Symmetries of type D+D− gravitational instantons

Abstract

Gravitational instantons are considered important in analyzing such quantum gravitational effects as the evaporation and condensation of black holes and the stability of the vacuum at finite temperature. Unfortunately, not every positive definite solution of the field equations is derivable from the Lorentzian regime by analytic continuation. As a first step in the investigation of these objects, it is proven that all Petrov-type D+D− spaces admit an Abelian, orthogonally transitive, two dimensional group of isometries. This is accomplished by using a Euclidean version of the Newman–Penrose formalism to classify the spaces according to the twist and expansion of their Debever–Penrose vectors. In each class one can then infer the existence of Killing spinors, vectors, and tensors required to yield the result.