Penrose Transform for Gravitation
Deformation theory of complex manifolds is applied to construct a class of anti-self-dual solutions of Einstein’s vacuum equations, following the work of Penrose and Ward. The hard part of the analysis is to find the holomorphic cross-sections of a deformed complex manifold, and the corresponding conformal structure of an anti-self-dual space-time. This calculation is repeated in detail, using complex analysis and two-component spinor techniques.
If no assumption about anti-self-duality is made, twistor theory is by itself insufficient to characterize geometrically a solution of the full Einstein equations. After a brief review of alternative ideas based on the space of complex null geodesics of complex space-time, and Einstein-bundle constructions, attention is focused on the recent attempt by Penrose to define twistors as charges for massless spin-3/2 fields. This alternative definition is considered since a vanishing Ricci tensor provides the consistency condition for the existence and propagation of massless spin-3/2 fields in curved space-time, whereas in Minkowski space-time the space of charges for such fields is naturally identified with the corresponding twistor space.
The two-spinor analysis of the Dirac form of such fields in Minkowski space-time is carried out in detail by studying their two potentials with corresponding gauge freedoms. The Rarita-Schwinger form is also introduced, and self-dual vacuum Maxwell fields are obtained from massless spin-3/2 fields by spin-lowering. In curved space-time, however, the local expression of spin-3/2 field strengths in terms of the second of these potentials is no longer possible, unless one studies the self-dual Ricci-flat case. Thus, much more work is needed to characterize geometrically a Ricci-flat (complex) space-time by using this alternative concept of twistors.
KeywordsTwistor Space Spinor Field Gauge Freedom Twistor Theory Vacuum Einstein Equation
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