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A Rigorous Treatment of Energy Extraction from a Rotating Black Hole

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Abstract

The Cauchy problem is considered for the scalar wave equation in the Kerr geometry. We prove that by choosing a suitable wave packet as initial data, one can extract energy from the black hole, thereby putting supperradiance, the wave analogue of the Penrose process, into a rigorous mathematical framework. We quantify the maximal energy gain. We also compute the infinitesimal change of mass and angular momentum of the black hole, in agreement with Christodoulou’s result for the Penrose process. The main mathematical tool is our previously derived integral representation of the wave propagator.

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References

  1. Andersson N., Laguna P., Papadopoulos P.: Dynamics of scalar fields in the background of rotating black holes II: a note on superradiance. Phys. Rev. D58, 087503 (1998)

    Google Scholar 

  2. Chandrasekhar S.: The Mathematical Theory of Black Holes. Oxford University Press, Oxford (1983)

    MATH  Google Scholar 

  3. Christodoulou D.: Reversible and irreversible transformations in black hole physics. Phys. Rev. Lett. 25, 1596–1597 (1970)

    Article  ADS  Google Scholar 

  4. Deruelle N., Ruffini R.: Quantum and classical relativistic energy states in stationary geometries. Phys. Lett. 52B, 437 (1974)

    ADS  Google Scholar 

  5. Deruelle N., Ruffini R.: Klein Paradox in a Kerr Geometry. Phys. Lett. 57B, 248 (1975)

    Google Scholar 

  6. Finster F., Kamran N., Smoller J., Yau S.-T.: The long-time dynamics of Dirac particles in the Kerr-Newman black hole geometry. Adv. Theor. Math. Phys. 7, 25–52 (2003)

    MathSciNet  Google Scholar 

  7. Finster F., Kamran N., Smoller J., Yau S.-T.: Decay rates and probability estimates for masssive Dirac particles in the Kerr-Newman black hole geometry. Commun. Math. Phys. 230, 201–244 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. Finster F., Kamran N., Smoller J., Yau S.-T.: An integral spectral representation of the propagator for the wave equation in the Kerr geometry. Commun. Math. Phys. 260, 257–298 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Finster, F., Kamran, N., Smoller, J., Yau, S.-T.: Decay of solutions of the wave equation in the Kerr geometry. Commun. Math. Phys. 264, 465–503 (2006), Erratum, Commun. Math. Phys. 280, 563–573 (2008)

    Google Scholar 

  10. Finster F., Smoller J.: A time independent energy estimate for outgoing scalar waves in the Kerr geometry. J. Hyperbolic Differ. Eq. 5, 221–255 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Frolov V.P., Novikov I.D.: Black Hole Physics. Basic Concepts and New Developments. Dordrecht, Kluwer Academic Publishers Group (1998)

    MATH  Google Scholar 

  12. Penrose R.: Gravitational collapse: The role of general relativity. Rev. del Nuovo Cimento 1, 252–276 (1969)

    ADS  Google Scholar 

  13. Starobinsky A.A.: Amplification of waves during reflection from a black hole. Sov. Phys. JETP 37, 28–32 (1973)

    ADS  Google Scholar 

  14. Teukolsky S., Press W.H.: Perturbations of a rotating black hole. III. Interaction of the hole with gravitational and electromagnetic radiation. Astrophys. J. 193, 443–461 (1974)

    Article  ADS  Google Scholar 

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

    MATH  Google Scholar 

  16. Zel’dovich Ya.B.: Amplification of cylindrical electromagnetic waves from a rotating body. Sov. Phys. JETP 35, 1085–1087 (1972)

    ADS  Google Scholar 

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Authors and Affiliations

  1. NWF I – Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    F. Finster

  2. Department of Math. and Statistics, McGill University, Montréal, Québec, Canada, H3A 2K6

    N. Kamran

  3. Mathematics Department, The University of Michigan, Ann Arbor, MI, 48109, USA

    J. Smoller

  4. Mathematics Department, Harvard University, Cambridge, MA, 02138, USA

    S. -T. Yau

Authors
  1. F. Finster
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  2. N. Kamran
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  3. J. Smoller
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  4. S. -T. Yau
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Corresponding author

Correspondence to F. Finster.

Additional information

Communicated by G. W. Gibbons

Research supported in part by the Deutsche Forschungsgemeinschaft.

Research supported by NSERC grant # RGPIN 105490-2004.

Research supported in part by the Humboldt Foundation and the National Science Foundation, Grant No. DMS-0603754.

Research supported in part by the NSF, Grant No. 33-585-7510-2-30.

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Finster, F., Kamran, N., Smoller, J. et al. A Rigorous Treatment of Energy Extraction from a Rotating Black Hole. Commun. Math. Phys. 287, 829–847 (2009). https://doi.org/10.1007/s00220-009-0730-7

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  • DOI: https://doi.org/10.1007/s00220-009-0730-7

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