Twistors in curved space time
This thesis is concerned with an investigation of twistorial structures present in curved Lorentzian space-times. Chapter 1 introduces the basic definitions and some theorems that will be used later in the text. Chapter 2 investigates generalised connections that arise in twister theory. First the Cartan con-formal connection is studied, and some of the geometry underlying it is shown to be that used by Fefferman and Graham C133. Also a condition that a space-time is conformal to vacuum is given. Secondly the theory of the Chern connection associated to a C.R. manifold is developed in such a way as to make the calculation of the connection associated to a twistor C.R. manifold straight forward. A new proof of the Chern theorem of existence and uniqueness is given. The Chern connection of a twistor C.R. manifold is then calculated, and discussed. In particular S-dimensionai C.R. manifolds arising as twistor C.R. manifolds are characterised. Canonical structures peculiar to the twister case are discussed. Applications of C.R. manifold theory to algebraically special space-times are suggested. Chapter three analyses how various twistorial structures behave in linearised general relativity. First, deformations of the space of complex null geodesies corresponding to variations of the conformal structure of space-time are shown to be generated by hami1tonians. Those that correspond to variations in the metric satisfying the field equations are given, along with hamiltonians corresponding to different fields and field equations. Beneralisations to nonlinear equations are discussed. These ideas are applied to hypersurface twisters in linearised theory, using fiat hypersurfaces and Cech cohoeology. Expressions are obtained for the deformation of the complex structure of the spaces and their evolution. The results are generalised to non flat hypersurfaces using Dolbeault cohomolcgy. It is shown that certain canonically defined forms on the spin bundle are preferred Dolbeault representatives for derivatives of the twister cohomology classes corresponding to the linearised field. In chapter four I generalise the results of chapter three to curved space using the Chern connection. In particular twistorial formulations of the constraint equations are given, and a formula for the evolution that satisfies the the vacuum evolution equations is given in terms of an "infinity" twistor and a "time" twister. This is then discussed. In chapter five I make some comments on the interpretation of a three form on the spin bundle discovered by B.A.J. Sparling as the gravitational hami1tonian. I then use this to show that one can give an interpretation of Penrose's quasi-local angular momentum twistor in terms of the canonical formalism.