Kerr metric, geodesic motion, and Flyby Anomaly in fourth-order Conformal Gravity

Research Article

Abstract

In this paper we analyze the Kerr geometry in the context of Conformal Gravity, an alternative theory of gravitation, which is a direct extension of General Relativity (GR). Following previous studies in the literature, we introduce an explicit expression of the Kerr metric in Conformal Gravity, which naturally reduces to the standard GR Kerr geometry in the absence of Conformal Gravity effects. As in the standard case, we show that the Hamilton–Jacobi equation governing geodesic motion in a space-time based on this geometry is indeed separable and that a fourth constant of motion—similar to Carter’s constant—can also be introduced in Conformal Gravity. Consequently, we derive the fundamental equations of geodesic motion and show that the problem of solving these equations can be reduced to one of quadratures. In particular, we study the resulting time-like geodesics in Conformal Gravity Kerr geometry by numerically integrating the equations of motion for Earth flyby trajectories of spacecraft. We then compare our results with the existing data of the Flyby Anomaly in order to ascertain whether Conformal Gravity corrections are possibly the origin of this gravitational anomaly. Although Conformal Gravity slightly affects the trajectories of geodesic motion around a rotating spherical object, we show that these corrections are minimal and are not expected to be the origin of the Flyby Anomaly, unless conformal parameters are drastically different from current estimates. Therefore, our results confirm previous analyses, showing that modifications due to Conformal Gravity are not likely to be detected at the Solar System level, but might affect gravity at the galactic or cosmological scale.

