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Geometry and topology of the Kerr photon region in the phase space

  • Carla Cederbaum
  • Sophia JahnsEmail author
Research Article

Abstract

We study the set of trapped photons of a subcritical (\(a<M\)) Kerr spacetime as a subset of the phase space. First, we present an explicit proof that the photons of constant Boyer–Lindquist coordinate radius are the only photons in the Kerr exterior region that are trapped in the sense that they stay away both from the horizon and from spacelike infinity. We then proceed to identify the set of trapped photons as a subset of the (co-)tangent bundle of the subcritical Kerr spacetime. We give a new proof showing that this set is a smooth 5-dimensional submanifold of the (co-)tangent bundle with topology \(SO(3)\times {\mathbb {R}}^2\) using results about the classification of 3-manifolds and of Seifert fiber spaces. Both results are covered by the rigorous analysis of Dyatlov (Commun Math Phys 335(3):1445–1485, 2015); however, the methods we use are very different and shed new light on the results and possible applications.

Keywords

Photon regions Kerr spacetime Phase space Classification of 3-manifolds Seifert fibered spaces 

Notes

Acknowledgements

We would like to thank Pieter Blue and András Vasy for useful comments. Furthermore, we thank Oliver Schön for generating the figures for this article. This work is supported by the Institutional Strategy of the University of Tübingen (Deutsche Forschungsgemeinschaft, ZUK 63).

References

  1. 1.
    Cederbaum, C.: Uniqueness of photon spheres in static vacuum asymptotically flat spacetimes. In: Complex Analysis & Dynamical Systems IV, Volume 667 of Contemporary Mathematics, pp. 86–99. AMS (2015)Google Scholar
  2. 2.
    Cederbaum, C., Galloway, G.J.: Uniqueness of photon spheres in electro-vacuum spacetimes. Class. Quantum Gravity 33(7), 075006 (2016)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Cederbaum, C., Galloway, G.J.: Uniqueness of photon spheres via positive mass rigidity. Commun. Anal. Geom. 25(2), 303–320 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dafermos, M., Rodnianski, I.: Lectures on black holes and linear waves (2008). arXiv:0811.0354 [gr-qc]
  5. 5.
    Dyatlov, S.: Asymptotics of linear waves and resonances with applications to black holes. Commun. Math. Phys. 335(3), 1445–1485 (2015)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Geiges, H., Lange, C.: Seifert fibrations of lens spaces. arXiv:gr-qc/0512066
  7. 7.
    Grenzebach, A.: The Shadow of Black Holes. An Analytic Description. Springer, Berlin (2016)zbMATHGoogle Scholar
  8. 8.
    Grenzebach, A., Perlick, V., Lämmerzahl, C.: Photon regions and shadows of Kerr–Newman–NUT black holes with a cosmological constant. Phys. Rev. D 89(12), 124004 (2014)ADSCrossRefGoogle Scholar
  9. 9.
    Grenzebach, A., Perlick, V., Lämmerzahl, C.: Photon regions and shadows of accelerated black holes. Int. J. Mod. Phys. D 24, 1542024 (2015)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Hatcher, A.: Notes on basic 3-manifold topology. http://pi.math.cornell.edu/~hatcher/3M/3Mdownloads.html (2007). Accessed 18 June 2018
  11. 11.
    O’Neill, B.: The Geometry of Kerr Black Holes. Dover Publications, Mineola, New York (2014)zbMATHGoogle Scholar
  12. 12.
    Orlik, P.: Seifert Manifolds, Volume 291 of Lecture Notes in Mathematics. Springer, Berlin (1972)Google Scholar
  13. 13.
    Paganini, C., Ruba, B., Oancea, M.A.: Characterization of Null Geodesics on Kerr Spacetimes (2016). arXiv:1611.06927 [gr-qc]
  14. 14.
    Perlick, V.: Gravitational lensing from a spacetime perspective. Living Rev. Relativ. 7(1), 9 (2004)ADSCrossRefGoogle Scholar
  15. 15.
    Perlick, V.: On totally umbilic submanifolds of semi-Riemannian manifolds. Nonlinear Anal. 63(5–7), 511–518 (2005)CrossRefGoogle Scholar
  16. 16.
    Scott, P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15(5), 401–487 (1983)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Shoom, A.A.: Metamorphoses of a photon sphere. Phys. Rev. D 96, 084056 (2017)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Shoom, A.A., Walsh, C., Booth, I.: Geodesic motion around a distorted static black hole. Phys. Rev. D 93, 064019 (2016)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Sommers, P.: On Killing tensors and constants of motion. J. Math. Phys. 14(6), 787–790 (1973)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Tangherlini, F.R.: Schwarzschild field in \(n\) dimensions and the dimensionality of space problem. Nuovo Cim. 27, 636–651 (1963)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Teo, E.: Spherical photon orbits around a Kerr black hole. Gen. Relativ. Gravit. 35(11), 1909–1926 (2003)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Yazadjiev, S., Lazov, B.: Uniqueness of the static Einstein–Maxwell spacetimes with a photon sphere. Class. Quantum Gravity 32(16), 165021 (2015)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Yazadjiev, S.S.: Uniqueness of the static spacetimes with a photon sphere in Einstein-scalar field theory. Phys. Rev. D 91(12), 123013 (2015)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Yazadjiev, S.S., Lazov, B.: Classification of the static and asymptotically flat Einstein–Maxwell-dilaton spacetimes with a photon sphere. Phys. Rev. D 93(8), 083002 (2016)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Yoshino, H.: Uniqueness of static photon surfaces: perturbative approach. Phys. Rev. D 95, 044047 (2017)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.University of TübingenTübingenGermany

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