Article

Communications in Mathematical Physics

, Volume 66, Issue 3, pp 291-310

First online:

Classification of Gravitational Instanton symmetries

  • G. W. GibbonsAffiliated withDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge
  • , S. W. HawkingAffiliated withDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge

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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.