Abstract
Classification of conformally flat n-dimensional pseudo-Riemannian spaces via Plebanski's method is discussed. It is based on embedding these spaces into a flat (n + 2)-dimensional space and on finding their minimal isometry groups which are subgroups of the conformal group. In particular, the case n = 4 is given in full detail and compared with incomplete results known in the literature. The found conformally flat spacetimes are identified with the associated solutions of the Einstein equations and with the spacetimes used in various cosmological considerations.
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References
Lie S.: Theorie der Transformations-Gruppen, 2nd edn. Chelsea, New York, 1970, Chapter 18.
Fulton T., Rohrlich F., and Witten L.: Rev. Mod. Phys. 34 (1962) 442. Carruthers P.: Phys. Rep. C 1 (1971). Mickelsson J. and Niederle J.: Ann. Inst. Henri Poincare XXIII (1975) 277. Kotecký R. and Niederle J.: Rep. Math. Phys. 7 (1975) 95. Todorov I.T., Mintchev N.C., Petkova V.B.: Conformal Invariance in Quantum Field Theory, Scuola Normale Superiore, Pisa, 1978. Todorov I.T. and Stanev Y.S.: Chiral Current Algebras and 2-Dimensional Conformal Models, Lecture notes, Univ. de Laussanne, 1992. Barut A.O., Budinich P., Niederle J., and Rączka R.: Found. Phys. 24 (1994) 1461. Halpern M.B., Kiritsis E., Obers N.A., and Clubok K.: Irrational Conformal Field Theory, Preprint UCB-PTH-95/02, University of California, Berkeley.
Plebanski J.F.: in Gravitation and Geometry (Eds. W. Rindler and A. Trautman), Bibliopolis, Naples, 1987, p. 363.
Levine J.: Bull. Amer. Math. Soc. 42 (1936) 418.
Levine J.: Bull. Amer. Math. Soc. 45 (1939) 766.
Eisenhart L.P.: Riemannian Geometry, Princeton Univ. Press, Princeton (N.J.), 1927.
Kramer D., Stephani H., MacCallum M., and Herlt E.: Exact solutions of Einstein's field equations, Cambridge Univ. Press, 1980.
Ryan M.P. and Shapley L.C.: Homogeneous Relativistic Cosmologies, Princeton Univ. Press, 1975.
Weinberg S.: Gravitation and Cosmology, J. Wiley & Sons, New York, 1972.
Hawking S. and Ellis G.: The Large Scale Structure of Space-Time, Cambridge Univ. Press, 1973.
Bayin S.S.: Phys. Rev. D 19 (1979) 2838.
Siemieniec-Ozieblo G.: Acta Phys. Polon. B 14 (1983) 465.
Broadhurst T.J., Ellis R.S., Koo D.C., and Szalay A.S.: Nature 343 (1990) 726. Broadhurst T.J. et al.: in Proc. VI Marcel Grossman Meeting (Editor H. Sato), World Scientific, Singapore, 1992. Geller M.J. and Huchra J.P.: Science 246 (1989) 989. Da Costa L.N. et al.: Astrophysics J. Letters 424 (1994) L1.
Krasinski A.: Physics in an Homogeneous Universe, Cambridge Univ. Press, Cambridge, 1996.
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Bačkovský, P., Niederle, J. On classification of conformally flat spaces. Czechoslovak Journal of Physics 47, 1001–1017 (1997). https://doi.org/10.1023/A:1021181106125
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DOI: https://doi.org/10.1023/A:1021181106125