On stationary black holes of the Einstein conformally invariant scalar system

Journal of Mathematical Physics

Volumn 39, Issue 12, 1998, Pages 6651-6667

  Zannias, T ^a  

^a UNIVERSIDAD MICHOACANA DE SAN NICOLÁS DE HIDALGO   (Mexico)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032344451     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532647     Document Type: Article

References (62)

1. T. Zannias, J. Math. Phys. 9, 2643 (1995).
2. There has been a significant amount of work aiming to explore consequences of the coupling of the scalar field to background scalar curvature at lower than four, four, and higher than four dimensions. See, for instance, C. Martinez et al., Phys. Rev. D 54, 3830 (1996); C. Klimcic, J. Math. Phys. 34, 1914 (1993); B. C. Xanthopoulos and A. Dialynas ibid. 33, 1463 (1992); J. D. Bekenstein, Ann. Phys. (N.Y.) 82, 535 (1974); 91, 72 (1975); B. C. Xanthopoulos and T. Zannias, J. Math. Phys. 32, 1875 (1991); 33, 1462 (1992); K. Bronnikov and K. Kireyev, Phys. Lett. A 67, 95 (1978).
3. There has been a significant amount of work aiming to explore consequences of the coupling of the scalar field to background scalar curvature at lower than four, four, and higher than four dimensions. See, for instance, C. Martinez et al., Phys. Rev. D 54, 3830 (1996); C. Klimcic, J. Math. Phys. 34, 1914 (1993); B. C. Xanthopoulos and A. Dialynas ibid. 33, 1463 (1992); J. D. Bekenstein, Ann. Phys. (N.Y.) 82, 535 (1974); 91, 72 (1975); B. C. Xanthopoulos and T. Zannias, J. Math. Phys. 32, 1875 (1991); 33, 1462 (1992); K. Bronnikov and K. Kireyev, Phys. Lett. A 67, 95 (1978).
4. There has been a significant amount of work aiming to explore consequences of the coupling of the scalar field to background scalar curvature at lower than four, four, and higher than four dimensions. See, for instance, C. Martinez et al., Phys. Rev. D 54, 3830 (1996); C. Klimcic, J. Math. Phys. 34, 1914 (1993); B. C. Xanthopoulos and A. Dialynas ibid. 33, 1463 (1992); J. D. Bekenstein, Ann. Phys. (N.Y.) 82, 535 (1974); 91, 72 (1975); B. C. Xanthopoulos and T. Zannias, J. Math. Phys. 32, 1875 (1991); 33, 1462 (1992); K. Bronnikov and K. Kireyev, Phys. Lett. A 67, 95 (1978).
5. There has been a significant amount of work aiming to explore consequences of the coupling of the scalar field to background scalar curvature at lower than four, four, and higher than four dimensions. See, for instance, C. Martinez et al., Phys. Rev. D 54, 3830 (1996); C. Klimcic, J. Math. Phys. 34, 1914 (1993); B. C. Xanthopoulos and A. Dialynas ibid. 33, 1463 (1992); J. D. Bekenstein, Ann. Phys. (N.Y.) 82, 535 (1974); 91, 72 (1975); B. C. Xanthopoulos and T. Zannias, J. Math. Phys. 32, 1875 (1991); 33, 1462 (1992); K. Bronnikov and K. Kireyev, Phys. Lett. A 67, 95 (1978).
6. There has been a significant amount of work aiming to explore consequences of the coupling of the scalar field to background scalar curvature at lower than four, four, and higher than four dimensions. See, for instance, C. Martinez et al., Phys. Rev. D 54, 3830 (1996); C. Klimcic, J. Math. Phys. 34, 1914 (1993); B. C. Xanthopoulos and A. Dialynas ibid. 33, 1463 (1992); J. D. Bekenstein, Ann. Phys. (N.Y.) 82, 535 (1974); 91, 72 (1975); B. C. Xanthopoulos and T. Zannias, J. Math. Phys. 32, 1875 (1991); 33, 1462 (1992); K. Bronnikov and K. Kireyev, Phys. Lett. A 67, 95 (1978).
7. There has been a significant amount of work aiming to explore consequences of the coupling of the scalar field to background scalar curvature at lower than four, four, and higher than four dimensions. See, for instance, C. Martinez et al., Phys. Rev. D 54, 3830 (1996); C. Klimcic, J. Math. Phys. 34, 1914 (1993); B. C. Xanthopoulos and A. Dialynas ibid. 33, 1463 (1992); J. D. Bekenstein, Ann. Phys. (N.Y.) 82, 535 (1974); 91, 72 (1975); B. C. Xanthopoulos and T. Zannias, J. Math. Phys. 32, 1875 (1991); 33, 1462 (1992); K. Bronnikov and K. Kireyev, Phys. Lett. A 67, 95 (1978).
