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On the existence of conformal Killing horizons in LRS spacetimes

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Abstract

Let M be a locally rotationally symmetric spacetime, and \(\xi ^a\) a conformal Killing vector for the metric on M, lying in the subspace spanned by the unit timelike direction and the preferred spatial direction, and with non-constant components. Under the assumption that the divergence of \(\xi ^a\) has no critical point in M, we obtain the necessary and sufficient condition for \(\xi ^a\) to generate a conformal Killing horizon. It is shown that \(\xi ^a\) generates a conformal Killing horizon if and only if either of the components (which coincide on the horizon) is constant along its orbits. That is, a conformal Killing horizon can be realized as the set of critical points of the variation of the component(s) of the conformal Killing vector along its orbits. Using this result, a simple mechanism is provided by which to determine if an arbitrary vector in an expanding LRS spacetime is a conformal Killing vector that generates a conformal Killing horizon. In specializing the case for which \(\xi ^a\) is a special conformal Killing vector, provided that the gradient of the divergence of \(\xi ^a\) is non-null, it is shown that LRS spacetimes cannot admit a special conformal Killing vector field, thereby ruling out conformal Killing horizons generated by such vector fields.

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References

  1. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973)

    Book  Google Scholar 

  2. Wald, R.: General Relativity. University of Chicago Press, Chicago (1984)

    Book  Google Scholar 

  3. Hayward, S.: General laws of black-hole dynamics. Phys. Rev. D 49, 6467 (1994)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  4. Ashtekar, A., Galloway, G.J.: Some uniqueness results for dynamical horizons. Adv. Theor. Math. Phys. 9, 1 (2005)

    Article  MathSciNet  Google Scholar 

  5. Ashtekar, A., Beetle, C., Fairhurst, S.: Isolated horizons: a generalization of black hole mechanics. Class. Quantum Grav. 16, L1-7 (1999)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  6. Ashtekar, A., Krishnan, B.: Dynamical horizons: energy, angular momentum, fluxes and balance laws. Phys. Rev. Lett. 89, 261101 (2002)

    Article  ADS  MathSciNet  PubMed  Google Scholar 

  7. Ashtekar, A., Krishnan, B.: Dynamical horizons and their properties. Phys. Rev. D 68, 104030 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  8. Booth, I.: Black hole boundaries. Can. J. Phys. 83, 1073 (2005)

    Article  ADS  CAS  Google Scholar 

  9. Booth, I., Brits, L., Gonzalez, J., Van Den Broeck, C.: Marginally trapped tubes and dynamical horizons. Class. Quantum Grav. 23, 413 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  10. Andersson, L., Mars, M., Simon, W.: Local existence of dynamical and trapping horizons. Phys. Rev. Lett. 92, 111102 (2005)

    Article  ADS  Google Scholar 

  11. Sherif, A.M., Dunsby, P.K.S.: Horizon area bound and MOTS stability in locally rotationally symmetric solutions. Class. Quantum Grav. 40, 045005 (2023)

    Article  ADS  MathSciNet  Google Scholar 

  12. Senovilla, J.: On the existence of horizons in spacetimes with vanishing curvature invariants. JHEP 2003, 046 (2003)

    Article  MathSciNet  Google Scholar 

  13. Nielsen, A.B., Shoom, A.A.: Conformal Killing horizons and their thermodynamics. Class. Quantum Grav. 35, 105008 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  14. Dyer, C.C., Honig, E.: Conformal Killing horizons. J. Math. Phys. 20, 409 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  15. Sultana, J., Dyer, C.C.: Conformal Killing horizons. J. Math. Phys. 45, 4764 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  16. Jacobson, T., Kang, G.: Conformal invariance of black hole temperature. Class. Quantum Grav. 10, L201 (1993)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  17. Pook-Kolb, D., Hennigar, R.A., Booth, I.: What happens to apparent horizons in a binary black hole merger? Phys. Rev. Lett. 127, 181101 (2021)

