Kaluza-Klein reductions on five dimensional action principles

Abstract

Since Einstein published his theory of general relativity, scientists wanted to construct a theory which describes both gravity and electromagnetism. Kaluza introduced a five dimensional manifold, which consists of spacetime plus an extra circle dimension. After he reduced the Einstein equations for the five dimensional manifold to four dimensional equations, he could unify gravity and the electromagnetic field in one theory. The derivation done by Kaluza is called the Kaluza-Klein reduction. In this reduction, Kaluza used an ansatz, called the cylindrical condition, which was motivated by Klein. In this thesis we will look at the Kaluza-Klein reduction and at the cylindrical condition. This reduction is a motivation to consider actions in the five dimensional spacetime and look at the results they give in four dimensional spacetime. We are going to look at the action for a(p−1)-form gauge field on a five dimensional manifold and we are going to reduce the action to four dimensions. To do this reduction, we cover the mathematics of differential forms. With this mathematics, we will look at Hodge theorem, which relates exact, closed and harmonic forms. With Hodge theorem we will better understand the action for a(p−1)-form gauge field.