The weight of light

In 1913, Max Planck visited Einstein in Zurich with the aim of persuading him to move to Berlin. In conversation, Einstein remarked to Planck that he was working on a new theory of gravity. Planck's response was forthright, but concerned:

Planck was only partly right. Einstein succeeded and his theory of ‘general relativity’ was believed, but, for the most part, his theory had little relevance to mainstream physics. It was not until after Einstein's death in 1955 that the new technological advances of the 1960s rekindled interest in general relativity. Whereas most advances in understanding Nature could have been made by several scientists working at the time instead of the actual discoverer, this is probably not true of general relativity.

Fields of force

Three pictures hung on the wall of Einstein's study – portraits of Isaac Newton, Michael Faraday and James Clerk Maxwell. These three physicists provided the inspiration for Einstein's great works. With the tools provided by Faraday and Maxwell, Einstein eventually overturned Newton's conception of the universe and the very fabric of space and time which had proved itself so successful for over 200 years.

Newton pictured atoms of matter as having various powers of attraction and repulsion, gravity being the most famous such property. In Newton's scheme of things, the Earth attracts the Moon and all other bodies – such as the famous apple -by virtue of its mass. Newton discovered how this attractive force diminished with distance and, by applying his force law to the Sun and the planets of the solar system, he was able to explain the orbits of the planets.

Encouraged by the success of The Quantum Universe, we have tried to adopt a similarly pragmatic approach in this sister volume on Albert Einstein's relativity. Our goal is not only to present the essential ideas of both special and general relativity as simply as possible, but also to demonstrate how the predictions of these theories are verified by the results of experiments. Special relativity is concerned with uniform motion, and does away with Isaac Newton's notion of 'absolute time': it makes startling predictions for objects and observers moving at very high speeds. General relativity, on the other hand, is concerned with accelerations: it turns out to be a theory of gravity which has had a profound impact on our modem view of the universe.

Since our aim is to introduce as many people as possible to the strange world of relativity we have deliberately used a minimal amount of mathematics in the text. Some simple derivations requiring no more than high school maths have been relegated to an appendix for the curious. Needless to say, any book about both special and general relativity must also be to some extent about the physicist who almost single-handedly created these theories. Einstein's legacy is truly remarkable – both inside and outside of physics – and we hope to have captured some flavour of the man through the quotations and stories that accompany the text.

Time dilation

In chapter 3, we promised more details on the derivation of relativistic ‘time dilation'. We considered a thought experiment with a simple clock and two 'observers’ – one at rest relative to the clock and one in motion. We shall now see that the stationary observer sees the moving clock running slow.

Our ‘clock’ consists of a box with a mirror at either end. A light pulse bounces back and forth between the two mirrors, and the time taken for one round trip is taken to be one ‘tick’ of the clock. For an observer at rest relative to this clock, all is fine. Consider how things look for an observer moving in a direction at right angles to the length of the clock (Figure AI). She will see the light pulse start out, and she will see the clock with its two mirrors move away from her with a constant speed. According to her, the light must travel further than just twice the distance between the mirrors. The argument is very similar to that used in the Michelson and Morley experiment. If the distance between the mirrors is denoted by L, and the speed at which the clock moves away from our observer is written as v, we can relate the times measured by the two observers: TR for the observer at rest with respect to the mirrors, and TM for the observer in motion.

The strange behaviour of the velocity of light

As we have seen, Roemer showed as long ago as I 676 that the velocity of light was not infinite. Subsequent measurements by Michelson and others now agree on a value for the speed of light of some 299 792 kilometres/second. This applies not only to the visible part of the electromagnetic spectrum but also to much longer-wavelength radio waves and much shorter-wavelength gamma rays, as expected from Maxwell's equations. Now, according to Newton's laws of motion, there is nothing special about the speed of light. There is nothing, in principle, to stop one accelerating an object – or indeed oneself – to any speed whatsoever. It was the problem of what one would see in a mirror if both observer and mirror were moving at the speed of light that set Einstein on his path to relativity.

It is sometimes said that Einstein showed little exceptional talent when he was at school. This may be true, but it is certain that few schoolboys could have formulated the key paradox of the mirror at the age of sixteen.