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  18. Booth, I., Hennigar, R.A., Pook-Kolb, D.: Ultimate fate of apparent horizons during a binary black hole merger. I. Locating and understanding axisymmetric marginally outer trapped surfaces. Phys. Rev. D 104, 84083 (2021)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  19. Pook-Kolb, D., Booth, I., Hennigar, R.A.: Ultimate fate of apparent horizons during a binary black hole merger. II. The vanishing of apparent horizons. Phys. Rev. D 104, 084084 (2021)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  20. Chatterjee, A., Ghosh, A.: Quasilocal conformal Killing horizons: Classical phase space and the first law. Phys. Rev. D 91, 064054 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  21. Chatterjee, A., Ghosh, A.: Quasilocal rotating conformal Killing horizons. Phys. Rev. D 92, 044003 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  22. De Lorenzo, T., Perez, A.: Light cone thermodynamics. Phys. Rev. D 97, 044052 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  23. Nielsen, A.B.: Revisiting Vaidya horizons. Galaxies 2, 62 (2014)

    Article  ADS  Google Scholar 

  24. Koh, S., Park, M., Sherif, A. M.: Thermodynamics with conformal Killing vector in the charged Vaidya metric. ArXiv, arXiv:2309.17398 (2023)

  25. Ellis, G.F.R.: Dynamics of pressure-free matter in general relativity. J. Math. Phys. 8, 1171 (1967)

    Article  ADS  Google Scholar 

  26. Stewart, J.M., Ellis, G.F.R.: Solutions of Einstein’s equations for a fluid which exhibit local rotational symmetry. J. Math. Phys. 9, 1072 (1968)

    Article  ADS  CAS  Google Scholar 

  27. Clarkson, C., Barrett, R.K.: Covariant perturbations of Schwarzschild black holes. Class. Quantum Grav. 20, 3855 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  28. Clarkson, C.: Covariant approach for perturbations of rotationally symmetric spacetimes. Phys. Rev. D 76, 104034 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  29. Semren, P.: Dissipative perturbations on LRS class II cosmologies using the 1+1+2 covariant split of spacetime, Masters Dissertation, Umeå University, (2020)

  30. Hansraj, C., Goswami, R.: A semi-tetrad decomposition of the Kerr spacetime. Eur. Phys. J. C 83, 321 (2023)

    Article  ADS  CAS  Google Scholar 

  31. Koh, S., Sherif, A.M., Tumurtushaa, T.: Existence of gradient CKV and gradient conformally stationary LRS spacetimes. Eur. Phys. J. C 84, 69 (2023)

    Article  ADS  Google Scholar 

  32. Sherif, A.M.: On the existence of marginally trapped tubes in spacetimes with local rotational symmetry. Eur. Phys. J. C 81, 440 (2021)

    Article  ADS  CAS  Google Scholar 

  33. Ellis, G.F.R., Goswami, R., Hamid, A.I.M., Maharaj, S.D.: Astrophysical black hole horizons in a cosmological context: Nature and possible consequences on Hawking radiation. Phys. Rev. D 90, 084013 (2014)

    Article  ADS  Google Scholar 

  34. Sherif, A.M.: Some results on cosmological and astrophysical horizons and trapped surfaces. Class. Quantum Grav. 36, 215001 (2019)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  35. Coley, A.A., Tupper, B.O.J.: Special conformal Killing vectors space-times and symmetry inheritance. J. Math. Phys. 30, 2616 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  36. Hall, G.S.: Special conformal symmetries in general relativity. Gen. relativ. Grav. 22, 203 (1990)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

The author gratefully acknowledges Miok Park of the IBS Center for Theoretical Physics of the Universe, for useful discussions during the early stages of this work. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of education (grant numbers) (NRF-2022R1I1A1A01053784) and (NRF-2021R1A2C1005748).

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  1. Department of Science Education, Jeju National University, Jeju, 63243, South Korea

    Abbas M. Sherif

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Sherif, A.M. On the existence of conformal Killing horizons in LRS spacetimes. Gen Relativ Gravit 56, 15 (2024). https://doi.org/10.1007/s10714-024-03197-w

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