##### Abstract

In this paper we aim to study a small step in the physicist marathon that is the unifica- tion of two field theories that, while similar in some senses, are rather different in other, namely gravity and electromagnetism. The man on the starting block was none other than Albert Einstein who, although ultimately unsuccessful, spent the later part of his life trying to formulate a unified theory of the two field theories. Having made the monu- mental achievement of General Relativity [1] and thus geometrized gravity, he sought to also geometrize electromagnetism. Some might object that, since we have the Einstein- Maxwell equations, there is already a unification of the two field theories. But this is not really the case, as these equations are more or less obtained by taking the Einstein equa- tions and slapping the Maxwell equations on top. Hardly a very profound nor intimate unification. Thus what one really is seeking, is a deeper, more fundamental unification. This search has a long tradition, and we are not the first since Einstein to attempt to investigate this. A few years after Einstein’s paper on General Relativity, Kaluza and Klein [2, 3] presented their theory of Einstein’s equations on a 5-dimensional space-time where this ”extra” dimension is curled up in a circle. This in fact leads to Einstein’s equations in 4D space-time together with Maxwell’s equations. But also together with a massless spin-0 particle that, sadly, doesn’t appear to exist. Our approach is to show that by imbuing ordinary 4D Minkowski space-time with a second property in addition to the metric, we obtain the theory of electromagnetism by geometric considerations. This second property is a field of two-dimensional spacelike planes. At each point in space-time we fix such a plane, which can be described by an anti-symmetric tensor of rank 2. The first works on this model are found in [4, 5, 6, 7, 8].

In Chapter 1 we will give a short introduction to the formalism of differential forms and tensors, and briefly discuss why this formalism is better suited for the more advanced calculations than vector formalism. For a more in-depth discussion on this formalism, [9] is a good place to look. In Chapter 2 we examine the model in three-dimensional Euclidean space, investigate some of its properties, and we also look at some examples of configurations for various electric and magnetic fields. This chapter is based on the work in [5]. What is new here is that we explicitly include electric fields in addition to magnetic fields, and discuss also examples of some specific electric field configurations. In Chapter 3 we extend the model to four-dimensional Minkowski space, examining the various properties and also how they relate to the three-dimensional model already dis- cussed. This is partially based on [6] as well as the matrix formulation of frame bundles found in [10]. We study in greater detail the properties of some of the general new struc- tures which emerge in four-dimensional space-time than what has been done before and how they may or may not relate to the electromagnetic field. As an example we take a look at the field from a single monopole moving at constant velocity, and various inter- pretations of the quantities in the model with respect to this specific configuration. In Chapter 4 we examine the field from a We try to understand and interpret the various

quantities and properties of the model in this context and if possible give a more de- tailed and explicit derivation. Finally in Chapter 5 we investigate the motion of charged particles in the presence of electromagnetic fields described by the plane-model we have introduced earlier, and how we can think of such a particle as a neutral, rotating particle with a constraint on its rotation. We first examine the case of the particle moving non- relativistically, which can be found in [5], and then we try to extend this to also cover particles moving relativistically in an electromagnetic field.