Tensors, Relativity, and Cosmology

By Nils Dalarsson and Mirjana Dalarsson

This book combines relativity, astrophysics, and cosmology in a single volume, providing an introduction to each subject that enables students to understand more detailed treatises as well as the current literature. The section on general relativity gives the case for a curved space-time, presents the mathematical background (tensor calculus, Riemannian geometry), discusses the Einstein equation and its solutions (including black holes, Penrose processes, and similar topics), and considers the energy-momentum tensor for various solutions. The next section on relativistic astrophysics discusses stellar contraction and collapse, neutron stars and their equations of state, black holes, and accretion onto collapsed objects. Lastly, the section on cosmology discusses various cosmological models, observational tests, and scenarios for the early universe.

* Clearly combines relativity, astrophysics, and cosmology in a single volume so students can understand more detailed treatises and current literature * Extensive introductions to each section are followed by relevant examples and numerous exercises * Provides an easy-to-understand approach to this advanced field of mathematics and modern physics by providing highly detailed derivations of all equations and results

• Language: English
• Publisher: Academic Press
• Release date: Apr 29, 2005
• ISBN: 9780080575438

Nils Dalarsson

Nils Dalarsson has been with the Royal Institute of Technology, Department of Theoretical Physics in Stockholm, Sweden, since 1999. His research and teaching experience spans 32 years. Former academic and private sector affiliations include University of Virginia, Uppsala University, FSB Corporation, France Telecom Corporation, Ericsson Corporation, and ABB Corporation. He holds a PhD in Theoretical Physics, an MSc in Engineering Physics, and an MSc in Education.

Book preview

Chapter 1

Introduction

The tensor calculus is a mathematical discipline of relatively recent origin. It is fair to say that, with few exceptions, the tensor calculus was developed during the twentieth century. It is also an area of mathematics that was developed for an immediate practical use in the theory of relativity, with which it is strongly interrelated. Later, however, the tensor calculus has proven to be useful in other areas of physics and engineering such as classical mechanics of particles and continuous media, differential geometry, electrodynamics, quantum mechanics, solid-state physics, and quantum field theory. Recently, it has been used even in electric circuit theory and some other purely engineering disciplines.

In the early twentieth century, at the same time when the tensor calculus was developed, a number of major breakthroughs in modern science were made. In 1905 the special theory of relativity was formulated, then in 1915 the general theory of relativity was developed, and in 1925 quantum mechanics took its present form. In the years to come quantum mechanics and the special theory of relativity were combined to develop the relativistic quantum field theory, which gives at least a partial explanation of the three fundamental forces of nature (strong, electromagnetic, and weak).

The remaining known fundamental force of nature, the force of gravity, is different from the other three fundamental forces. Although very weak on the small scale, gravity dominates the other three forces over cosmic distances. This dominance, due to gravity being a long-range force that cannot be screened, makes it the only available foundation for any cosmology. The other three fundamental forces are explained through particle interactions in the flat space-time of special relativity. However, gravity does not allow for such an explanation. In order to explain gravity, Einstein had to connect it with the geometry of space-time and formulate a relativistic theory of gravitation. For a long time, general relativity was separate from the other parts of physics, partly because of the mathematical framework of the theory (tensor calculus), which was not extensively used in any other discipline during that time.

The tensor calculus today is used in a number of other disciplines as well, and its extension to other areas of physics and engineering is a result of the simplification of the mathematical notation and in particular the possibility of natural extension of the equations to the relativistic case.

Today, physics and astronomy have joined forces to form the discipline called relativistic astrophysics. The major advances in cosmology, including the first attempts to formulate quantum cosmology, also increase the importance of general relativity. Finally, a number of attempts have been made to unify gravity with the other three fundamental forces of nature, thus introducing the tensor calculus and Riemannian geometry to the new exciting areas of physics such as the theory of superstrings.

In the first two parts of the book a pedagogical introduction to the tensor calculus is covered. Thereafter, an introduction to the special and general theories of relativity is presented. Finally an introduction to the modern theory of cosmology is discussed.

PART I

Tensor Algebra

Chapter 2

Notation and Systems of Numbers

2.1 Introduction and Basic Concepts

In order to get acquainted with the basic notation and concepts of the tensor calculus, it is convenient to use some well known concepts from linear algebra. The collection of N elements of a column matrix is often denoted by subscripts as x1, x2,…, xN. Using a lower index i = 1,2,…, N, we can introduce the following short-hand notation:

(2.1)

Sometimes, the same collection of N elements is denoted by corresponding superscripts as x¹, x²,…, xN. Using here an upper index i = 1,2,…, N, we can also introduce the following short-hand notation:

(2.2)

In general the choice of a lower or an upper index to denote the collection of N elements of a column matrix is fully arbitrary. However, it will be shown later that in the tensor calculus lower and upper indices are used to denote mathematical objects of different natures. Both types of indices are therefore essential for the development of tensor calculus as a mathematical discipline. In the definition (2.2) it should be noted that i is an upper index and not a power of x. Whenever there is a risk of confusion of an upper index and a power, such as when we want to write a square of xi, we will use parentheses as follows:

(2.3)

A collection of numbers, defined by just one (upper or lower) index, will be called a first-order system or a simple system. The individual elements of such a system will be called the elements or coordinates of the system. The introduction of the lower and upper indices provides a device to highlight the different nature of different first-order systems with equal numbers of elements. Consider, for example, the following linear form:

(2.4)

Introducing the labels ai = {a, b, c} and xi = {x, y, z}, the expression (2.4) can be written as

(2.5)

indicating the different nature of the two first-order systems. In order to emphasize the advantage of the proposed notation, let us consider a bilinear form created using two first-order systems xi and yi (i = 1,2,3).

(2.6)

Here we see that the short-hand notation on the right-hand side of Eq. (2.6) is quite compact. The system of parameters of the bilinear form

(2.7)

is labeled by two lower indices. This system has nine elements and they can be represented by the following 3 × 3 square matrix:

(2.8)

A system of quantities determined by two indices is called a second-order system.

Depending on whether the indices of a second-order system are upper or lower, there are three types of second-order systems:

(2.9)

A second-order system in N dimensions has N² elements. In a similar way we can define the third-order systems, which may be of one of four different types:

(2.10)

The most general system of order K is denoted by

(2.11)

and, depending on the position of the indices, it may be of one of several different types. The K th-order system in N dimensions has NK elements.

2.2 Symmetric and Antisymmetric Systems

Let us consider a second-order system in three dimensions

(2.12)

The system (2.12) is called a symmetric system with respect to the two lower indices if the elements of the system satisfy the