All static spherically symmetric perfect-fluid solutions of Einstein's equations

Physical Review D

Volumn 68, Issue 10, 2003, Pages

  Lake, Kayll ^a  

^a QUEEN S UNIVERSITY   (Canada)

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EID: 0347157058     PISSN: 05562821     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevD.68.104015     Document Type: Article

References (26)

1. See M. S. R. Delgaty and K. Lake, Comput. Phys. Commun. 115, 395 (1998).
2. The conditions used in [1] were (i) isotropy of the pressure (otherwise any metric is a "solution"), 00 regularity at the origin, (iii) positivity of the pressure and energy density at the origin, (iv) vanishing of the pressure at a finite boundary, (v) monotone decrease of the energy density to the boundary, and (vi) subluminal adiabatic sound speed. In addition to these, a monotone decrease in the subluminal adiabatic sound speed is desirable.
3. We use geometrical units throughout. The "curvature coordinates" used in Eq. (3) have the advantage that the metric functions have a clear invariant physical interpretation (but see also [14] below). The function m(r) is the effective gravitational mass. See W. C. Hernandez and C. W. Misner, Astrophys. J. 143, 452 (1965);
4. E. Poisson and W. Israel, Phys. Rev. D 41, 1796 (1990);
5. T. Zannias, Phys. Rev. D ibid. 41, 3252 (1990);
6. S. Hayward, Phys. Rev. D ibid. 53, 1938 (1996).
7. -Φ(r) is the effective potential for null geodesies [see, for example, M. Ishak, L. Chamandy, N. Neary, and K. Lake, Phys. Rev. D 64, 024005 (2001)].
8. K. Lake, gr-qc/0209063.
9. One can take the view that the Tolman-Oppenheimer-Volkoff equation is a consequence of the invariant statement (2).
10. 2[r-2m(r)]-r(d/drΦ(r))[d/drm(r))r+r -3m(r)]+3m(r)-(d/dr m(r)r=0.
11. The problem has also been reduced to a linear equation of first order by A. S. Berger, R. Hojman, and J. Santamarina, J. Math. Phys. 28, 2949 (1987). Recently G. Fodor (gr-qc/0011040) has reduced the problem to ah algebraic one with integration required only for one metric function but not the physical variables ρ and p.
12. D. Pollney, N. Pelavas, P. Musgrave, and K. Lake, Comput. Phys. Commun. 115, 381 (1998).
13. 3, which gives, uniquely, the Schwarzschild interior solution. See also H. A. Buchdahl, Am. J. Phys. 39, 158 (1971).
14. See T. W. Baumgarte and A. D. Rendall, Class. Quantum Grav. 10, 327 (1993);
15. M. Mars, M. Mercè Martin-Prats, and J. M. M. Senovilla, Phys. Lett. A 218, 147 (1996).
16. See, for example, P. Musgrave and K. Lake, Class. Quantum Grav. 13, 1885 (1996).
17. At an interior boundary surface p, but not ρ, must be continuous. Discontinuities in ρ are associated with phase transitions, which we do not consider here. For a discussion of interior phase transitions see, for example, L. Lindblom, Phys. Rev. D 58, 024008 (1998).
18. 2/3 which is usually dismissed under the erroneous assumption that C=0.
19. N. Neary, J. Lattimer, and K. Lake (in preparation).
20. 2. The form of condition (2) (given above in [6]) remains unchanged, as do the functional forms and physical meanings of Φ, m, ρ, and p.
21. 2/2 where the prime indicates d/dr.
22. S. Rahman and M. Visser, Class. Quantum Grav. 19, 935 (2002).
23. 2) with β and γ positive constants immediately gives Gold III.
24. It is also of interest to note that seven of the eleven previously known solutions of this type are special cases resulting from the two generating functions considered here.
25. M. Wyman, Phys. Rev. 75, 1930 (1949).
26. This is a package which runs within MAPLE. It is entirely distinct from packages distributed with MAPLE and must be obtained independently. The GRTENSOR II software and documentation is distributed freely on the World Wide Web from the address http://grtensor.org