- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Jebsen-Birkhoff theorem in alternative gravity

Valerio Faraoni*

  • Physics Department, Bishop’s University, Sherbrooke, Québec, Canada J1M 1Z7

  • *vfaraoni@ubishops.ca
PDF

Phys. Rev. D 81, 044002 – Published 1 February, 2010

DOI: https://doi.org/10.1103/PhysRevD.81.044002

Abstract

We discuss the validity, or lack thereof, of the Jebsen-Birkhoff theorem in scalar-tensor theories by generalizing it and regarding the Brans-Dicke-like scalar as effective matter. Both the Jordan and Einstein frames are discussed and an apparent contradiction between static spherical solutions of scalar-tensor gravity and Hawking’s theorem on Brans-Dicke black holes is clarified. The results are applied to metric and Palatini f(R) gravity.

View figure in article

Article Text

References (62)

  1. A. G. Riess et al., Astron. J. 116, 1009 (1998); 118, 2668 (1999); 560, 49 (2001); 607, 665 (2004); S. Perlmutter et al., Nature (London) 391, 51 (1998); Astrophys. J. 517, 565 (1999); J. L. Tonry et al., 594, 1 (2003); R. Knop et al., 598, 102 (2003); B. Barris et al., 602, 571 (2004).
  2. E. V. Linder, Am. J. Phys. 76, 197 (2008).
  3. S. Capozziello, Int. J. Mod. Phys. D 11, 483 (2002); S. Capozziello, S. Carloni, and A. Troisi, arXiv:astro-ph/0303041.
  4. S. M. Carroll, V. Duvvuri, M. Trodden, and M. S. Turner, Phys. Rev. D 70, 043528 (2004).
  5. D. N. Vollick, Phys. Rev. D 68, 063510 (2003).
  6. T. P. Sotiriou, Classical Quantum Gravity 23, 5117 (2006); arXiv:gr-qc/0611158; arXiv:0710.4438; T. P. Sotiriou and S. Liberati, Ann. Phys. (N.Y.) 322, 935 (2007); J. Phys. Conf. Ser. 68, 012 022 (2007).
  7. T. P. Sotiriou and V. Faraoni, arXiv:0805.1726 [Rev. Mod. Phys. (to be published)].
  8. S. Nojiri and S. D. Odintsov, Int. J. Geom. Methods Mod. Phys. 4, 115 (2007); S. Capozziello and M. Francaviglia, Gen. Relativ. Gravit. 40, 357 (2008); N. Straumann, arXiv:0809.5148; H.-J. Schmidt, Int. J. Geom. Methods Mod. Phys. 4, 209 (2007); V. Faraoni, arXiv:0810.2602; T. P. Sotiriou, J. Phys. Conf. Ser. 189, 012 039 (2009); S. Capozziello, M. De Laurentis, and V. Faraoni, arXiv:0909.4672.
  9. R. M. Wald, General Relativity (Chicago University Press, Chicago, 1984).
  10. C. M. Will, Theory and Experiment in Gravitational Physics (Cambridge University Press, Cambridge, England, 1993).
  11. F. Briscese, E. Elizalde, S. Nojiri, and S. D. Odintsov, Phys. Lett. B 646, 105 (2007); A. V. Frolov, Phys. Rev. Lett. 101, 061103 (2008); A. A. Appleby and R. A. Battye, Phys. Lett. B 654, 7 (2007); S. Nojiri and S. D. Odintsov, Phys. Rev. D 78, 046006 (2008).
  12. G. D. Birkhoff, Relativity and Modern Physics (Harvard University Press, Cambridge, Massachusetts, 1923).
  13. S. Deser and J. Franklin, Am. J. Phys. 73, 261 (2005).
  14. N. V. Johansen and F. Ravndal, Gen. Relativ. Gravit. 38, 537 (2006).
  15. S. Deser, Gen. Relativ. Gravit. 37, 2251 (2005).
  16. J. T. Jebsen, Ark. Mat. Ast. Fys. (Stockholm) 15, 18 (1921); reprinted in Gen. Relativ. Gravit. 37, 2253 (2005).
  17. P. W. Higgs, Nuovo Cimento 11, 816 (1959); P. Teyssandier and P. Tourrenc, J. Math. Phys. (N.Y.) 24, 2793 (1983); B. Whitt, Phys. Lett. 145B, 176 (1984); J. D. Barrow, Nucl. Phys. B296, 697 (1988); J. D. Barrow and S. Cotsakis, Phys. Lett. B 214, 515 (1988); D. Wands, Classical Quantum Gravity 11, 269 (1994); T. Chiba, Phys. Lett. B 575, 1 (2003).
  18. C. H. Brans, Phys. Rev. 125, 2194 (1962).
  19. M. Campanelli and C. Lousto, Int. J. Mod. Phys. D 2, 451 (1993).
  20. S. W. Hawking, Commun. Math. Phys. 25, 167 (1972).
  21. L. D. Landau and E. M. Lifschitz, The Classical Theory of Fields (Pergamon Press, Oxford, 1989).

