The embedding of relativistic space-times in flat spaces

Abstract

This dissertation is divided into the following five chapters: CHAPTER 1. The definition of a flat space is discussed from the point of view of the metric form in the space being reducable to one with constant coefficients for some coordinate system, and this is shown to be equivalent to the vanishing of the tensor. Some general results concerning the isometric embedding of Riemannian metrics in flat spaces are derived. CHAPTER 2. In this chapter, Kasner's proof that an embedding of any solution of Einstein's vacuum equations in a five dimensional flat space is impossible, is given in detail, with a short note regarding the signature of the embedding space. Kasner's minimal embedding of the Schwarzschild Exterior Solution is derived. The contents of this chapter are further discussed in chapter 4. CHAPTER 3. Fronsdal's embedding of the Schwarzschild Exterior Solution in E6 is treated in detail as an illustration of the use of the embedding method in general relativity. CHAPTER 4. The contents of this chapter are wholly based upon a systematic approach to the problem of finding the embedding transformations and minimal embedding spaces for certain relativistic metrics, by the use of sets of partial differential equations. The embeddings of the Schwarzschild Exterior Solution by Kasner and Fronsdal are slightly generalised. Kasner's result of Chapter 2 is criticized on the grounds that its initial assumptions are too restrictive, and give rise to misleading results. In view of this, it is proved that the Schwarzschild Exterior Solution is not embeddable in E5 by an independent method. Other embeddings of relativistic metrics are derived. CHAPTER 5. By way of motivation for the present reawakening of interest in the embedding problem, a proposed relation of the embedding space to the elementary particle symmetries is discussed in general terms.