Ashtekar's spinorial formulation of general relativity is used to study perturbations of gravitational instantons corresponding to finite-action solutions of the Euclidean Einstein equations (with a nonzero cosmological constant) possessing an anti-self-dual Weyl curvature tensor. It is shown that, with an appropriate "on-shell" form of infinitesimal gauge transformations, the space of solutions to the linearized instanton equation can be described in terms of an elliptic complex; the cohomology of the complex defines gauge-inequivalent perturbations. Using this elliptic complex we prove that there are no nontrivial solutions to the linearized instanton equation on conformally anti-self-dual Einstein spaces with a positive cosmological constant. Thus, the space of gravitational instantons is discrete when the cosmological constant is positive; i.e., the dimension of the gravitational moduli space in this case is zero. We discuss the issue of linearization stability as well as the feasibility of using the Atiyah-Singer index theorem to compute the dimension of the gravitational moduli space when the cosmological constant is negative.
Article
Perturbations of Gravitational Instantons
Physical Review D
Document Type
Article
Publisher
American Physical Society
Publication Date
1-1-1990
Disciplines
Abstract
Citation Information
C. Torre, “Perturbations of gravitational instantons,” Physical Review D, vol. 41(12), 1990, pp. 3620-3627.
Note: Charles Torre was affiliated with the Center for Space Science Research, Space Research Institute, Florida Institute of Technology, Melbourne, Florida at the time of publication.