Einstein’s theory in a three-dimensional space-time. (English) Zbl 0541.53021
After considering general relativity as either a theory of a Riemannian space-time or as canonical geometrodynamics, the strange behaviour of Einstein’s theory in three dimensions is discussed with some emphasis on properties of the break-down of the Newtonian limit. The discussion is extended to homogeneous isotropic cosmologies in n dimensions. Applications include collapsing clouds of dust and static stars in three dimensions.
Reviewer: J.Spicker
MSC:
| 53B50 | Applications of local differential geometry to the sciences |
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 83F05 | Relativistic cosmology |
| 83E05 | Geometrodynamics and the holographic principle |
Keywords:
Einstein’s theory in three dimensions; Newtonian limit; homogeneous isotropic cosmologies; collapsing clouds of dust; static starsReferences:
| [1] | 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). · doi:10.1093/bjps/VI.21.13 |
| [2] | t’Hooft, G., and Veltman, M. (1972).Nucl. Phys.,B44, 189. · doi:10.1016/0550-3213(72)90279-9 |
| [3] | Schwinger, J. (1962).Phys. Rev.,128, 2425. · Zbl 0118.44001 · doi:10.1103/PhysRev.128.2425 |
| [4] | Glimm, J., and Jaffe, A. (1981).Quantum Physics, A Functional Integral Point of View (Springer, Berlin). · Zbl 0461.46051 |
| [5] | Deser, S. and van Nieuwehuizen, P. (1974).Phys. Rev. D,10, 401. · doi:10.1103/PhysRevD.10.401 |
| [6] | Kuchař, K. (1981). InQuantum Gravity 2: A Second Oxford Symposium. C. J. Isham, R. Penrose, and D. W. Sciama, eds. (Clarendon Press, Oxford). |
| [7] | Leutwyler, H. (1966).Nuovo Cim.,42, 159. · doi:10.1007/BF02856201 |
| [8] | 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). |
| [9] | 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. |
| [10] | Cartan, E. (1922).J. Math. Pure Appl,1, 141. |
| [11] | Vermeil, H. (1917).Nachr. Ges. Wiss. Gottingen,334. |
| [12] | Weyl, H. (1921).Raum, Zeit, Materie (Springer, Berlin). |
| [13] | Lovelock, D. (1971).J. Math. Phys.,12, 498. · Zbl 0213.48801 · doi:10.1063/1.1665613 |
| [14] | Infeld, L., and Schild, A. (1949).Revs. Mod. Phys.,21, 408; Taub, A. (1964).J. Math. Phys.,5, 112. · Zbl 0036.42604 · doi:10.1103/RevModPhys.21.408 |
| [15] | Kuchař, K. (1976).J. Math. Phys.,17, 792. · doi:10.1063/1.522977 |
| [16] | Kuchař, K. (1974).J. Math. Phys.,15, 708; Hojman, S., Kuchař, K., and Teitelboim, C. (1974).Ann. Phys. (N. Y.),96, 88. · doi:10.1063/1.1666715 |
| [17] | Eisenhart, L. P. (1949).Riemannian Geometry (Princeton University Press, Princeton, New Jersey), p. 92. · Zbl 0041.29403 |
| [18] | Kuchař, K. (1978).J. Math. Phys.,19, 390. · doi:10.1063/1.523684 |
| [19] | Allen, M., and Kuchař, K. Work in progress. |
| [20] | 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). · doi:10.1016/0003-4916(82)90164-6 |
| [21] | Misner, C., Thorne, K., and Wheeler, J. (1973).Gravitation (W. H. Freeman, San Francisco), Chap. 18. |
| [22] | Misner, C., Thorne, K., and Wheeler, J. (1973). Gravitation (W. H. Freeman, San Francisco), pp. 728, 729. |
| [23] | Giddings, S., to appear inAm. J. Phys. |
| [24] | Bondi, H. (1960).Cosmology, 2nd ed. (Cambridge University Press, Cambridge), p. 73. · Zbl 0046.20807 |
| [25] | Israel, W. (1966),Il Nuovo Cim.,X44, 1; Israel, W. (1967).Il Nuovo Cim.,X48, 463; Israel, W. (1967).Phys. Rev.,153, 1388. |
| [26] | Staruszkiewicz, A. (1963).Acta. Phys. Polon.,24, 735. |
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