Abstract
Flows of a perfect fluid in which the flow-lines form a time-like shear-free normal congruence are investigated. The space-time is quite severely restricted by this condition on the flow: it must be of Petrov Type I and is either static or degenerate. All the degenerate fields are classified and the field equations solved completely, except in one class where one ordinary differential equation remains to be solved. This class contains the spherically symmetric non-uniform density fields and their analogues with planar or hyperbolic symmetry. The type D fields admit at least a one-parameter group of local isometries with space-like trajectories. All vacuum fields which admit a time-like shear-free normal congruence are shown to be static. Finally, shear-free perfect fluid flows which possess spherical or a related symmetry are considered, and all uniform density solutions and a few non-uniform density solutions are found. The exact solutions are tabulated in section 7.
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References
Trümper, M. (1962).Z. Phys.,168, 55.
Ehlers, J. (1961).Akad. Wiss. Mainz, Math-Nat. Kl., No. 11.
Kundt, W. and Trümper, M. (1962).Akad. Wiss. Mainz, Math-Nat. Kl., No. 12.
Hawking, S.W. (1966).Singularities and the Geometry of Space-Time, (Essay submitted for the Adams Prize, 1966).
Hawking, S.W. (1966).Astrophys. J.,145, 544.
Synge, J.L. (1937).Proc. Lond. Math. Soc.,43, 376.
Eisenhart, L.P. (1949).Riemannian Geometry, (Princeton University Press, Princeton, N.J.), p. 92.
Robertson H.T. and Noonan, T.W. (1968).Relativity and Cosmology, (W.B. Saunders Co., Philadelphia, Pa.), p. 219, 325.
Thompson, I.H. and Whitrow, G.J. (1967).M. Not. R. Astr. Soc.,136, 207;ibid,139, 499.
Schwarzschild, K. (1916).Sitzber. Preuss. Akad. Wiss., p.189 and 424.
Estabrook, F. and Wahlquist, H. (1967).Phys. Rev.,156, 1359.
Stepanyuk, N.M. (1968).Sov. Phys. JETP,26, 369.
Barnes, A. (1972).J. Phys. A,5, 374.
Jordan, P., Ehlers, J. and Kundt, W. (1960).Akad. Wiss. Mainz. Math-Nat. Kl., No. 2.
Barnes, A. (1971). Thesis, (University of London).
Trümper, M. (1965).J. Math. Phys.,6, 584.
Levi-Civita, T. (1917–19).Rend. Acc. Lincei,26–28, (nine articles).
Birkhoff, G.D. (1927).Relativity and Modern Physics, (Cambridge University Press).
Tolman, R.C. (1939).Phys. Rev.,55, 364.
Bondi, H. (1964).Proc. Roy. soc. A,282, 303.
Harrison, B.K., Thorne, K.S., Wakano, M. and Wheeler, J.A. (1965).Gravitational Theory and Gravitational Collapse, (Chicago University Press).
Bonnor, W.B. and Faulkes, M.C. (1967).Mon. Not. R. Astr. soc.,137, 239.
Bondi, H. (1969).Mon. Not. R. Astr. Soc.,142, 333.
Whittaker, E.T. and Watson, G.N. (1969).A Course in Modern Analysis, (Cambridge University Press), pp. 429–461.
Cahill, M.E. and McVittie, G.C. (1970).J. Math. Phys.,11, 1382.
Faulkes, M.C. (1970).Prog. Theor. Phys.,42, 1139.
Nariai, H. (1969).Prog. Theor. Phys.,40, 1013.
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Supported by a Science Research Council Research Studentship and by a Turner and Newall Research Fellowship.
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Barnes, A. On shear free normal flows of a perfect fluid. Gen Relat Gravit 4, 105–129 (1973). https://doi.org/10.1007/BF00762798
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DOI: https://doi.org/10.1007/BF00762798