Bi-conformal symmetry and static Green functions in the Schwarzschild-Tangherlini spacetimes

Open Access
Regular Article - Theoretical Physics

Abstract

We study a static massless minimally coupled scalar field created by a source in a static D-dimensional spacetime. We demonstrate that the corresponding equation for this field is invariant under a special transformation of the background metric. This transformation consists of the static conformal transformation of the spatial part of the metric accompanied by a properly chosen transformation of the red-shift factor. Both transformations are determined by one function Ω of the spatial coordinates. We show that in a case of higher dimensional spherically symmetric black holes one can find such a bi-conformal transformation that the symmetry of the D-dimensional metric is enhanced after its application. Namely, the metric becomes a direct sum of the metric on a unit sphere and the metric of 2D anti-de Sitter space. The method of the heat kernels is used to find the Green function in this new space, which allows one, after dimensional reduction, to obtain a static Green function in the original space of the static black hole. The general useful representation of static Green functions is obtained in the Schwarzschild-Tangherlini spacetimes of arbitrary dimension. The exact explicit expressions for the static Green functions are obtained in such metrics for D < 6. It is shown that in the four dimensional case the corresponding Green function coincides with the Copson solution.

Keywords

Conformal and W Symmetry Field Theories in Higher Dimensions Black Holes 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    W.G. Unruh, Selfforce on charged particles, Proc. Roy. Soc. Lond. A 348 (1976) 447 [INSPIRE].ADSCrossRefMATHGoogle Scholar
  2. [2]
    A.G. Smith and C.M. Will, Force on a static charge outside a Schwarzschild black hole, Phys. Rev. D 22 (1980) 1276 [INSPIRE].ADSMathSciNetGoogle Scholar
  3. [3]
    A. Zelnikov and V. P. Frolov, The influence of gravitation, acceleration, and temperature on the self-energy of charged particles (in Russian), Proc. Lebedev Phys. Inst. 152 (1983) 96.Google Scholar
  4. [4]
    A. Zelnikov and V. Frolov, Influence of gravitation on the self-energy of charged particles, Sov. Phys. JETP 55 (1982) 191.Google Scholar
  5. [5]
    E.T. Copson, On electrostatics in a gravitational field, Proc. Roy. Soc. London A 118 (1928) 184.ADSCrossRefMATHGoogle Scholar
  6. [6]
    M.J.S. Beach, E. Poisson and B.G. Nickel, Self-force on a charge outside a five-dimensional black hole, Phys. Rev. D 89 (2014) 124014 [arXiv:1404.1031] [INSPIRE].ADSGoogle Scholar
  7. [7]
    V. Frolov and A. Zelnikov, Charged particles in higher dimensional homogeneous gravitational field: self-energy and self-force, JHEP 1410 (2014) 68 [arXiv:1407.3323] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    B. Linet, Electrostatics and magnetostatics in the Schwarzschild metric, J. Phys. A 9 (1976) 1081 [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    B. Linet, Scalar or electric charge at rest in a black hole space-time, Compt. Rend. Math. 284 (1977) 215.Google Scholar
  10. [10]
    A.C. Ottewill and P. Taylor, Static Kerr Green’s function in closed form and an analytic derivation of the self-force for a static scalar charge in Kerr space-time, Phys. Rev. D 86 (2012) 024036 [arXiv:1205.5587] [INSPIRE].ADSGoogle Scholar
  11. [11]
    V.P. Frolov and A. Zelnikov, Scalar and electromagnetic fields of static sources in higher dimensional Majumdar-Papapetrou spacetimes, Phys. Rev. D 85 (2012) 064032 [arXiv:1202.0250] [INSPIRE].ADSGoogle Scholar
  12. [12]
    A. Garcia-Parrado and J.M.M. Senovilla, Bi-conformal vector fields and their applications, Class. Quant. Grav. 21 (2004) 2153 [math-ph/0311014] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    A.G.P. Gomez-Lobo, Bi-conformal vector fields and the local geometric characterization of conformally separable pseudo-riemannian manifolds I, J. Geom. Phys. 56 (2006) 1069.ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    V.P. Frolov and A. Zelnikov, Self-energy of a scalar charge near higher-dimensional black holes, Phys. Rev. D 85 (2012) 124042 [arXiv:1204.3122] [INSPIRE].ADSGoogle Scholar
  15. [15]
    V.P. Frolov and A. Zelnikov, Classical self-energy and anomaly, Phys. Rev. D 86 (2012) 044022 [arXiv:1205.4269] [INSPIRE].ADSGoogle Scholar
  16. [16]
    V.P. Frolov and A. Zelnikov, Anomaly and the self-energy of electric charges, Phys. Rev. D 86 (2012) 104021 [arXiv:1208.5763] [INSPIRE].ADSGoogle Scholar
  17. [17]
    V.P. Frolov, A.A. Shoom and A. Zelnikov, Self-energy anomaly of an electric pointlike dipole in three-dimensional static spacetimes, Phys. Rev. D 88 (2013) 024032 [arXiv:1303.1816] [INSPIRE].ADSGoogle Scholar
  18. [18]
    M. Demianski and I. Novikov, Electric charge in the kruskal space-time and the jeans conjecture, Gen. Rel. Grav. 14 (1982) 1115.ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    V. Frolov and I. Novikov, Black hole physics: basic concepts and new developments, Fundamental Theories of Physics vol. 96, Kluwer Academic Publishers, Dordrecht Netherlands (1998).Google Scholar
  20. [20]
    E. Poisson, A. Pound and I. Vega, The motion of point particles in curved spacetime, Living Rev. Rel. 14 (2011) 7 [arXiv:1102.0529] [INSPIRE].CrossRefMATHGoogle Scholar
  21. [21]
    R. Camporesi, Harmonic analysis and propagators on homogeneous spaces, Phys. Rept. 196 (1990) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    L.A. Kofman and V. Sahni, A new selfconsistent solution of the Einstein equations with one loop quantum gravitational corrections, Phys. Lett. B 127 (1983) 197 [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    A.C. Ottewill and P. Taylor, Quantum field theory on the Bertotti-Robinson space-time, Phys. Rev. D 86 (2012) 104067 [arXiv:1209.6080] [INSPIRE].ADSGoogle Scholar
  24. [24]
    V.P. Frolov and A.I. Zel’nikov, The massless scalar field around a static black hole, J. Phys. A 13 (1980) L345.MathSciNetGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Theoretical Physics Institute, Department of PhysicsUniversity of AlbertaEdmontonCanada

Personalised recommendations