  22. Alternatively, one can require that the Lie derivatives of Tab(m) along the directions of the two Killing vectors ta (timelike) and ψc (spacelike) vanish [23].

  23. K. Schleich and D. M. Witt, arXiv:0908.4110; arXiv:0910.5194.
  24. J. Ehlers and A. Krasinski, Gen. Relativ. Gravit. 38, 1329 (2006).
  25. H. Stephani et al., Exact Solutions of the Einstein Field Equations (Cambridge University Press, Cambridge, England, 2003).
  26. V. P Frolov and I. D. Novikov, Black Hole Physics (Kluwer Academic, Dordrecht, 1998).

  27. The local nature of the Jebsen-Birkhoff theorem is emphasized in [23].

  28. A. Das, Prog. Theor. Phys. 24, 915 (1960).
  29. G. C. McVittie, Mon. Not. R. Astron. Soc. 93, 325 (1933).
  30. E. Ayon-Beato, C. Martinez, R. Troncoso, and J. Zanelli, Phys. Rev. D 71, 104037 (2005).
  31. L. M. Sokolowski, Acta Phys. Pol. B 35, 587 (2004); E. Ayon-Beato, C. Martinez, and J. Zanelli, Gen. Relativ. Gravit. 38, 145 (2006); D. C. Robinson, 38, 153 (2006); D. A. Demir and B. Pulice, Phys. Lett. B 638, 1 (2006); N. Banerjee, R. K. Jain, and D. P. Jaktar, Gen. Relativ. Gravit. 40, 93 (2008).
  32. P. Jordan, Schwerkraft und Weltall (F. Vieweg & Sohn, Braunschweig, 1952).
  33. C. H. Brans and R. H. Dicke, Phys. Rev. 124, 925 (1961).
  34. P. G. Bergmann, Int. J. Theor. Phys. 1, 25 (1968); R. V. Wagoner, Phys. Rev. D 1, 3209 (1970); K. Nordvedt, Astrophys. J. 161, 1059 (1970).
  35. E. Schücking, Z. Phys. 148, 72 (1957).
  36. J. O’Hanlon and B. O. J. Tupper, Nuovo Cimento B 7, 305 (1972).
  37. D. R. K. Reddy, J. Phys. A 6, 1867 (1973).
  38. D. R. K. Reddy, J. Phys. A 10, 185 (1977).
  39. K. D. Krori and D. Nandy, J. Phys. A 10, 993 (1977).
  40. T. Singh and L. N. Rai, Indian J. Pure Appl. Math. 10, 1432 (1979).
  41. S. B. Dutta Choudhury and D. Battacharyya, J. Phys. A 13, 3643 (1980).
  42. T. Singh, Phys. Rev. D 34, 646 (1986).
  43. T. Singh, J. Math. Phys. (N.Y.) 27, 1074 (1986).
  44. D. R. K. Reddy, M. B. Avadhanulu, and R. Venkateswarlu, Astrophys. Space Sci. 134, 201 (1987).
  45. D. R. K. Reddy, Astrophys. Space Sci. 140, 85 (1988).
  46. R. Venkateswarlu and D. R. K. Reddy, Astrophys. Space Sci. 159, 173 (1989).
  47. K. Nordtvedt, Phys. Rev. 169, 1017 (1968).

