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Expanding spherically symmetric models without shear. (English) Zbl 0848.35134

Summary: The integrability properties of the field equation \(L_{xx}= F(x) L^2\) of a spherically symmetric shear-free fluid are investigated. A first integral, subject to an integrability condition on \(F(x)\), is found, giving a new class of solutions which contains the solutions of Stephani and Srivastava as special cases. The integrability condition on \(F(x)\) is reduced to quadrature which is expressible in terms of elliptic integrals in general. There are three classes of solution and in general the solution of \(L_{xx}= F(x) L^2\) can only be written in parametric form.
The case for which \(F= F(x)\) can be explicitly given corresponds to the solution of Stephani. A Lie analysis of \(L_{xx}= F(x) L^2\) is also performed. If a constant \(\alpha\) vanishes, then the solutions of Kustaanheimo and Qvist and of this paper are regained. For \(\alpha\neq 0\) we reduce the problem to a simpler, autonomous equation. The applicability of the Painlevé analysis is also briefly considered.

MSC:

35Q75 PDEs in connection with relativity and gravitational theory
83C15 Exact solutions to problems in general relativity and gravitational theory
83F05 Relativistic cosmology
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[1] Chakravarty, N., Dutta Choudhury, S. B., and Banerjee, A. (1976).Austral. J. Phys. 29, 113.
[2] de Oliveira, A. K. G., Santos, N. O., and Kolassis, C. A. (1985).Mon. Not. R. Astr. Soc. 216, 1001.
[3] Dyer, C. C., McVittie, G. C., and Oates, L. M. (1987).Gen. Rel. Grow. 19, 887 · Zbl 0626.53052 · doi:10.1007/BF00759293
[4] Gradshteyn, I. S., and Ryzhik, I. M. (1994).Table of Integrals, Series and Products (Academic Press, New York). · Zbl 0918.65002
[5] Herrera, L., and Ponce de Leon, J. (1985).J. Math. Phys. 26 778,2018,2847. · doi:10.1063/1.526567
[6] Kramer, D., Stephani, H., MacCallum, M. A. H., and Herlt, E. (1980).Exact Solutions of Einstein’s Field Equations (Cambridge University Press, Cambridge). · Zbl 0449.53018
[7] Krasiński, A. (1989).J. Math. Phys. 30, 438
[8] Kustaanheimo, P., and Qvist, B. (1948).Soc. Sci. Fennica, Commentationes Physico-Mathematicae XIII, 16.
[9] Leach, P. G. L. (1981).J. Math. Phys. 22, 465. · Zbl 0459.70024 · doi:10.1063/1.524932
[10] Leach, P. G. L., Maartens, R., and Maharaj, S. D. (1992).Int. J. Nonlin. Mech. 27, 575 · Zbl 0760.34005 · doi:10.1016/0020-7462(92)90062-C
[11] Lie, S. (1912).Vorlesungen über Differentialgleichungen (Teubner, Leipzig/Berlin). · JFM 43.0373.01
[12] Maartens, R., and Maharaj, S. D. (1990).J. Math. Phys. 31, 151. · Zbl 0706.76143 · doi:10.1063/1.528853
[13] Maharaj, S. D., Leach, P. G. L., and Maartens, R. (1991).Gen. Rel. Grav. 23, 261. · Zbl 0715.53053 · doi:10.1007/BF00762289
[14] McVittie, G. C. (1933).Mon. Not. R. Astr. Soc. 93, 325.
[15] McVittie, G. C. (1967).Ann. Inst. H. Poincaré 6, 1.
[16] McVittie, G. C. (1984).Ann. Inst. H. Poincaré 40, 325.
[17] Ramani, A., Grammaticos, B., and Bountis, T. (1989).Phys. Rep. 180, 159. · doi:10.1016/0370-1573(89)90024-0
[18] Santos, N. O. (1985).Mon. Not. R. Astr. Soc. 216, 403.
[19] Srivastava, D. C. (1987).Class. Quant. Grav. 4, 1093. · Zbl 0649.76070 · doi:10.1088/0264-9381/4/5/012
[20] Stephani, H. (1983).J. Phys. A: Math. Gen. 16, 3529. · Zbl 0541.76179 · doi:10.1088/0305-4470/16/15/017
[21] Sussman, R. A. (1989).Gen. Rel. Grav. 12, 1281. · Zbl 0682.53082 · doi:10.1007/BF00763315
[22] Wyman, M. (1976).Can. Math. Bull. 19, 343. · Zbl 0355.53014 · doi:10.4153/CMB-1976-052-0
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