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Hawking, S.W. And G.W Gibbons. "Classification of Gravitational Instanton Symmetries", in Communications in Mathematical Physics, 1979, volume 66, pp 291-310, offered in the full volume of vols 65 & 66 bound together, 312+311pp. Very nice copy, ex-library, bound in a solid red cloth. There are library stamps on the top and bottom of the textblock, otherwise there are no external marks. There are a few library stamps in the text plus a bookplate. Very crisp and fresh copy. There are also 20 other papers in the two volumes. [++] [++] "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." abstract from the paper. [++] "In mathematical physics and differential geometry, a gravitational instanton is a four-dimensional complete Riemannian manifold satisfying the vacuum Einstein equations. They are so named because they are analogues in quantum theories of gravity of instantons in Yang Mills theory. In accordance with this analogy with self-dual Yang Mills instantons, gravitational instantons are usually assumed to look like four dimensional Euclidean space at large distances, and to have a self-dual Riemann tensor. Mathematically, this means that they are asymptotically locally Euclidean (or perhaps asymptotically locally flat) hyperkähler 4-manifolds, and in this sense, they are special examples of Einstein manifolds. From a physical point of view, a gravitational instanton is a non-singular solution of the vacuum Einstein equations with positive-definite, as opposed to Lorentzian, metric. There are many possible generalizations of the original conception of a gravitational instanton: for example one can allow gravitational instantons to have a nonzero cosmological constant or a Riemann tensor which is not self-dual. One can also relax the boundary condition that the metric is asymptotically Euclidean. There are many methods for constructing gravitational instantons, including the Gibbons Hawking Ansatz, twistor theory, and the hyperkähler quotient construction."--Wikipedia, "gravitational instantons". [++] "In physics, in the theory of general relativity, spacetimes with at least a 1-parameter group of isometries can be classified according to the fixed point-sets of the action. Isolated fixed points are called nuts. The other possibility is that the fixed point set is a metric 2-sphere, called bolt. The number of nuts and bolts can also be related to topological invariants, such as the Euler characteristic. This classification is widely used in the analysis of gravitational instantons."--Wikipedia, "Nuts and Bolts" [++] Hawking had earlier visited this topic in "Gravitational Instanton", found in "Physics Letters A", Volume 60, Issue 2, 7 February 1977, Pages 81-8.
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