On Spinors, Twistors and Six-Dimensional General Relativity

Abstract

In this dissertation we review the article "On the Six-dimensional Kerr Theorem and Twistor Equation" [1], introducing advanced topics from general relativity along the way. We start by introducing how the concept of a spinor appears in physics from the simple requirement of Lorentz invariance on a field theory that would describe the electron. We then introduce ideas from representation theory so that we can approach the spinor concept from a more mathematical point of view and with that know its properties so we can apply the formalism as we wish. We then show the correspondence between tensors and spinors in four dimensions and rewrite General Relativity in terms of spinors, and we see that this allows us to simplify several equations in terms of the Newman-Penrose formalism. The concept of congruences is introduced, and its study in turn leads us to the Kerr Theorem and the notion of a twistor. Then, we move on to six-dimensional space-time, where we see that the correspondence between tensors and spinors in six dimensions is quite different than the one we described for the four-dimensional case, but group theory allows us to easily obtain the spinor form of the desired tensors. The Kerr Theorem is obtained in six dimensions, and its generalization to generic even dimensions is discussed. We then study the six-dimensional twistor equation and see that it imposes an algebraic constraint on the form of the Weyl tensor, and a family of examples that exhibits this property is discussed.