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Comments on the static spherically symmetric cosmologies of Ellis, Maartens, and Nel

### Abstract

Ellis, Maartens, and Nel have discussed the viability of static spherically symmetric (SSS) cosmologies in general relativity, and in doing so they have studied some of the mathematical aspects of the field equations in that situation. We investigate further these mathematical aspects. Since the field equations correspond to those studied for stellar models, this question is related to previous investigations in that context. In particular, it is shown that conditions at the center of symmetry do not always uniquely determine the space‐time geometry; this has relevance to the numerical investigation of stellar systems. Finally, in view of the need to generalize SSS models, some remarks are made on the possibility of relaxing the staticity condition in the case of models that are shear‐free.

© 1983 American Institute of Physics

Received 09 November 1981
Accepted 06 August 1982

/content/aip/journal/jmp/24/1/10.1063/1.525595

1.

1.O. F. R. Ellis, R. Maartens, and S. D. Nel, Mon. Not. R. Astron. Soc. 184, 439 (1978).

2.

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.

3.S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space‐Time (Cambridge U.P., Cambridge, 1973).

4.

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.

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.

6.C. B. Collins, J. Math. Phys. 18, 1374 (1977).

7.

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.

8.C. W. Misner and H. S. Zapolsky, Phys. Rev. Lett. 12, 635 (1964).

9.

9.S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (Wiley, New York, 1972).

10.

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.

11.M. Wyman, Phys. Rev. 75, 1930 (1949).

12.

12.S. Chandrasekhar, An Introduction to the Study of Stellar Structure (Dover, New York, 1969).

13.

13.B. Schmidt, Kugelsymmetrische statische Materielösungen der Einsteinschen Feldgleichungen (Diplomarbeit, Hamburg University, 1966).

14.

14.J. B. Hartle, “Relativistic Stars, Gravitational Collapse and Black Holes” in Relativity, Astrophysics and Cosmology, edited by W. Israel (Reidel, Boston, 1973).

15.

15.G. Birkhoff and G.‐C. Rota, Ordinary Differential Equations (Blaisdell, Waltham, Mass., 1969).

16.

16.R. Mansouri, Ann. Inst. Henri Poincaré 27, 175 (1977).

17.

17.E. N. Glass, J. Math. Phys. 20, 1508 (1979).

18.

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.

19.M. Wyman, Phys. Rev. 70, 396 (1946).

20.

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.

21.A. Barnes, Gen. Relativ. Gravit. 4, 105 (1973).

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