# Topology and the Causality of Spacetime

While preparing for tomorrow’s first day of office for 2011, I stumbled upon a hard-copy of my undergraduate thesis. I realized that I don’t have any backup of this piece of work. Thanks to the JotNot application of iPhone, I was able to scan my thesis and convert it to PDF.

Below are samples of the scanned pages. Click on each thumbnail to get the full resolution.

So, if you want to understand how Stephen Hawking and Roger Penrose arrived at their Singularity Theorems, you can read it from the PDF version of my thesis:

Topology And The Causality Of Spacetime

N.B., the file is 40 MB so have some patience.

Einstein’s Theory of General Relativity is one ofthe greatest intellectual achievements in this century. It radically changed our way of looking at the world. lt particularly freed us from our usual Euclidean way of looking at the universe, However, even without the use of Einstein’s equation we can already say something about the universe we live in. It has something to do with the manifold structure of spacetime and the causality conditions imposed bythe speed of light, in that no signal could travel at speeds greater than the speed of light. At this topological level, we impose certain conditions that our spacetime should posses in order for it to be called a reasonable model of our universe.

This thesis is a review article of the causal structure of spacetime. By causal structure, we mean those properties a spacetime should possess so that two pairs of points in it can be joined by a causal curve. A basic notion such as this can already give many results in terms of our understanding of the universe.

When we say that a given model of the universe is reasonable , we mean the following: (i) a lorentz metric can be defined on it,(ii) it shoud be orientable in space and time, (iii) it does not contain closed timelike curves and those curves, though not closed but are “almost closed” , which will be made more precise later, (iii) it possesses a surface( achronal) where we can define initial data set that will determine the evolution of the universe. `

In Chapter ll, we are going to review those aspects of mathematics that are useful in causal analysis. We will start with a review of manifold theory. We will define what we mean by a manifold and all other defined structures on it. We will notice that even without the metric, many consequences of the conditions enumerated above will be derived The only thing we assume on our manifold is that it allows the existence of a Lorentz metric.

In Chapter III, we will show that not all manifolds admit a Lorentz metric. In fact it does not make sense to talk of the surface of the sphere as a two dimensional model of spcetime since a Lorentz metric cannot be defined on the sphere. We will also show that compact spacetimes are of little interest to cosmology because they allow the existence of closed timelike curves.

Chapter IV is a brief chapter that applies the concepts introduced in Chapter III to the proof of Hawking’s Singularity Theorem. Here we will define what we mean by a singularity and learn that a generic class of solutions of Einstein’s equation is singular. Singularities do not tell us that the theory itself is defective, but that they are really physical entities that exist.

Although causal analysis is already a mature subject, it has only been available to a handful of people because it’s nature is in a way abstract. The purpose therefore of this review is to present more details of the subject and providing proofs to some statements that have been omitted in standards texts.