Dynamics and Relativity

Jeffrey Forshaw, Gavin Smith

• Publisher: Wiley
• Year: 2014
• ISBN:
  9781118933299
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Mathematical & Computational Physics

A new title in the Manchester Physics Series, this introductory text emphasises physical principles behind classical mechanics and relativity. It assumes little in the way of prior knowledge, introducing relevant mathematics and carefully developing it within a physics context. Designed to provide a logical development of the subject, the book is divided into four sections, introductory material on dynamics, and special relativity, which is then followed by more advanced coverage of dynamics and special relativity. Each chapter includes problems ranging in difficulty from simple to challenging withsolutions for solving problems.

• Includessolutions for solving problems
• Numerous worked examples included throughout the book
• Mathematics is carefully explained and developed within a physics environment
• Sensitive to topics that can appear daunting or confusing

Part I

Introductory Dynamics

1

Space, Time and Motion

1.1 DEFINING SPACE AND TIME

If there is one part of physics that underpins all others, it is the study of motion. The accurate description of the paths of celestial objects, of planets and moons, is historically the most celebrated success of a classical mechanics underpinned by Newton’s laws¹. The range of applicability of these laws is vast, encompassing a scale that extends from the astronomical to the microscopic. We have come to understand that many phenomena not previously associated with motion are in fact linked to the movement of microscopic objects. The absorption and emission spectra of atoms and molecules arise as a result of transitions made by their constituent electrons, and the random motion of ensembles of atoms and molecules forms the basis for the modern statistical description of thermodynamics. Although atomic and subatomic objects are properly described using quantum mechanics, an understanding of the principles of classical mechanics is essential in making the conceptual leap from continuous classical systems with which we are most familiar, to the discretised quantum mechanical systems, which often behave in a manner at odds with our intuition. Indeed, the calculational techniques that are routinely used in quantum mechanics have their roots in the classical mechanics of particles and waves; a close familiarity with their use in classical systems is an asset when facing problems of an inherently quantum mechanical nature.

As we shall see in the second part of this book, when objects move at speeds approaching the speed of light classical notions about the nature of space and time fail us. As a result, the classical mechanics of Newton should be viewed as a low-velocity approximation to the more accurate relativistic theory of Einstein². To look carefully at the differences between relativistic and non-relativistic theories forces us to recognise that our intuitive ideas about how things move are often incorrect. At the most fundamental level, mechanics of either the classical or the quantum kind, in either the relativistic or non-relativistic limit, is a study of motion and to study motion is to ask some fundamental questions about the nature of space and time. In this book we will draw out explicitly the different underlying structures of space and time used in the approaches of Newton and Einstein.

1.1.1 Space and the classical particle

We all have strong intuitive ideas about space, time and motion and it is precisely because of this familiarity that we must take special care in our attempts to define these fundamental concepts, so as not to carry too many unrecognised assumptions along with us as we develop the physics. So let us start by picking apart what we mean by position. We can usually agree what it means for London to be further away than Inverness and we all know that in order to go to London from Inverness we must also know the direction in which to travel. It may also seem to be fairly uncontentious that an object, such as London, has a position that can be specified, i.e. it is assumed that given enough information there will be no ambiguity about where it is. Although this seems reasonable, there is immediately a problem: day-to-day objects such as tennis balls and cities have finite size; there are a number of ‘positions’ for a given object that describe different parts of the object. Having directions to London may not be enough to find Kings Cross station, and having directions to Kings Cross station may not be enough to find platform number nine. To unambiguously give the position of an object is therefore only possible if the object is very small – vanishingly small, in fact. This hypothetical, vanishingly small object is called a particle. It might be suggested that with the discovery of the substructure of the atom, true particles, with mass but no spatial extent, have been identified. However, at this level, the situation becomes complicated by quantum uncertainty which makes the simultaneous specification of position and momentum impossible. The classical particle is therefore an idealisation, a limit in which the size of an object tends to zero but in which we ignore quantum phenomena. Later we shall see that it is possible to define a point called the centre of mass of an extended object and that this point behaves much like a classical particle. The collection of all possible positions for a particle forms what we call space.

The mathematical object possessing the properties we require for the description of position is called the vector. A vector has both magnitude and direction and we must be careful to distinguish it from a pure number which has a magnitude, but no directional properties. The paradigm for the vector comes from the displacement of a particle from point A to point B as shown in Figure 1.1. The displacement from A to B is represented by the directed-line-segment AB. We can imagine specifying the displacement as, for example, “start at A and move 3 km to the northeast” or “start at A and go 1 parsec in the direction of Alpha Centuri”. Once we have specified a displacement between the two points A and B we can imagine sliding each end of the line segment in space until it connects another two points C and D. To do this, we move each end through the same distance and in the same direction, an operation that is known as parallel transport. Now the displacement is denoted CD but its direction and magnitude are the same. It should be clear that there is an infinity of such displacements that may be obtained by parallel transport of the directed line segment. The displacement vector a has the magnitude and direction common to this infinite set of displacements but is not associated with a particular position in space. This is an important point which sometimes causes confusion since vectors are illustrated as directed line segments, which appear to have a well defined beginning and an end in space: A vector has magnitude and direction but not location. The position of a particle in space may be given generally by a position vector r only in conjunction with a fixed point of origin.

Figure 1.1 Displacement of a particle from point A to point B is illustrated by the directed line segment AB. Parallel transport of this line gives the displacement from point C to point D. The displacement vector a is not associated with any particular starting point.

Now, all of this assumes that we understand what it means for lines to be parallel. At this point we assume that we are working in Euclidean space, which means that parallel lines remain equidistant everywhere, i.e. they never intersect. In non-Eucl...

Table of contents

• Cover Page
• Contents
• Series page
• Title Page
• Copyright Page
• Dedication
• Editors’ Preface to the Manchester Physics Series
• Authors’ Preface
• Part I Introductory Dynamics
• Part II Introductory Special Relativity
• Part III Advanced Dynamics
• Part IV Advanced Special Relativity
• Appendix A Deriving the Geodesic Equation
• Appendix B Solutions to Problems
• Index