Alpha Surfaces for Complex Space-Times with Torsion

TL;DR

This paper investigates the conditions for alpha-surfaces in complex space-times with torsion, revealing how torsion influences integrability and identifying special solutions with specific geometric properties.

Contribution

It introduces a new integrability condition for alpha-surfaces that incorporates torsion effects, extending previous complex relativity frameworks.

Findings

Derived an integrability condition involving torsion spinor

Identified conformally right-flat and torsion-free solutions

Showed torsion's nonlinear impact on alpha-surface existence

Abstract

This paper studies necessary conditions for the existence of alpha-surfaces in complex space-time manifolds with nonvanishing torsion. 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. Interestingly, a particular solution of the integrability condition is given by conformally right-flat and right-torsion-free space-times.