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Black hole perturbation under a 2+2 decomposition in the action

Justin L. Ripley* and Kent Yagi†

  • Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

  • *jripley@princeton.edu
  • †kyagi@princeton.edu
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Phys. Rev. D 97, 024009 – Published 9 January, 2018

DOI: https://doi.org/10.1103/PhysRevD.97.024009

Abstract

Black hole perturbation theory is useful for studying the stability of black holes and calculating ringdown gravitational waves after the collision of two black holes. Most previous calculations were carried out at the level of the field equations instead of the action. In this work, we compute the Einstein-Hilbert action to quadratic order in linear metric perturbations about a spherically symmetric vacuum background in Regge-Wheeler gauge. Using a 2+2 splitting of spacetime, we expand the metric perturbations into a sum over scalar, vector, and tensor spherical harmonics, and dimensionally reduce the action to two dimensions by integrating over the two sphere. We find that the axial perturbation degree of freedom is described by a two-dimensional massive vector action, and that the polar perturbation degree of freedom is described by a two-dimensional dilaton massive gravity action. Varying the dimensionally reduced actions, we rederive covariant and gauge-invariant master equations for the axial and polar degrees of freedom. Thus, the two-dimensional massive vector and massive gravity actions we derive by dimensionally reducing the perturbed Einstein-Hilbert action describe the dynamics of a well-studied physical system: the metric perturbations of a static black hole. The 2+2 formalism we present can be generalized to m+n-dimensional spacetime splittings, which may be useful in more generic situations, such as expanding metric perturbations in higher dimensional gravity. We provide a self-contained presentation of m+n formalism for vacuum spacetime splittings.

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References (46)

  1. T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957).
  2. C. V. Vishveshwara, Phys. Rev. D 1, 2870 (1970).
  3. F. J. Zerilli, Phys. Rev. Lett. 24, 737 (1970).
  4. V. Moncrief, Ann. Phys. (N.Y.) 88, 323 (1974).
  5. U. H. Gerlach and U. K. Sengupta, Phys. Rev. D 19, 2268 (1979).
  6. U. H. Gerlach and U. K. Sengupta, Phys. Rev. D 22, 1300 (1980).
  7. S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford Classic Texts in the Physical Sciences (Oxford University Press, Oxford, 2002).
  8. K. Martel and E. Poisson, Phys. Rev. D 71, 104003 (2005).
  9. T. Kobayashi, H. Motohashi, and T. Suyama, Phys. Rev. D 85, 084025 (2012).
  10. T. Kobayashi, H. Motohashi, and T. Suyama, Phys. Rev. D 89, 084042 (2014).
  11. A. De Felice, T. Suyama, and T. Tanaka, Phys. Rev. D 83, 104035 (2011).
  12. H. Motohashi and T. Suyama, Phys. Rev. D 84, 084041 (2011).
  13. H. Motohashi and T. Suyama, Phys. Rev. D 85, 044054 (2012).
  14. H. Ogawa, T. Kobayashi, and T. Suyama, Phys. Rev. D 93, 064078 (2016).
  15. K. Takahashi, T. Suyama, and T. Kobayashi, Phys. Rev. D 93, 064068 (2016).
  16. K. Takahashi and T. Suyama, Phys. Rev. D 95, 024034 (2017).
  17. R. Kase, L. A. Gergely, and S. Tsujikawa, Phys. Rev. D 90, 124019 (2014).
  18. W. G. Unruh, Phys. Rev. D 14, 870 (1976).
  19. J. A. Harvey and A. Strominger, in String Theory and Quantum Gravity ’92, Proceedings of the Trieste Spring School and Workshop, Trieste, Italy, 1992, edited by J. Harvey, R. Iengo, K. S. Narain, S. Randjbar-Daemi, and H. Verlinde (World Scientific, Singapore, 1993), p. 122.
  20. E. C. Vagenas, Mod. Phys. Lett. A 17, 609 (2002).
  21. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W.H. Freeman, San Francisco 1973).
  22. P. R. Brady, S. Droz, W. Israel, and S. M. Morsink, Classical Quantum Gravity 13, 2211 (1996).
  23. J. M. Bardeen, Phys. Rev. D 22, 1882 (1980).
  24. H. Motohashi, T. Suyama, and K. Takahashi, Phys. Rev. D 94, 124021 (2016).
  25. V. Frolov and I. Novikov, Black Hole Physics: Basic Concepts and New Developments (Springer Science and Business Media, New York, 2012), Vol. 96.
  26. O. Sarbach and M. Tiglio, Phys. Rev. D 64, 084016 (2001).
  27. O. Sarbach, Ph.D. thesis, Universität Zürich, 2000.
  28. K. Martel, Ph.D. thesis, The University of Guelph, 2003.
  29. M. Fierz and W. Pauli, Proc. R. Soc. A 173, 211 (1939).
  30. C. de Rham, G. Gabadadze, and A. J. Tolley, Phys. Rev. Lett. 106, 231101 (2011).
  31. K. Hinterbichler, Rev. Mod. Phys. 84, 671 (2012).
  32. C. de Rham, Living Rev. Relativity 17, 7 (2014).
  33. R. H. Price, Phys. Rev. D 5, 2419 (1972).
  34. R. M. Wald, J. Math. Phys. (N.Y.) 20, 1056 (1979).
  35. M. Dafermos, G. Holzegel, and I. Rodnianski, arXiv:1601.06467.
  36. E. Newman and R. Penrose, J. Math. Phys. (N.Y.) 3, 566 (1962).
  37. S. Ray, Ph.D. thesis, University of Massachusetts-Amherst, 2008.
  38. B. Carter, J. Geom. Phys. 8, 53 (1992).
  39. E. Gourgoulhon, Phys. Rev. D 72, 104007 (2005).
  40. T. Damour, Second Marcel Grossmann meeting, in Proceedings of the Conference, Trieste, Italy, 1979, edited by R. Ruffini (North-Holland Publication Co., Amsterdam, 1982), p. 587.
  41. T. Damour, Phys. Rev. D 18, 3598 (1978).
  42. T. Damour, Ph.D. thesis, Université Pierre et Marie Curie, Paris 6, 1979.
  43. T. Padmanabhan, Phys. Rev. D 83, 044048 (2011).
  44. E. Poisson, A Relativist’s Toolkit: The Mathematics of Black Hole Mechanics (Cambridge University Press, Cambridge, 2004).
  45. K. Hinterbichler, M. Trodden, and D. Wesley, Phys. Rev. D 82, 124018 (2010).
  46. J. D. E. Grant and I. G. Moss, Phys. Rev. D 56, 6284 (1997).

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Phys. Rev. D 97, 024009– Published 9 January, 2018
  • Received 19 May 2017
Article part of CHORUS
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