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The mathematical foundations of general relativity theory are the concepts of semi-Riemannian geometry. In order to describe cosmological models one uses 4-dimensional semi-Riemannian manifolds. These models must have certain symmetries to be physically viable. Such symmetries are represented by Killing vector fields on manifolds. An important class of cosmic models are the Gödel type solutions. In 1949 Gödel published a metric that models a rotating universe with vanishing shear and acceleration as well as vanishing expansion. The Killing vectors of this metric are well known. In 2009 M. Gürses, M. Plaue, M. Scherfner, T. Schönfeld and L. A. M. de Sousa published two generalizations of the Gödel metric. The Killing vectors of these generalized metrics have not been calculated so far. The aim of this work is to introduce the mathematical concepts of Killing vector fields and their role in general relativity theory to a reader with basic knowledge in differential geometry. These concepts are then applied to calculate the Killing vectors of the generalized Gödel metrics mentioned above. Chapter 1 contains a short repetition of the basics in semi-Riemannian geometry including sections on tensor fields and Lie algebras. These concepts are fundamental for a profound understanding of the material. Section 1.4 shortly explains how semi-Riemannian geometry is applied in general relativity theory. In Chapter 2 the definition of a Killing vector field is introduced followed by important theorems and examples. Then three sections explain how 2-, 3- and 4-dimensional Lie algebras such as Killing vector fields can be classified. This turns out to be of great importance in Chapter 3 where the relevance of Killing vector fieelds to several concepts in general relativity theory is explained in detail. In Chapter 4 an approach by T. Chrobok to calculate the Killing vectors of the Gödel metric is explained carefully. Then a slightly different approach for calculating the Killing vectors of the generalized Gödel metrics mentioned above is presented in detail.
