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On the linearization stability of the conformally (anti-) self-dual Einstein equations

1990; American Institute of Physics; Volume: 31; Issue: 12 Linguagem: Inglês

10.1063/1.528951

1527-2427

Charles G Torre,

Cosmology and Gravitation Theories

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.