8. There has been a significant amount of work aiming to explore consequences of the coupling of the scalar field to background scalar curvature at lower than four, four, and higher than four dimensions. See, for instance, C. Martinez et al., Phys. Rev. D 54, 3830 (1996); C. Klimcic, J. Math. Phys. 34, 1914 (1993); B. C. Xanthopoulos and A. Dialynas ibid. 33, 1463 (1992); J. D. Bekenstein, Ann. Phys. (N.Y.) 82, 535 (1974); 91, 72 (1975); B. C. Xanthopoulos and T. Zannias, J. Math. Phys. 32, 1875 (1991); 33, 1462 (1992); K. Bronnikov and K. Kireyev, Phys. Lett. A 67, 95 (1978).
9. There has been a significant amount of work aiming to explore consequences of the coupling of the scalar field to background scalar curvature at lower than four, four, and higher than four dimensions. See, for instance, C. Martinez et al., Phys. Rev. D 54, 3830 (1996); C. Klimcic, J. Math. Phys. 34, 1914 (1993); B. C. Xanthopoulos and A. Dialynas ibid. 33, 1463 (1992); J. D. Bekenstein, Ann. Phys. (N.Y.) 82, 535 (1974); 91, 72 (1975); B. C. Xanthopoulos and T. Zannias, J. Math. Phys. 32, 1875 (1991); 33, 1462 (1992); K. Bronnikov and K. Kireyev, Phys. Lett. A 67, 95 (1978).
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15. P. T. Chruściel In Journées Relativistas, Ascona (1996) and gr-qc 9610010.
16. P. T. Chruściel, gr-qc 9610011.
17. Condition (c), i.e., the spherical like cross sections of the horizon, actually follows from other considerations of the CT. In particularly the CT assumes that the domain of outer communication is simply connected. Such property is guaranteed as long as the Ricci tensor satisfies the null convergence condition, and additionally the null infinity satisfies some additional conditions referred as regularity conditions (or some equivalent version as those for example spell out in the Ref. 10 below.) As a byproduct of the satisfaction of the null convergence condition and regularity of the null infinity it then follows that the event horizon necessarily possess spherical like cross sections. For a proof see P. T. Chruściel and R. M. Wald in Ref. 6 above as well as the following Ref. 10.
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19. M. Heusler, gr-qc 96077001.
20. R. Beig and P. T. Chruściel, gr-qc 9510015, in particular notice the assumptions underlying Theorem (4.1). Prerequisite for the timelike character of the ADM four momenta is the existence of appropriate spacelike slices where matter and geometry exhibit special fall of rates.
21. M. zum Hagen, Proc. Cambridge Philos. Soc. 67, 415 (1970); 68, 187 (1970).
22. M. zum Hagen, Proc. Cambridge Philos. Soc. 67, 415 (1970); 68, 187 (1970).
23. J. Estevez Delgado and T. Zannias (in preparation).
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25. B. Carter, in Gravitation in Astrophysics, edited by B. Carter and J. B. Hartle (Kluwer, Gargese, 1986).
26. The construction of 〈〈M〉〉 used in Carter's work is to be contrasted with the corresponding definitions in Refs. 7 and 8. Chrusciel makes heavy use of the concept of completeness of trajectories of the asymptotically timelike Killing vector field.
27. There is no loss of generality in imposing the commuting property of the two actions. The justification for this can be found in B. Carter, Commun. Math. Phys. 17, 233 (1970), In fact a slightly more general conclusion can be drawn whenever axisymmetry action is replaced by the closely related but not identical concept of cyclic symmetry. See, for instance, discussion in the following Ref. 19.
28. L. B. Szabados, J. Math. Phys. 28, 2688 (1987).
29. B. Carter, J. Math. Phys. 10, 70 (1969).
30. mnV(Φ) to the right-hand side of (1a).
31. R. M. Wald, General Relativity (Chicago University Press, Chicago, 1984).
32. In the case where the gradient of Φ is also identically vanishing one can still apply the method of the Appendix A to compute the Ricci tensor. In such event Eq. (A4) of Appendix A is differentiated twice, and one follows the same steps leading to (A7). Note in that event, the resulting Ricci contains fourth-order derivatives of the field Φ.
