Communications in Mathematical Physics

, Volume 33, Issue 4, pp 323–334 | Cite as

A variational principle for black holes

  • S. W. Hawking


It is shown that the initial data which gives rise to stationary black hole solutions extremizes the mass for a given angular momentum and area of the horizon. The only extremum of the mass for a given area of the horizon but arbitrary angular momentum is the Schwarzschild solution. In this case, and when the angular momentum is small, the extremum of the mass is a local minimum. This suggests that the initial data for the Schwarzschild solution has a smaller mass than any other initial data with the same area of the horizon. If this is the case, there is no possibility of proving the occurrence of naked singularities by methods suggested by Penrose and Gibbons. Together with Carter's theorem, the fact that the extremum is a local minimum indicates that the Kerr solutions are stable against axisymmetric perturbations.


Neural Network Black Hole Statistical Physic Angular Momentum Complex System 
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  1. 1.
    Bondi, H., van der Burg, M. G. J., Metzner, A. W. K.: Proc. Roy. Soc. A269, 21 (1962)Google Scholar
  2. 2.
    Penrose, R.: Phys. Rev. Lett.10, 66–68 (1963)Google Scholar
  3. 3.
    Hawking, S. W., Hartle, J. B.: Commun. math. Phys.27, 283–290 (1972)Google Scholar
  4. 4.
    Hawking, S. W.: The Event Horizon in Black Holes Ed. DeWitt and DeWitt, New York, Paris, London: Gordon and Breach 1973Google Scholar
  5. 5.
    Hawking, S. W.: Commun. math. Phys.25, 152–166 (1972)Google Scholar
  6. 6.
    Hawking, S. W., Ellis, G. F. R.: The Large Scale Structure of Spacetime. London: Cambridge University Press 1973Google Scholar
  7. 7.
    Carter, B.: J. Math. Phys.10, 70–81 (1969)Google Scholar
  8. 8.
    Carter, B.: The General Theory of Stationary Black Hole States in Black Holes Ed. DeWitt and DeWitt, New York, Paris, London: Gordon and Breach 1973Google Scholar
  9. 9.
    Gibbons, G. W.: Commun. math. Phys.27, 87 (1972)Google Scholar
  10. 10.
    Choquet-Bruhat, Y., Geroch, R. P.: Commun. math. Phys.14, 329–335 (1969)Google Scholar
  11. 11.
    Bardeen, J. M., Carter, B., Hawking, S. W.: Commun. math. Phys.31, 161–170 (1973)Google Scholar
  12. 12.
    Bardeen, J. M.: Ap. J.162, 64 (1970)Google Scholar
  13. 13.
    Carter, B.: Phys. Rev. Lett.26, 331–332 (1971)Google Scholar
  14. 14.
    Chandrasekhar, S., Friedman, J. L.: Ap. J.177, 745 (1972)Google Scholar

Copyright information

© Springer-Verlag 1973

Authors and Affiliations

  • S. W. Hawking
    • 1
  1. 1.Institute of AstronomyUniversity of CambridgeCambridgeEngland

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