https://doi.org/10.1007/BF00762914
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DOI: https://doi.org/10.1007/BF00762914