Gourgoulhon - Special Relativity in General Frames From Particles to Astrophysics

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ric Gourgoulhon Graduate Texts in Physics Special Relativity in General Frames 2013 From Particles to Astrophysics 10.1007/978-3-642-37276-6_1

Springer-Verlag Berlin Heidelberg 2013

1. Minkowski Spacetime

ric Gourgoulhon 1

(1)

Laboratoire Univers et Thories, Observatoire de Paris, CNRS, Universit Paris Diderot, Meudon, France

Abstract

The concept of Minkowski spacetime is introduced as the mathematical framework to expose special relativity. After discussing the general concept of a four-dimensional spacetime, the fundamental structure that characterizes relativistic physics is introduced, namely, the metric tensor. Various related basic concepts are defined, among which vector orthogonality, orthonormal bases, spacetime diagrams, null cone and time arrow. The spacetime orientation is presented via the LeviCivita tensor. Finally the duality between vectors and linear forms induced by the metric tensor and used throughout the book is discussed.

1.1 Introduction

This first chapter is purely mathematical: there is no direct mention of physical objects. The aim is to set the geometrical framework for special relativity, i.e. to introduce Minkowski spacetime . Later on, when dealing with physics, the outcomes of measurements will be modelled as mathematical operations in that space, such as scalar products.

Let us point out that the mathematics required for the foundations of special relativity are rather elementary. They involve linear algebra at the level of the first two years of university. For the benefit of the reader, the definitions of the basic algebraic structures are recalled in Appendix A.

1.2 The Four Dimensions

1.2.1 Spacetime as an Affine Space

Relativity has performed the fusion of space and time , two entirely distinct concepts in Galilean mechanics. Four numbers are required to determine an event in the space-and-time continuum: three for its spatial position (for instance its Cartesian coordinates ( x , y , z ) or the spherical ones ) and one for its date. The general mathematical structure corresponding to such a four-dimensional continuum is a manifold . Without entering into technical details, let us say that, given an integer , a manifold of dimension n is a set that locally resembles (in the present case n =4), but may differ from at a global scale. Regarding the dimension n =2, standard examples of manifolds are the plane, the cylinder, the sphere and the torus.

As far as special relativity is concerned, the chosen manifold is the simplest that one could imagine, namely, an affine space of dimension 4. We are familiar with the structure of affine space of dimension 3. It involves the notion of points that can be joined two by two by vectors . More precisely (cf. Fig.

(1.1)

that obeys the following two properties:

• For any point , the function

  

  (1.2)

  is bijective.
• For any triplet ( A , B , C ) of elements of , Chasles relation holds:

  

  (1.3)

Fig. 1.1

Affine space and the underlying vector space E (for graphical purposes, the dimension of is reduced to 2, whereas the actual dimension of spacetime is 4)

The elements of are called points and E is called the vector space underlying .

Example 1.1.

An affine space of dimension 1 is a straight line and an affine space of dimension 2 is a plane. Still in dimension 2, a counterexample is a sphere.

Choosing for spacetime a structure as simple as an affine space is sufficient to treat electromagnetism, hydrodynamics and relativistic quantum field theory. On the other side, it does not allow one to incorporate gravitation into relativity in a satisfactory manner. We will see in Chap.).

Accordingly, in what follows, we shall call spacetime , and denote by , an affine space of dimension 4 on . We shall note E the underlying vector space, which is isomorphic to . The elements of are called events and those of E are called vectors , or four-vectors , abridged as 4-vectors .

Remark 1.1.

The term four-vector or 4-vector introduced by the physicist stands for nothing but a vector for the mathematician, that is to say the element of a vector space ( E in the present case). The prefix 4- simply recalls that such a vector belongs to a vector space of dimension 4 on . These vectors are hence distinguished from the vectors of three-dimensional vector spaces usually manipulated by the non-relativist physicist. Since in this book the framework is four-dimensional from the very beginning, we shall not use the word 4-vector and shall refer to the elements of E simply as vectors .

1.2.2 A Few Notations

Vectors in E are denoted by boldface characters with an arrow above them, for instance: , and . The components of a vector with respect to a basis of E are denoted with an index placed at the top right of the vector symbol, ranging from 0 to 3 (and not from 1 to 4). Hence if

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