The Geometry of Complex Space-Times with Torsion

Details The Geometry of Complex Space-Times with Torsion The Geometry of Complex Space-Times with Torsion

1995-08-14

arxiv.org/abs/gr-qc/9508030v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

3 pages, plain-tex, published in Proceedings of the X Italian Conference on General Relativity and Gravitational Physics, edit

Scientific paper

The necessary and sufficient condition for the existence of $\alpha$-surfaces in complex space-time manifolds with nonvanishing torsion is derived. For these manifolds, Lie brackets of vector fields and spinor Ricci identities contain explicitly the effects of torsion. This leads to an integrability condition for $\alpha$-surfaces which does not involve just the self-dual Weyl spinor, as in complex general relativity, but also the torsion spinor, in a nonlinear way, and its covariant derivative. A similar result also holds for four-dimensional, smooth real manifolds with a positive-definite metric. Interestingly, a particular solution of the integrability condition is given by conformally right-flat and right-torsion-free space-times.

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