The expanding universe

After his stunning success with general relativity, Einstein began to think about the implications of his theory for the universe as a whole. In 1917, he wrote a paper that began a new field of physics, that would be called ‘relativistic cosmology’. He wrote to his friend Paul Ehrenfest:

Although Einstein had the nerve to provide the first mathematical model of the universe, he had also, in a sense, lost his nerve at the crucial moment. Instead of predicting Hubble's discovery of the expansion of the universe, a solution that followed naturally from his own field equations, Einstein chose to modify gravity and introduce a new repulsive force.

Phlogiston and caloric

Before we look at Einstein's famous equation, we had better set the scene by describing how scientists had arrived at the concepts of energy and mass – the ‘E’ and ‘m’ in the equation. The gradual evolution of our present understanding of energy began with two ideas at least as curious as the infamous aether – ‘phlogiston’ and ‘caloric’. It provides us with some insight into the way that science progresses to look at why these two theories were invented and subsequently discarded.

Phlogiston was introduced towards the end of the seventeenth century by Georg Stahl, a German professor of medicine and chemistry, in an attempt to understand fire. Even in the latter part of the eighteenth century, many scientists still regarded fire as an element. Combustible materials were supposed to be made up of two parts -the calx, or ash, and the ‘phlogiston’. It was thought that, when a substance burned, the phlogiston was liberated, leaving the ash behind.

Prologue

In this chapter, we shall explore the application of special relativity to atomic physics. In order to make accurate quantitative predictions for the atomic structure underlying both physics and chemistry, it turns out to be essential to take into account ‘relativistic corrections’ to the standard quantum mechanical picture of the atom. Such applications of relativity form an important part of the experimental evidence supporting Einstein's theory, and to ignore them would give a distorted impression of its success. Thus, although this is a book about relativity, in order to appreciate this area of applied relativity, it is necessary to give a brief overview of our present understanding of atomic physics. A companion volume, The Quantum Universe, gives a fuller account of the development of quantum theory and its application to the modern world.

We begin our overview with an account of the great debate about the existence of atoms. In 1905, the same year that he published his paper on special relativity, the young Einstein made a crucial contribution to the debate with an atomic explanation of Brownian motion. At the same time as this debate was raging, the first discoveries about radioactivity were being made by Roentgen, Bequerel and the Curies. A major puzzle of the era was the origin of the energy released in radioactive decays.

Prologue

In this chapter, we explore me application of special relativity to nuclear physics. As in chapter 6, the reader need only read the following overview to gain an impression of the impact Einstein's theory has had on our understanding of nuclear structure and nuclear reactions. The story of Einstein's development of the theory of general relativity is taken up in chapter 8: the rest of this chapter is not essential for an understanding of the remainder of this book.

The story begins with Ernest Rutherford and Frederick Soddy quantifying the amount of energy released in radioactive decays and their joint realization that such a huge energy source could be a mixed blessing for humanity. Francis Aston, working in Cambridge with Rutherford, invented the ‘mass spectrograph’ and was able to separate different ‘isotopes’ of many elements. There was much confusion about the nature of ‘isotopes’ until James Chadwick discovered the neutron in 1932. Aston had also realized that the neutrons and protons when bound in the nucleus weigh less than in their free state. The difference arises from the nuclear binding energy, which results from the strong nuclear forces holding the nucleus together. This is a direct example of Einstein's mass-energy relation.

The scope of this review

Ever since the 1930s, it has been conventional wisdom in cosmology that the Friedmann (1922, 1924)–Lemaître (1927, 1931)–Robertson (1929, 1933)–Walker (1935) (FLRW) models describe the large-scale properties of our observed Universe faithfully. At the same time, it has been conventional wisdom in relativity theory that finding exact solutions of the Einstein equations is extremely difficult and possible only for exceptionally simple cases. Both these views were challenged repeatedly by lone rebels, but a few generations of physicists and astronomers have been educated with these conventional wisdoms solidly incorporated into their minds. As a result of this situation, a large body of literature has come into existence in which exact solutions generalizing FLRW have been derived and applied to the description of our observed Universe, but most of it remains unknown to the physics community and is not being introduced into textbooks. This book is intended to achieve the following two objectives:

1. To list all the independently derived cosmological solutions of the Einstein equations and to reveal all the interconnections between them.

2. To compile an encyclopaedia of physics in an inhomogeneous Universe by gathering together all physical conclusions drawn from such solutions.

An exact solution of the Einstein equations is termed “cosmological” if it can reproduce a FLRW metric when its arbitrary constants or functions assume certain values or limits. This requirement will be discussed in Section 1.2. The solutions are organized into a few families.