  48. The Sen-Dunn theory [49] is a scalar-tensor theory in which the combination of second derivatives ∇a∇bϕ−gab□ϕ on the right-hand side of the field equations (17) vanishes identically, making a massless ϕ satisfy the weak energy condition in the Jordan frame.

  49. D. K. Sen and K. A. Dunn, J. Math. Phys. (N.Y.) 12, 578 (1971).
  50. R. H. Dicke, Phys. Rev. 125, 2163 (1962).
  51. E. E. Flanagan, Classical Quantum Gravity 21, 417 (2004).
  52. V. Faraoni and S. Nadeau, Phys. Rev. D 75, 023501 (2007).
  53. J. D. Bekenstein, Ann. Phys. (N.Y.) 82, 535 (1974).
  54. A. Bhadra and K. Sarkar, Gen. Relativ. Gravit. 37, 2189 (2005).
  55. S. W. Hawking, Commun. Math. Phys. 25, 152 (1972).
  56. J. D. Bekenstein, Phys. Rev. D 11, 2072 (1975); L. H. Ford, 35, 2339 (1987); M. Madsen, Classical Quantum Gravity 5, 627 (1988); W. A. Hiscock, 7, L35 (1990); C. Barcelo and M. Visser, 17, 3843 (2000); L. H. Ford and T. A. Roman, Phys. Rev. D 64, 024023 (2001).
  57. E. Barausse, T. P. Sotiriou, and J. D. Miller, Classical Quantum Gravity 25, 062 001 (2008); 25, 105 008 (2008); EAS Publication Series 30, 189 (2008).
  58. V. Faraoni, Phys. Lett. B 665, 135 (2008).
  59. N. Lanahan-Tremblay and V. Faraoni, Classical Quantum Gravity 24, 5667 (2007); V. Faraoni and N. Lanahan-Tremblay, Phys. Rev. D 78, 064017 (2008).
  60. J. Sultana and C. C. Dyer, Gen. Relativ. Gravit. 37, 1347 (2005); V. Faraoni, Phys. Rev. D 80, 044013 (2009); M. L. McClure and C. C. Dyer, Classical Quantum Gravity 23, 1971 (2006); Gen. Relativ. Gravit. 38, 1347 (2006); V. Faraoni and A. Jacques, Phys. Rev. D 76, 063510 (2007); H. Saida, T. Harada, and H. Maeda, Classical Quantum Gravity 24, 4711 (2007); C. Gao, X. Chen, V. Faraoni, and Y-G. Shen, Phys. Rev. D 78, 024008 (2008); M. Nozawa and H. Maeda, Classical Quantum Gravity 25, 055 009 (2008); H. Maeda, T. Harada, and B. J. Carr, Phys. Rev. D 77, 024023 (2008); V. Faraoni, C. Gao, X. Chen, and Y.-G. Shen, Phys. Lett. B 671, 7 (2009).
  61. T. Clifton, D. F. Mota, and J. D. Barrow, Mon. Not. R. Astron. Soc. 358, 601 (2005).
  62. T. Clifton, Classical Quantum Gravity 23, 7445 (2006); V. Faraoni, 26, 195 013 (2009).

Outline

Information

Phys. Rev. D 81, 044002– Published 1 February, 2010
  • Received 18 December 2009

© 2010 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation