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The two-spinor formalism in the study of the Bach equations and the massless free-field equations

Forbes, H. (2021). The two-spinor formalism in the study of the Bach equations and the massless free-field equations. (Unpublished Doctoral thesis, City, University of London)

Abstract

This thesis applies the two-component spinor (two-spinor) calculus to study the Bach equations and the massless free-field equations. We first provide some context for the following chapters by outlining the history of the spinor concept and the motivation for using two-spinor methods as opposed to the better-known 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 two-spinor calculus in practical calculations, where explicit solutions are sought, we introduce the Newman-Penrose spin-coefficient formalism and its compacted version.

We proceed by applying the compacted spin-coefficient formalism to the Bach equations, which is an example of a conformal theory of gravity. We reconstruct two known exact solutions, namely the PP-wave spacetime and the static spherically-symmetric 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 two-spinor methods, we introduce Lanczos potential theory, including the Lanczos spinor and the Weyl-Lanczos 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 free-field 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 three-surface integral over a conserved current density. Moreover, from the conformal invariance of the massless free-field 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
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