Keywords

Conformal gravity Kerr metric Geodesics Flyby Anomaly 

References

  1. 1.
    Clowe, D., Bradac, M., Gonzalez, A.H., Markevitch, M., Randall, S.W., et al.: Astrophys. J. 648, L109 (2006). astro-ph/0608407 ADSCrossRefGoogle Scholar
  2. 2.
    Aguilar, M., et al.: (AMS Collaboration). Phys. Rev. Lett. 110, 141102 (2013)Google Scholar
  3. 3.
    Milgrom, M.: Astrophys. J. 270, 365 (1983)ADSCrossRefGoogle Scholar
  4. 4.
    Milgrom, M.: Astrophys. J. 270, 371 (1983)ADSCrossRefGoogle Scholar
  5. 5.
    Bekenstein, J.D.: Phys. Rev. D 70, 083509 (2004). astro-ph/0403694 ADSCrossRefGoogle Scholar
  6. 6.
    Moffat, J.: Phys. Lett. B 355, 447 (1995). gr-qc/9411006 ADSCrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Mannheim, P.D.: Prog. Part. Nucl. Phys. 56, 340 (2006). astro-ph/0505266 ADSCrossRefGoogle Scholar
  8. 8.
    Schmidt, H.-J.: Int. J. Geom. Meth. Phys. 4, 209 (2007). gr-qc/0602017 CrossRefMATHGoogle Scholar
  9. 9.
    Clifton, T., Ferreira, P.G., Padilla, A., Skordis, C.: Phys. Rept. 513, 1 (2012). arXiv:1106.2476 [astro-ph.CO]
  10. 10.
    Varieschi, G.U.: Gen. Relativ. Gravit. 42, 929 (2010). arXiv:0809.4729 [gr-qc]
  11. 11.
    Varieschi, G.U.: ISRN Astron. Astrophys. 2011, 806549 (2011). arXiv:0812.2472 [astro-ph]
  12. 12.
    Varieschi, G.U.: Phys. Res. Int. 2012, 469095 (2012). arXiv:1010.3262 [astro-ph.CO]
  13. 13.
    Weyl, H.: Math. Z. 2, 384 (1918)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Weyl, H.: Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1918, 465 (1918)Google Scholar
  15. 15.
    Weyl, H.: Ann. Phys. 59, 101 (1919)CrossRefMATHGoogle Scholar
  16. 16.
    Bach, R.: Math. Z. 9, 110 (1921)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Lanczos, C.: Ann. Math. 39, 842 (1938)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Mannheim, P.D., Kazanas, D.: Astrophys. J. 342, 635 (1989)ADSCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kazanas, D., Mannheim, P.D.: Astrophys. J. Suppl. 76, 431 (1991)ADSCrossRefGoogle Scholar
  20. 20.
    Mannheim, P.D., Kazanas, D.: Phys. Rev. D 44, 417 (1991)ADSCrossRefMathSciNetGoogle Scholar
  21. 21.
    Mannheim, P.D.: Astrophys. J. 419, 150 (1993). hep-ph/9212304ADSCrossRefGoogle Scholar
  22. 22.
    Mannheim, P.D.: Astrophys. J. 479, 659 (1997). astro-ph/9605085 ADSCrossRefGoogle Scholar
  23. 23.
    Mannheim, P.D., O’Brien, J.G.: Phys. Rev. Lett. 106, 121101 (2011). arXiv:1007.0970 [astro-ph.CO]
  24. 24.
    Mannheim, P.D., O’Brien, J.G.: Phys. Rev. D 85, 124020 (2012). arXiv:1011.3495 [astro-ph.CO]
  25. 25.
    O’Brien, J.G., Mannheim, P.D.: Mon. Not. R. Astron. Soc. 421, 1273 (2012). arXiv:1107.5229 [astro-ph.CO]
  26. 26.
    Mannheim, P.D., O’Brien, J.G.: J. Phys. Conf. Ser. 437, 012002 (2013). arXiv:1211.0188 [astro-ph.CO]
  27. 27.
    Mannheim, P.D.: Phys. Rev. D 75, 124006 (2007). gr-qc/0703037
  28. 28.
    Mannheim, P.D.: Private communication (2011)Google Scholar
  29. 29.
    Mannheim, P.D.: Private communication (2014)Google Scholar
  30. 30.
    Mannheim, P.D., Kazanas, D.: Gen. Relativ. Gravit. 26, 337 (1994)ADSCrossRefGoogle Scholar
  31. 31.
    Carter, B.: In: DeWitt, C., DeWitt, B.S. (eds.) Black holes. Lectures delivered at the Summer School of Theoretical Physics of the University of Grenoble at Les Houches. New York, NY (USA) Gordon and Breach, 12 + 552 + 176 p. (1973)Google Scholar
  32. 32.
    Mannheim, P.D.: Private communication (2010)Google Scholar
  33. 33.
    Varieschi, G.U., Burstein, Z.: ISRN Astron. Astrophys. 2013, 482734 (2013). arXiv:1208.3706 [gr-qc]
  34. 34.
    Kerr, R.P.: Phys. Rev. Lett. 11, 237 (1963)ADSCrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Carter, B.: Phys. Rev. 174, 1559 (1968)ADSCrossRefMATHGoogle Scholar
  36. 36.
    Carter, B.: Commun. Math. Phys. 10, 280 (1968)MATHGoogle Scholar
  37. 37.
    Chandrasekhar, S.: The Mathematical Theory of Black Holes. Oxford University Press, New York (1992)Google Scholar
  38. 38.
    Hawking, S.W., Hunter, C.J., Taylor-Robinson, M.M.: Phys. Rev. D 59, 064005 (1999). hep-th/9811056 ADSCrossRefMathSciNetGoogle Scholar
  39. 39.
    Cardoso, V., Dias, O.J.: Phys. Rev. D 70, 084011 (2004). hep-th/0405006 ADSCrossRefMathSciNetGoogle Scholar
  40. 40.
    Berti, E., Cardoso, V., Starinets, A.O.: Class. Quantum Grav. 26, 163001 (2009). arXiv:0905.2975 [gr-qc]
  41. 41.
    Said, J.L., Sultana, J., Adami, K.Z.: Phys. Rev. D 88, 087504 (2013). arXiv:1401.2898 [gr-qc]ADSCrossRefGoogle Scholar
  42. 42.
    Kraniotis, G.: Class. Quantum Grav. 21, 4743 (2004). gr-qc/0405095
  43. 43.
    Hackmann, E., Lammerzahl, C., Kagramanova, V., Kunz, J.: Phys. Rev. D 81, 044020 (2010). arXiv:1009.6117 [gr-qc]
  44. 44.
    Poudel, P., Khanal, U.: (2013). arXiv:1309.1685 [gr-qc]
  45. 45.
    Antreasian, P.G., Guinn, J.R.: Am. Inst. Aeronaut. Astronaut. Paper No. 98–4287, 1 (1998). http://www.issibern.ch/teams/Pioneer/pa-literature.htm
  46. 46.
    Morley, T., Budnik, F.: Proc. Int. Symp. Space Technol. Sci. 25, 593 (2006)Google Scholar
  47. 47.
    Anderson, J.D., Campbell, J.K., Nieto, M.M.: New Astron. 12, 383 (2007). astro-ph/0608087
  48. 48.
    Lämmerzahl, C., Preuss, O., Dittus, H.: In: Dittus, H., Lammerzahl, C., Turyshev, S.G. (eds.) Lasers, Clocks and Drag-Free Control: Exploration of Relativistic Gravity in Space. (2008), Astrophysics and Space Science Library, vol. 349, p. 75, gr-qc/0604052
  49. 49.
    Anderson, J.D., Nieto, M.M.: In: Klioner, S.A., Seidelmann, P.K., Soffel M.H. (eds.) IAU Symposium, vol. 261, pp. 189–197 (2010). arXiv:0907.2469 [gr-qc]
  50. 50.
    Nieto, M.M., Anderson, J.D.: Phys. Today 62N10, 76 (2009). arXiv:0910.1321 [gr-qc]
  51. 51.
    Turyshev, S.G., Toth, V.T.: Space Sci. Rev. 148, 169 (2010). arXiv:0907.4184 [gr-qc]
  52. 52.
    Anderson, J.D., Campbell, J.K., Ekelund, J.E., Ellis, J., Jordan, J.F.: Phys. Rev. Lett. 100, 091102 (2008)ADSCrossRefGoogle Scholar
  53. 53.
    Iorio, L.: (2013). arXiv:1311.4218 [gr-qc]
  54. 54.
    Shibata, M., Sasaki, M.: Phys. Rev. D 58, 104011 (1998). gr-qc/9807046
  55. 55.
    Berti, E., White, F., Maniopoulou, A., Bruni, M.: Mon. Not. R. Astron. Soc. 358, 923 (2005). gr-qc/0405146
  56. 56.
    Benhar, O., Ferrari, V., Gualtieri, L., Marassi, S.: Phys. Rev. D72, 044028 (2005). gr-qc/0504068
  57. 57.
    Sultana, J., Kazanas, D., Said, J.L.: Phys. Rev. D 86, 084008 (2012)ADSCrossRefGoogle Scholar
  58. 58.
    Said, J.L., Sultana, J., Adami, K.Z.: Phys. Rev. D 85, 104054 (2012). arXiv:1201.0860 [gr-qc]
  59. 59.
    Said, J.L., Sultana, J., Adami, K.Z.: Phys. Rev. D 86, 104009 (2012). arXiv:1207.2108 [gr-qc]

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of PhysicsLoyola Marymount UniversityLos AngelesUSA

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