Forbes, H. (2021). The twospinor formalism in the study of the Bach equations and the massless freefield equations. (Unpublished Doctoral thesis, City, University of London)
Abstract
This thesis applies the twocomponent spinor (twospinor) calculus to study the Bach equations and the massless freefield equations. We first provide some context for the following chapters by outlining the history of the spinor concept and the motivation for using twospinor methods as opposed to the betterknown tensor calculus. We also give a summary of the original results and introduce the notation to be used throughout the thesis. We continue with the rules of spinor algebra, which are indispensable for carrying out calculations involving spinors, followed by a description of the geometrical picture of a spinor, which provides a useful mental aid to the more abstract algebraic approach. Next, in close analogy to the tensor formalism, we discuss spinor analysis by introducing the spinor covariant derivative and spinor curvature. Then in order to apply the twospinor calculus in practical calculations, where explicit solutions are sought, we introduce the NewmanPenrose spincoefficient formalism and its compacted version.
We proceed by applying the compacted spincoefficient formalism to the Bach equations, which is an example of a conformal theory of gravity. We reconstruct two known exact solutions, namely the PPwave spacetime and the static sphericallysymmetric spacetime. In order to better understand the relationship between solutions of the Bach equations and the Einstein equations, we present the necessary and sufficient conditions, for a certain class of spacetimes to be a conformal Einstein space. As a further application of twospinor methods, we introduce Lanczos potential theory, including the Lanczos spinor and the WeylLanczos equations. We proceed to solve these equations for the Bach spacetime solutions found earlier.
Next, we discuss duality rotations and helicity in the context of the massless freefield equations. Using concepts from symplectic geometry, we derive an expression for the helicity of an integer spin s field. The helicity expression is given in terms of a threesurface integral over a conserved current density. Moreover, from the conformal invariance of the massless freefield equations, we show that it is conformally invariant. We also utilise concepts from twistor theory in order to better understand the relationship between duality rotations and helicity. We finish with a summary of results and outline future directions related to this work.
Publication Type:  Thesis (Doctoral) 

Subjects:  Q Science > QA Mathematics 
Departments:  Doctoral Theses Doctoral Theses > School of Mathematics, Computer Science and Engineering Doctoral Theses School of Mathematics, Computer Science & Engineering 

Text
 Accepted Version
Download (2MB)  Preview 
Export
Downloads
Downloads per month over past year