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

DOI

10.1063/1.528951

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.

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.

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