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Comments on the static spherically symmetric cosmologies of Ellis, Maartens, and Nel
1.O. F. R. Ellis, R. Maartens, and S. D. Nel, Mon. Not. R. Astron. Soc. 184, 439 (1978).
2.We choose geometrical units, in which where G is the Newtonian gravitational constant, and c is the velocity of light in a vacuum. The signature of the space‐time metric is (−++), and the conventions for the Riemann tensor, Ricci tensor, and Ricci scalar are defined, respectively, by and where is any (sufficiently differentiable) vector field. The tensors and are, respectively, the metric tensor, the Einstein tensor, and the energy‐momentum tensor.
3.S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space‐Time (Cambridge U.P., Cambridge, 1973).
4.D. Kramer, H. Stephani, M. A. H. MacCallum, and E. Herlt, Exact Solutions of Einstein’s Field Equations (Cambridge U.P., Cambridge, New York, New Rochelle, Melbourne, and Sydney, 1980), and references cited.
5.G. F. R. Ellis, “Relativistic Cosmology” in General Relativity and Cosmology, Proc. Int. Sch. Phys. “Enrico Fermi” Course XLVII, edited by R. K. Sachs (Academic, New York and London, 1971).
6.C. B. Collins, J. Math. Phys. 18, 1374 (1977).
7.Equation (2.2a), with X = 0, corrects a misprint in Eq. (3.1) of Ref. 6. We have changed the notation to conform with that of Refs. 1 and 4.
8.C. W. Misner and H. S. Zapolsky, Phys. Rev. Lett. 12, 635 (1964).
9.S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (Wiley, New York, 1972).
10.R. C. Tolman, Phys. Rev. 55, 364 (1939). The special “Misner‐Zapolsky” solution is obtained by putting and (with ) in Eq. (4.6) of Tolman’s paper.
11.M. Wyman, Phys. Rev. 75, 1930 (1949).
12.S. Chandrasekhar, An Introduction to the Study of Stellar Structure (Dover, New York, 1969).
13.B. Schmidt, Kugelsymmetrische statische Materielösungen der Einsteinschen Feldgleichungen (Diplomarbeit, Hamburg University, 1966).
14.J. B. Hartle, “Relativistic Stars, Gravitational Collapse and Black Holes” in Relativity, Astrophysics and Cosmology, edited by W. Israel (Reidel, Boston, 1973).
15.G. Birkhoff and G.‐C. Rota, Ordinary Differential Equations (Blaisdell, Waltham, Mass., 1969).
16.R. Mansouri, Ann. Inst. Henri Poincaré 27, 175 (1977).
17.E. N. Glass, J. Math. Phys. 20, 1508 (1979).
18.C. B. Collins and J. Wainwright, “On the Role of Shear in General Relativistic Cosmological and Stellar Models,” preprint, University of Waterloo, Canada (1982).
19.M. Wyman, Phys. Rev. 70, 396 (1946).
20.Note that the special case in Eq. (14.35) of Ref. 4 is considered. However, we are concerned here with solutions which not only have but also possess an equation of state In Ref. 4, the final reduction in the case is valid only if another solution (not given) is arrived at. Further details are provided in Ref. 18.
21.A. Barnes, Gen. Relativ. Gravit. 4, 105 (1973).
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