Advertisement

Gravitational lensing beyond geometric optics: I. Formalism and observables

  • Abraham I. HarteEmail author
Research Article

Abstract

The laws of geometric optics and their corrections are derived for scalar, electromagnetic, and gravitational waves propagating in generic curved spacetimes. Local peeling-type results are obtained, where different components of high-frequency fields are shown to scale with different powers of their frequencies. Additionally, finite-frequency corrections are identified for a number of conservation laws and observables. Among these observables are a field’s energy and momentum densities, as well as several candidates for its corrected “propagation directions”.

Keywords

Wave propagation Gravitational lensing Gravitational waves Geometric optics 

Notes

Acknowledgements

I thank Yi-Zen Chu, Sam Dolan, and Justin Vines for valuable discussions.

References

  1. 1.
    Schneider, P., Ehlers, J., Falco, E .E.: Gravitational Lenses. Springer, Berlin (1992)Google Scholar
  2. 2.
    Wambsganss, J.: Living Rev. Relativ. 1, 12 (1998)ADSCrossRefGoogle Scholar
  3. 3.
    Perlick, V.: Living Rev. Relativ. 7, 9 (2004)ADSCrossRefGoogle Scholar
  4. 4.
    Bartelmann, M.: Class. Quantum Gravity 27, 233001 (2010)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Nye, J.: Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations. IOP Publishing, Bristol (1999)zbMATHGoogle Scholar
  6. 6.
    Born, M., Wolf, E.: Principles of Optics. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  7. 7.
    Thorne, K.S., Blandford, R.D.: Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics. Princeton University Press, Princeton (2017)zbMATHGoogle Scholar
  8. 8.
    Goodman, J.J., Romani, R.W., Blandford, R.D., Narayan, R.: Mon. Not. R. Astron. Soc. 229, 73 (1987)ADSCrossRefGoogle Scholar
  9. 9.
    Turyshev, S.G., Toth, V.T.: Phys. Rev. D 96, 024008 (2017)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Ehlers, J.: Z. Naturforsch. A 22, 1328 (1967)ADSCrossRefGoogle Scholar
  11. 11.
    Anile, A.M.: J. Math. Phys. 17, 576 (1976)ADSCrossRefGoogle Scholar
  12. 12.
    Isaacson, R.A.: Phys. Rev. 166, 1263 (1968)ADSCrossRefGoogle Scholar
  13. 13.
    Nakamura, T.T.: Phys. Rev. Lett. 80, 1138 (1998)ADSCrossRefGoogle Scholar
  14. 14.
    Nakamura, T.T., Deguchi, S.: Prog. Theor. Phys. Suppl. 133, 137 (1999)ADSCrossRefGoogle Scholar
  15. 15.
    Takahashi, R.: Astrophys. J. 644, 80 (2006)ADSCrossRefGoogle Scholar
  16. 16.
    Rahvar, S.: Mon. Not. R. Astron. Soc. 479, 406 (2018)ADSCrossRefGoogle Scholar
  17. 17.
    Wald, R .M.: General Relativity. University Of Chicago Press, Chicago (1984)CrossRefGoogle Scholar
  18. 18.
    Teitelboim, C., Villarroel, D., van Weert, C.G.: Riv. Nuovo Cim. 3, 1 (1980)CrossRefGoogle Scholar
  19. 19.
    Hogan, P .A., Ellis, G .F .R.: Ann. Phys. (N.Y.) 210, 178 (1991)ADSCrossRefGoogle Scholar
  20. 20.
    Nolan, B.C.: Proc. R. Irish Acad. A 97, 31 (1997)Google Scholar
  21. 21.
    Günther, P.: Huygens’ Principle and Hyperbolic Equations. Academic Press, New York (1988)zbMATHGoogle Scholar
  22. 22.
    Belger, M., Schimming, R., Wünsch, V.: Z. Anal. Anwend. 16, 9 (1997)CrossRefGoogle Scholar
  23. 23.
    Sommerfeld, A., Runge, J.: Ann. Phys. (Leipzig) 340, 277 (1911)ADSCrossRefGoogle Scholar
  24. 24.
    Friedlander, F .G.: The Wave Equation on a Curved Space-Time. Cambridge University Press, Cambridge (2010)Google Scholar
  25. 25.
    Keller, J.B., Lewis, R.M.: Asymptotic methods for partial differential equations: the reduced wave equation and Maxwell’s equations. In: Keller, J.B., McLaughlin, D.W., Papanicolaou, G.C. (eds.) Surveys in Applied Mathematics, p. 1. Springer, Berlin (1995)CrossRefGoogle Scholar
  26. 26.
    Kline, M., Kay, I .W.: Electromagnetic Theory and Geometrical Optics. Interscience Publishers, Geneva (1965)zbMATHGoogle Scholar
  27. 27.
    Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W. H. Freeman, San Francisco (1973)Google Scholar
  28. 28.
    Quinn, T.C.: Phys. Rev. D 62, 064029 (2000)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Burko, L.M., Harte, A.I., Poisson, E.: Phys. Rev. D 65, 124006 (2002)ADSCrossRefGoogle Scholar
  30. 30.
    Harte, A.I.: Motion in classical field theories and the foundations of the self-force problem. In: Puetzfeld, D.L., Lämmerzahl, C., Schutz, B. (eds.) Equations of Motion in Relativistic Gravity, Fundamental Theories of Physics, vol. 179, p. 327. Springer, Berlin (2015)CrossRefGoogle Scholar
