All Physics Faculty Publications
Document Type
Article
Journal/Book Title/Conference
Journal of Mathematical Physics
Volume
31
Issue
12
Publisher
American Institute of Physics
Publication Date
1990
First Page
2983
Last Page
2986
Abstract
The Einstein equations with a cosmological constant, when restricted to Euclidean space‐times with anti‐self‐dual Weyl tensor, can be replaced by a quadratic condition on the curvature of an SU(2) (spin) connection. As has been shown elsewhere, when the cosmological constant is positive and the space‐time is compact, the moduli space of gauge‐inequivalent solutions to this equation is discrete, i.e., zero dimensional; when the cosmological constant is negative, the dimension of the moduli space is essentially controlled by the Atiyah–Singer index theorem provided the field equations are linearization stable. It is shown that linearization instability occurs whenever the unperturbed geometry possesses a Killing vector and/or a ‘‘harmonic Weyl spinor.’’ It is then proven that while there are no Killing vectors on compact conformally anti‐self‐dual Einstein spaces with a negative cosmological constant, it is possible to have harmonic Weyl spinors. Therefore, the conformally anti‐self‐dual Einstein equations on a compact Euclidean manifold are linearization stable when the cosmological constant is negative provided the unperturbed geometry admits no harmonic Weyl spinors.
Recommended Citation
C.G. Torre, “On the linearization stability of the conformally (anti-) self-dual Einstein equations,” Journal of Mathematical Physics, vol. 31(12), 1990, p. 2983.
Comments
Originally published by the American Institute of Physics. Publisher's PDF can be accessed through the Journal of Mathematical Physics. Note: At time of publication, Charles Torre was affiliated with theCenter for Space Science Research, Space Research Institute, Florida Institute of Technology, Melbourne, Florida.