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Conditions for non-existence of static or stationary, Einstein–Maxwell, non-inheriting black-holes


We consider asymptotically-flat, static and stationary solutions of the Einstein equations representing Einstein–Maxwell space–times in which the Maxwell field is not constant along the Killing vector defining stationarity, so that the symmetry of the space-time is not inherited by the electromagnetic field. We find that static degenerate black hole solutions are not possible and, subject to stronger assumptions, nor are static, non-degenerate or stationary black holes. We describe the possibilities if the stronger assumptions are relaxed.

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  1. 1.

    MacCallum, M.A.H.,Van den Bergh, N.: Noninheritance of static symmetry by Maxwell fields. In: MacCallum, M.A.H., (ed.) Galaxies, Axisymmetric Systems and Relativity. Cambridge University Press, Cambridge (1985)

  2. 2.

    Stephani H., Kramer D., MacCallum M., Hoenselaers C. and Herlt E. (2003). Exact Solutions of Einstein’s Field Equations, 2nd ed. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge

    Google Scholar 

  3. 3.

    Chruściel P.T. and Wald R.M. (1994). On the topology of stationary black holes. Class. Quant. Grav. 11: L147–L152

    Article  Google Scholar 

  4. 4.

    Michalski H. and Wainwright J. (1975). Killing vector fields and the Einstein–Maxwell field equations in general relativity. Gen. Rel. Grav. 6: 289–318

    MATH  Article  ADS  MathSciNet  Google Scholar 

  5. 5.

    Heusler, M. Black Hole Uniqueness Theorems. In: Cambridge Lecture Notes in Physics, vol.~6. Cambridge University Press, Cambridge(1996)

  6. 6.

    Misner C.W. and Wheeler J.A. (1957). Classical physics as geometry: gravitation, electromagnetism, unquantized charge and mass as properties of curved empty space. Ann. Phys. NY. 2: 525–603

    MATH  Article  ADS  MathSciNet  Google Scholar 

  7. 7.

    Racz I. and Wald R.M. (1992). Extensions of space–times with Killing horizons. Class. Quant. Grav. 9: 2643–2656

    MATH  Article  ADS  MathSciNet  Google Scholar 

  8. 8.

    Bartrum P.C. (1967). Null electromagnetic fields in the form of spherical radiation. J. Math. Phys. 8: 1464–1467

    Article  ADS  Google Scholar 

  9. 9.

    Chruściel P.T., Reall H.S. and Tod P. (2006). On non-existence of static vacuum black holes with degenerate components of the event horizon. Class. Quant. Grav. 23: 549–554

    Article  ADS  MATH  Google Scholar 

  10. 10.

    Müller zum Hagen H. (1970). On the analyticity of static vacuum solutions of Einstein’s equations. Proc. Cambridge Philos. Soc. 67: 415–421

    MATH  MathSciNet  Article  Google Scholar 

  11. 11.

    Müller zum Hagen H. (1970). On the analyticity of stationary vacuum solutions of Einstein’s equation. Proc. Cambridge Philos. Soc. 68: 199–201

    MATH  MathSciNet  Google Scholar 

  12. 12.

    Chruściel P.T. (2005). On analyticity of static vacuum metrics at non-degenerate Killing horizons. Acta Phys. Polon. B36: 17–26

    ADS  Google Scholar 

  13. 13.

    Banerji A. (1970). Null electromagnetic fields in general relativity admitting time-like or null Killing vectors. Jour. Math. Phys. 11: 51–55

    Article  ADS  Google Scholar 

  14. 14.

    Melrose R.B. (1995). Geometric Scattering Theory. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  15. 15.

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

  16. 16.

    Newman, E.T., Tod, K.P. In: Held, A., (ed.) Asymptotically-Flat Space–Times in General Relativity. Plenum Press (1980)

  17. 17.

    Penrose R. and Rindler W. (1984). Spinors and Space-Time I. Cambridge University Press, Cambridge

    Google Scholar 

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Correspondence to Paul Tod.

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Tod, P. Conditions for non-existence of static or stationary, Einstein–Maxwell, non-inheriting black-holes. Gen Relativ Gravit 39, 111–127 (2007).

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  • Black Hole
  • Killing Vector
  • Stationary Black Hole
  • Killing Horizon
  • Maxwell Space