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

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

Dec 1990

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1990jmp....31.2983t&link_type=abstract

Journal of Mathematical Physics, Vol. 31, No. 12, p. 2983 - 2986

Mathematics

Logic

4

Cosmological Constant

Scientific paper

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.

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