  31. 31.
    Flanagan, É.É., Wald, R.M.: Phys. Rev. D 54, 6233 (1996)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Dolan, S.R.: arXiv:1801.02273
  33. 33.
    Mashhoon, B.: Phys. Lett. A 122, 299 (1987)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C .L .U., Herlt, E.: Exact Solutions of Einstein’s field equations. Cambridge University Press, Cambridge (2009)zbMATHGoogle Scholar
  35. 35.
    Frolov, V.P.: The Newman–Penrose method in the theory of general relativity. In: Basov, N.G. (ed.) Problems in the General Theory of Relativity and Theory of Group Representations, p. 73. Springer, Berlin (1979)CrossRefGoogle Scholar
  36. 36.
    Penrose, R.: Proc. R. Soc. A 284, 159 (1965)ADSGoogle Scholar
  37. 37.
    Hogan, P.A., Ellis, G.F.R.: J. Math. Phys. 30, 233 (1989)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    Frolov, V.P., Shoom, A.A.: Phys. Rev. D 84, 044026 (2011)ADSCrossRefGoogle Scholar
  39. 39.
    Yoo, C.M.: Phys. Rev. D 86, 084005 (2012)ADSCrossRefGoogle Scholar
  40. 40.
    Duval, C., Schücker, T.: Phys. Rev. D 96, 043517 (2017)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Bailyn, M., Ragusa, S.: Phys. Rev. D 15, 3543 (1977)ADSCrossRefGoogle Scholar
  42. 42.
    Bailyn, M., Ragusa, S.: Phys. Rev. D 23, 1258 (1981)ADSCrossRefGoogle Scholar
  43. 43.
    Gosselin, P., Bérard, A., Mohrbach, H.: Phys. Rev. D 75, 084035 (2007)ADSCrossRefGoogle Scholar
  44. 44.
    Duval, C., Horváth, Z., Horváthy, P.A.: Phys. Rev. D 74, 021701 (2006)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    Synge, J .L.: Relativity: The Special Theory. Interscience Publishers, Geneva (1956)zbMATHGoogle Scholar
  46. 46.
    Hall, G .S.: Symmetries and Curvature Structure in General Relativity. World Scientific, Singapore (2004)CrossRefGoogle Scholar
  47. 47.
    Chandrasekhar, S.: The Mathematical Theory of Black Holes. Oxford University Press, Oxford (1998)zbMATHGoogle Scholar
  48. 48.
    Cherubini, C., Bini, D., Bruni, M., Perjes, Z.: Class. Quantum Grav. 21, 4833 (2004)ADSCrossRefGoogle Scholar
  49. 49.
    Araneda, B., Dotti, G.: Class. Quantum Gravity 32, 195013 (2015)ADSCrossRefGoogle Scholar
  50. 50.
    Robinson, I.: J. Math. Phys. 2, 290 (1961)ADSCrossRefGoogle Scholar
  51. 51.
    Anco, S.C., Pohjanpelto, J.: Acta Appl. Math. 69, 285 (2001)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Bergqvist, G., Eriksson, I., Senovilla, J.M.M.: Class. Quantum Gravity 20, 2663 (2003)ADSCrossRefGoogle Scholar
  53. 53.
    Andersson, L., Bäckdahl, T., Blue, P.: J. Diff. Geom. 105, 163 (2017)CrossRefGoogle Scholar
  54. 54.
    Szekeres, P.: Ann. Phys. (N.Y.) 64, 599 (1971)ADSCrossRefGoogle Scholar
  55. 55.
    Ehlers, J., Prasanna, A.R., Breuer, R.A.: Class. Quantum Gravity 4, 253 (1987)ADSCrossRefGoogle Scholar
  56. 56.
    Ehlers, J., Prasanna, A.R.: Class. Quantum Gravity 13, 2231 (1996)ADSCrossRefGoogle Scholar
  57. 57.
    Isaacson, R.A.: Phys. Rev. 166, 1272 (1968)ADSCrossRefGoogle Scholar
  58. 58.
    Burnett, G.A.: J. Math. Phys. 30, 90 (1989)ADSMathSciNetCrossRefGoogle Scholar
  59. 59.
    Senovilla, J.M.M.: Class. Quantum Gravity 17, 2799 (2000)ADSMathSciNetCrossRefGoogle Scholar
  60. 60.
    Bonilla, M.Á.G., Senovilla, J.M.M.: Gen. Relativ. Gravit. 29, 91 (1997)ADSCrossRefGoogle Scholar
  61. 61.
    Clifton, T., Ellis, G.F.R., Tavakol, R.: Class. Quantum Gravity 30, 125009 (2013)ADSCrossRefGoogle Scholar
  62. 62.
    Goswami, R., Ellis, G.F.R.: Class. Quantum Gravity 35, 165007 (2018)ADSCrossRefGoogle Scholar
  63. 63.
    Bern, Z., Dennen, T., tin Huang, Y., Kiermaier, M.: Phys. Rev. D 82, 065003 (2010)ADSCrossRefGoogle Scholar
  64. 64.
    Bahjat-Abbas, N., Luna, A., White, C.D.: J. High Energy Phys. 2017, 4 (2017)CrossRefGoogle Scholar
  65. 65.
    González, M.C., Penco, R., Trodden, M.: J. High Energ. Phys. 2018, 28 (2018)CrossRefGoogle Scholar
  66. 66.
    Xanthopoulos, B.C.: J. Math. Phys. 19, 1607 (1978)ADSMathSciNetCrossRefGoogle Scholar
  67. 67.
    Harte, A.I., Vines, J.: Phys. Rev. D 94, 084009 (2016)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Centre for Astrophysics and Relativity, School of Mathematical SciencesDublin City UniversityGlasnevin, Dublin 9Ireland

Personalised recommendations