Complex Space-Times with Torsion
Theories of gravity with torsion are relevant since torsion is a naturally occurring geometric property of relativistic theories of gravitation, the gauge theory of the Poincaré group leads to its presence, the constraints are second-class and the occurrence of cosmological singularities can be less generic than in general relativity. In a space-time manifold with non-vanishing torsion, the Riemann tensor has 36 independent real components at each point, rather than 20 as in general relativity. The information of these 36 components is encoded in three spinor fields and in a scalar function, having 5,9,3 and 1 complex components respectively. If space-time is complex, this means that, with respect to a holomorphic coordinate basis x a , the metric is a 4 × 4 matrix of holomorphic functions of x a , and its determinant is nowhere-vanishing. Hence the connection and Riemann are holomorphic as well, and the Ricci tensor becomes complex-valued.
After a two-component spinor analysis of the curvature and of spinor Ricci identities, the necessary condition for the existence of □-surfaces in complex space-time manifolds with non-vanishing 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 □-surfaces which does not involve just the self-dual Weyl spinor, as in complex general relativity, but also the torsion spinor, in a non-linear 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 right conformally flat and right-torsion-free space-times.
KeywordsIntegrability Condition Twistor Space Spinor Field Torsion Tensor Weyl Spinor
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