Communications in Mathematical Physics

, Volume 88, Issue 3, pp 295–308 | Cite as

Positive mass theorems for black holes

  • G. W. Gibbons
  • S. W. Hawking
  • Gary T. Horowitz
  • Malcolm J. Perry
Article

Abstract

We extend Witten's proof of the positive mass theorem at spacelike infinity to show that the mass is positive for initial data on an asymptotically flat spatial hypersurface Σ which is regular outside an apparent horizonH. In addition, we prove that if a black hole has electromagnetic charge, then the mass is greater than the modulus of the charge. These results are also valid for the Bondi mass at null infinity. Finally, in the case of the Einstein equation with a negative cosmological constant, we show that a suitably defined mass is positive for data on an asymptotically anti-de Sitter surface Σ which is regular outside an apparent horizon.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Penrose, R.: Gravitational collapse and spacetime singularities. Phys. Rev. Lett.14, 57–59 (1965)Google Scholar
  2. 2.
    Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge: Cambridge University Press 1973Google Scholar
  3. 3.
    Hawking, S.W.: The event horizon in: Black holes, pp. 1–55. deWitt, B.S., deWitt, C.M. (eds.) New York: Gordon and Breach 1973Google Scholar
  4. 4.
    Schoen, R., Yau, S.-T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys.65, 45–76 (1979)Google Scholar
  5. 5.
    Schoen, R., Yau, S.-T.: Proof of the positive mass theorem. II. Commun. Math. Phys.79, 231–260 (1981)Google Scholar
  6. 6.
    Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys.80, 381–402 (1981)Google Scholar
  7. 7.
    Carter, B.: The general theory of black hole properties. (eds.). In: General relativity, pp. 294–369. Hawking, S.W., Israel, W., Cambridge: Cambridge University Press 1979Google Scholar
  8. 8.
    Hawking, S.W.: The conservation of matter in general relativity. Commun. Math. Phys.18, 301–306 (1970)Google Scholar
  9. 9.
    Gibbons, G.W., Hull, C.M.: A Bogomolny bound for general relativity and solutions inN=2 supergravity. Phys. Lett.109 B, 190–194 (1982)Google Scholar
  10. 10.
    Horowitz, G.T., Perry, M.J.: Gravitational energy cannot become negative. Phys. Rev. Lett.48, 371–374 (1982)Google Scholar
  11. 11.
    Abbott, G., Deser, S.: Stability of gravity with a cosmological constant. Nucl. Phys. B195, 76–96 (1982)Google Scholar
  12. 12.
    Ashtekar, A., Horowitz, G.T.: Energy-momentum of isolated systems cannot be null. Phys. Lett.89 A, 181–183 (1982)Google Scholar
  13. 13.
    Parker, T., Taubes, C.H.: On Witten's proof of the positive energy theorem. Commun. Math. Phys.84, 223–238 (1982)Google Scholar
  14. 13a.
    Reula, O.: Existence theorem for solutions of Witten's equations and nonnegativity of total mass. J. Math. Phys.23, 810–814 (1982)Google Scholar
  15. 14.
    Hormander, L.: Linear partial differential operators. New York: Academic Press 1963Google Scholar
  16. 15.
    Geroch, R., Held, A., Penrose, R.: A space-time calculus based on pairs of null directions. J. Math. Phys.14, 874–881 (1973)Google Scholar
  17. 16.
    Gibbons, G.W., Hawking, S.W.: Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D15, 2738–2751 (1977)Google Scholar
  18. 17.
    Reula, O., Tod, P.: Private communicationGoogle Scholar
  19. 18.
    Freedman, D.Z.: private communication.Google Scholar
  20. 19.
    Pirani, F.A.E.: Introduction to gravitational radiation theory. In: Lectures on General Relativity, pp. 249–373. Trautman, A., Pirani, F.A.E., Bondi, H. (eds.). Englewood Cliffs, N.J.: Prentice-Hall 1965Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • G. W. Gibbons
    • 1
  • S. W. Hawking
    • 1
  • Gary T. Horowitz
    • 2
  • Malcolm J. Perry
    • 3
  1. 1.D.A.M.T.P.University of CambridgeCambridgeEngland
  2. 2.Institute for Advanced StudyPrincetonUSA
  3. 3.Dept. of PhysicsPrinceton UniversityUSA

Personalised recommendations