Communications in Mathematical Physics

, Volume 66, Issue 3, pp 291–310 | Cite as

Classification of Gravitational Instanton symmetries

  • G. W. Gibbons
  • S. W. Hawking


We classify the action of one parameter isometry groups of Gravitational Instantons, complete non singular positive definite solutions of the Einstein equations with or without Λ term. The fixed points of the action are of 2-types, isolated points which we call “nuts” and 2-surfaces which we call “bolts”. We describe all known gravitational instantons and relate the numbers and types of the nuts and bolts occurring in them to their topological invariants. We perform a 3+1 decomposition of the field equations with respect to orbits of the isometry group and exhibit a certain duality between “electric” and “magnetic” aspects of gravity. We also obtain a formula for the gravitational action of the instantons in terms of the areas of the bolts and certain nut charges and potentials that we define. This formula can be interpreted thermodynamically in several ways.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Field Equation 
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  1. 1.
    Polyakov, A.: Phys. Lett59B, 82 (1975)Google Scholar
  2. 2.
    't Hooft, G.: Phys. Rev. Lett.37, 8 (1976)Google Scholar
  3. 3.
    Belavin, A., Polyakov, A., Schwarz, A., Tyupkin, B.: Phys. Lett.59B, 85 (1975)Google Scholar
  4. 4.
    Jackiw, R., Rebbi, C.: Phys. Lett.37, 172 (1976)Google Scholar
  5. 5.
    Callan, C., Dashen, R., Gross, D.: Phys. Lett.63B, 334 (1976)Google Scholar
  6. 6.
    Hawking, S.W.: Nucl. Phys. B144, 349 (1978)Google Scholar
  7. 7.
    Zumino, B.: Ann. N.Y. Acad. Sci.302, 545 (1977)Google Scholar
  8. 8.
    Hartle, J., Hawking, S.W.: Phys. Rev. D13, 2188 (1976)Google Scholar
  9. 9.
    Gibbons, G.W., Perry, M.J.: Phys. Rev. Lett.36, 985 (1976)Google Scholar
  10. 10.
    Gibbons, G.W., Perry, M.J.: Proc. R. Soc. A358, 467 (1978)Google Scholar
  11. 11.
    Hawking, S.W.: Phys. Lett.60A, 81 (1977)Google Scholar
  12. 12.
    Gibbons, G.W., Hawking, S.W.: Phys. Rev. D15, 2752 (1977)Google Scholar
  13. 13.
    Hawking, S.W.: Commun. Math. Phys.43, 199 (1975)Google Scholar
  14. 14.
    Eguchi, T., Freund, P.G.O.: Phys. Rev. Lett.37, 1251 (1976)Google Scholar
  15. 15.
    Gibbons, G.W., Pope, C.N.: Commun. Math. Phys.61, 239 (1978)Google Scholar
  16. 16.
    Eguchi, T. Hanson, A.: Phys. Lett.74B, 249 (1978)Google Scholar
  17. 17.
    Belinski, V.A., Gibbons, G.W., Page, R.N., Pope, C.N.: Phys. Lett.76B, 433 (1978)Google Scholar
  18. 18.
    Page, D.N.: Phys. Lett.78B, 249 (1978)Google Scholar
  19. 19.
    Page, D.N.: Phys. Lett.79B, 235–238 (1978)Google Scholar
  20. 20.
    Gibbons, G.W., Hawking, S.W.: Phys. Rev. D15, 2738 (1977)Google Scholar
  21. 21.
    Gibbons, G.W., Hawking, S.W.: Phys. Lett.78B, 430 (1978)Google Scholar
  22. 22.
    Hitchin, N.J.: Polygons and gravitons. Preprint, Oxford University. Math. Proc. Camb. Phil. Soc. (in press)Google Scholar
  23. 23.
    Milnor, J.: Morse theory. Princeton: Princeton University Press (1963)Google Scholar
  24. 24.
    Chern, S.S.: Ann. Math.46, 674 (1945)Google Scholar
  25. 25.
    Atiyah, M.F., Patodi, V.K., Singer, I.M.: Proc. Cambridge Philos. Soc.77, 43 (1975);78, 405 (1975)Google Scholar
  26. 26.
    Geroch, R.P.: J. Math. Phys.9, 1739 (1968)Google Scholar
  27. 27.
    Atiyah, M.F., Bott, R.: Ann. Math.87, 451 (1968)Google Scholar
  28. 28.
    Bott, R.: Mich. Math. J.14, 231–244 (1967)Google Scholar
  29. 29.
    Atiyah, M.F., Singer, I.M.: Ann. Math.87, 546–604 (1968)Google Scholar
  30. 30.
    Baum, P., Cheeger, J.: Topology8, 173–193 (1969)Google Scholar
  31. 31.
    Hawking, S.W.: Commun. Math. Phys.25, 152–166 (1972)Google Scholar
  32. 32.
    Hawking, S.W., Pope, C.N.: Phys. Lett.73B, 42 (1978)Google Scholar
  33. 33.
    Back, A., Freund, P.G.O., Forger, M.: Phys. Lett.77B, 181 (1978)Google Scholar
  34. 34.
    Ehlers, J.: In: Les theories relativistes de la gravitation. Paris: CNRS 1959Google Scholar
  35. 35.
    Geroch, R.: J. Math. Phys.13, 394 (1972)Google Scholar
  36. 36.
    Gibbons, G.W., Hawking, S.W., Perry, M.J.: Nucl. Phys. B138, 141 (1978)Google Scholar
  37. 37.
    Page, D.N.: Phys. Rev. D18, 2733–2738 (1978)Google Scholar
  38. 38.
    Gibbons, G.W., Pope, C.N.: Commun. Math. Phys. (in press)Google Scholar
  39. 39.
    Hawking, S.W.: Phys. Rev. D18, 1747 (1978)Google Scholar
  40. 40.
    Gibbons, G.W., Hawking, S.W.: Proof of the positive action conjecture and the nature of the gravitational Action. Unpublished ReportGoogle Scholar
  41. 41.
    Shoen, R.M., Yau, S.T.: Phys. Rev. Lett.42, 547–548 (1979)Google Scholar
  42. 42.
    Hawking, S.W., Pope, C.N.: Nucl. Phys. B146, 381–392 (1978)Google Scholar
  43. 43.
    Hawking, S.W.: Euclidean quantum gravity. Cargese Summer School Lectures 1978, New York, London: Plenum Press (in press)Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • G. W. Gibbons
    • 1
  • S. W. Hawking
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeEngland

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