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Penrose Transform for Gravitation

Part of the Fundamental Theories of Physics book series (FTPH, volume 69)

Abstract

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.

Keywords

Twistor Space Spinor Field Gauge Freedom Twistor Theory Vacuum Einstein Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Kluwer Academic Publishers 2002

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