Weyl's Theory Of Non-Riemannian Geometry And Relativity

Abstract

The non-Riemannian geometry of Weyl is an outgrowth of Levi- Civita’s concept of parallelism. It is based on the concept of linear displacement. In Weyl’s geometry length is non-transferable and, in light of this indeterminateness that surrounds the comparison of lengths in different places, we must confine ourselves to the comparison of lengths at any one place or at points separated by infintesmal intervals. We must therefore fix at every point of space certain measuring rods which are to serve as a unit of length when we measure lengths situated by their side. The totality of these unit rods constitute what is known as a gauge system. These gauges may be selected arbitrarily. We will give a detailed discussion of the gauge system later in the thesis. Moreover, since Weyl’s geometry is a generalization of Riemann’s, we might first consider some of the foundations of that geometry which are applicable to the former. This we shall do by a process of abstraction. We shall proceed from a Euclidean space defined on the more familiar Cartesian co-ordinates to one defined on the general or curvilinear co-ordinates of three dimensions. We then proceed to a space of n dimensions and to the Riemann space. We shall discuss the theory of tensors from an intuitive standpoint using these geometrical foundations. The body of the thesis will involve a detailed discussion of the geometrical foundations of Weylian geometry and some of its important geometrical properties. The conclusion will be synthesis of the most important physical implications of Weylian geometry to the general theory of relativity.The non-Riemannian geometry of Weyl is an outgrowth of Levi- Civita’s concept of parallelism. It is based on the concept of linear displacement. In Weyl’s geometry length is non-transferable and, in light of this indeterminateness that surrounds the comparison of lengths in different places, we must confine ourselves to the comparison of lengths at any one place or at points separated by infintesmal intervals. We must therefore fix at every point of space certain measuring rods which are to serve as a unit of length when we measure lengths situated by their side. The totality of these unit rods constitute what is known as a gauge system. These gauges may be selected arbitrarily. We will give a detailed discussion of the gauge system later in the thesis. Moreover, since Weyl’s geometry is a generalization of Riemann’s, we might first consider some of the foundations of that geometry which are applicable to the former. This we shall do by a process of abstraction. We shall proceed from a Euclidean space defined on the more familiar Cartesian co-ordinates to one defined on the general or curvilinear co-ordinates of three dimensions. We then proceed to a space of n dimensions and to the Riemann space. We shall discuss the theory of tensors from an intuitive standpoint using these geometrical foundations. The body of the thesis will involve a detailed discussion of the geometrical foundations of Weylian geometry and some of its important geometrical properties. The conclusion will be synthesis of the most important physical implications of Weylian geometry to the general theory of relativity.