33. In the case of vanishing gradients, even though the resulting form of Eq. (9) is more complex it nevertheless maintains similar structure. An inhomogeneous term is also appearing.
34. In this regard it is of interest to repeat the calculations for arbitrary coupling of the scalar field Φ to scalar curvature. In this case one expects, if ξ is chosen to be negative circularity ought to hold although this has to be verified by detailed computations.
35. A more extensive discussion about this transformation can be found in J. D. Bekenstein in Ref. 2; B. C. Xanthopoulos and A. Dialynas in Ref. 2; T. Zannias in Ref. 1. We may parenthetically add here that because of this property of the Einstein conformal system, asymptotically flat nonsingular initial should exist, possessing positive ADM mass and timelike four momenta.
36. We could have assumed that those hypersurfaces merely intersect null infinity. Their asymptotic identification to some retarded time coordinates simply makes the evaluation of integral of Eq. (17) cited below an easier task.
37. Although we are working in the physical space-time occasionally we shall use terminology associated with the conformal completion of the physical space-time.
38. W. Israel, Phys. Rev. 164, 1776 (1967).
39. W. Israel, Commun. Math. Phys. 8, 245 (1968).
40. I. Racz and R. M. Wald, Class. Quantum Grav. 9, 2643 (1992).
41. +. In turn for nonvacuum spaces that requires, besides the peeling property of the Weyl tensor, certain asymptotic fall of rates for the Ricci and scalar curvature. For a relevant discussion consult Refs. 33 and 34 below.
42. See, for instance, E. Newman and R. Penrose, J. Math. Phys. 5, 566 (1962).
43. For a discussion concerning the asymptotic decay of the curvature for a vacuo, asymptotically flat at null infinity space see Ref. 33, while for electrovac, or general arbitrary nonvacuo space-times consult the following Ref., respectively; A. R. Exton, E. Newman, and R. Penrose, J. Math. Phys. 10, 1566 (1969); G. Ludwig, Gen. Relativ. Gravit. 7, 293 (1976).
44. For a discussion concerning the asymptotic decay of the curvature for a vacuo, asymptotically flat at null infinity space see Ref. 33, while for electrovac, or general arbitrary nonvacuo space-times consult the following Ref., respectively; A. R. Exton, E. Newman, and R. Penrose, J. Math. Phys. 10, 1566 (1969); G. Ludwig, Gen. Relativ. Gravit. 7, 293 (1976).
45. i) = 0.
46. For an overview, as well as an update on the various black hole uniqueness theorems, see, for instance, the recent monograph: M. Heusler, Black Hole Uniqueness Theorems (Cambridge Univ. Press, Cambridge, 1996).
47. See also T. Zannias, Talk delivered in, The Second Mexican School on Gravitation and Mathematical Physics, edited by D. Nunez et al. (1997).
48. D. Sudarsky and R. M. Wald, Phys. Rev. D 46, 1453 (1992); 47, R5209 (1993).
49. D. Sudarsky and R. M. Wald, Phys. Rev. D 46, 1453 (1992); 47, R5209 (1993).
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51. J. D. Bekenstein, Phys. Rev. D 5, 2941 (1972).
52. J. E. Chase, Commun. Math. Phys. 19, 276 (1970).
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61. This object has been examined at length in, J. L. Synge, Relativity. The General Theory (North Holland, Amsterdam, 1960); Further applications of it, as well as the idea of covariant type Taylor series expansion can be found in: B. S. Dewitt, in Relativity Groups and Topology (Les Houche 1963 Gordon-Breach).
62. This object has been examined at length in, J. L. Synge, Relativity. The General Theory (North Holland, Amsterdam, 1960); Further applications of it, as well as the idea of covariant type Taylor series expansion can be found in: B. S. Dewitt, in Relativity Groups and Topology (Les Houche 1963 Gordon-Breach).