Abstract
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.
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References
Polyakov, A.: Phys. Lett59B, 82 (1975)
't Hooft, G.: Phys. Rev. Lett.37, 8 (1976)
Belavin, A., Polyakov, A., Schwarz, A., Tyupkin, B.: Phys. Lett.59B, 85 (1975)
Jackiw, R., Rebbi, C.: Phys. Lett.37, 172 (1976)
Callan, C., Dashen, R., Gross, D.: Phys. Lett.63B, 334 (1976)
Hawking, S.W.: Nucl. Phys. B144, 349 (1978)
Zumino, B.: Ann. N.Y. Acad. Sci.302, 545 (1977)
Hartle, J., Hawking, S.W.: Phys. Rev. D13, 2188 (1976)
Gibbons, G.W., Perry, M.J.: Phys. Rev. Lett.36, 985 (1976)
Gibbons, G.W., Perry, M.J.: Proc. R. Soc. A358, 467 (1978)
Hawking, S.W.: Phys. Lett.60A, 81 (1977)
Gibbons, G.W., Hawking, S.W.: Phys. Rev. D15, 2752 (1977)
Hawking, S.W.: Commun. Math. Phys.43, 199 (1975)
Eguchi, T., Freund, P.G.O.: Phys. Rev. Lett.37, 1251 (1976)
Gibbons, G.W., Pope, C.N.: Commun. Math. Phys.61, 239 (1978)
Eguchi, T. Hanson, A.: Phys. Lett.74B, 249 (1978)
Belinski, V.A., Gibbons, G.W., Page, R.N., Pope, C.N.: Phys. Lett.76B, 433 (1978)
Page, D.N.: Phys. Lett.78B, 249 (1978)
Page, D.N.: Phys. Lett.79B, 235–238 (1978)
Gibbons, G.W., Hawking, S.W.: Phys. Rev. D15, 2738 (1977)
Gibbons, G.W., Hawking, S.W.: Phys. Lett.78B, 430 (1978)
Hitchin, N.J.: Polygons and gravitons. Preprint, Oxford University. Math. Proc. Camb. Phil. Soc. (in press)
Milnor, J.: Morse theory. Princeton: Princeton University Press (1963)
Chern, S.S.: Ann. Math.46, 674 (1945)
Atiyah, M.F., Patodi, V.K., Singer, I.M.: Proc. Cambridge Philos. Soc.77, 43 (1975);78, 405 (1975)
Geroch, R.P.: J. Math. Phys.9, 1739 (1968)
Atiyah, M.F., Bott, R.: Ann. Math.87, 451 (1968)
Bott, R.: Mich. Math. J.14, 231–244 (1967)
Atiyah, M.F., Singer, I.M.: Ann. Math.87, 546–604 (1968)
Baum, P., Cheeger, J.: Topology8, 173–193 (1969)
Hawking, S.W.: Commun. Math. Phys.25, 152–166 (1972)
Hawking, S.W., Pope, C.N.: Phys. Lett.73B, 42 (1978)
Back, A., Freund, P.G.O., Forger, M.: Phys. Lett.77B, 181 (1978)
Ehlers, J.: In: Les theories relativistes de la gravitation. Paris: CNRS 1959
Geroch, R.: J. Math. Phys.13, 394 (1972)
Gibbons, G.W., Hawking, S.W., Perry, M.J.: Nucl. Phys. B138, 141 (1978)
Page, D.N.: Phys. Rev. D18, 2733–2738 (1978)
Gibbons, G.W., Pope, C.N.: Commun. Math. Phys. (in press)
Hawking, S.W.: Phys. Rev. D18, 1747 (1978)
Gibbons, G.W., Hawking, S.W.: Proof of the positive action conjecture and the nature of the gravitational Action. Unpublished Report
Shoen, R.M., Yau, S.T.: Phys. Rev. Lett.42, 547–548 (1979)
Hawking, S.W., Pope, C.N.: Nucl. Phys. B146, 381–392 (1978)
Hawking, S.W.: Euclidean quantum gravity. Cargese Summer School Lectures 1978, New York, London: Plenum Press (in press)
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Communicated by R. Geroch
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Gibbons, G.W., Hawking, S.W. Classification of Gravitational Instanton symmetries. Commun.Math. Phys. 66, 291–310 (1979). https://doi.org/10.1007/BF01197189
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DOI: https://doi.org/10.1007/BF01197189