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.

This appendix contains a list of papers which, in the opinion of this author, played a crucial role in the development of inhomogeneous cosmological models. It must be stressed that, except for a few, the papers listed below have never been properly appreciated, and many of them are virtually unknown even today. The list is thus a call for historical justice (based on a personal assessment by this author) rather than a presentation of development of the field.

Lemaître (1933a) – the pioneering paper, and probably the most underappreciated one. The author introduced the Lemaître–Tolman model, and in addition presented or solved a few problems commonly associated now with names and papers younger by a whole generation. Examples: the definition of mass for a spherically symmetric perfect fluid, a proof that the Schwarzschild horizon is not a singularity (by a coordinate transformation to a system of freely falling observers), a preliminary statement of a singularity theorem illustrated by a Bianchi I model.

McVittie (1933) – presented a superposition of the Schwarzschild and FLRW metrics which is a perfect fluid solution. A remarkably bold and early entry, but the solution has still not been satisfactorily interpreted.

Dingle (1933) – a preliminary investigation of spherically symmetric shearfree perfect fluid solutions, later completed by Kustaanheimo and Qvist (1948). The paper is remarkable for the author's strong criticism of the cosmological principle and an explicit call for inhomogeneous models (see Appendix C).

Every review article or book raises the obvious question about its completeness. In order to give the readers an idea about the degree of completeness of this review, the method used to compile the bibliography is described briefly below. These were the essential steps:

1. The author has been interested in the subject since about 1980. Until 1988, when systematic compilation was begun, I studied every newly published research or review article on inhomogeneous cosmological models, and followed each reference whose context of citation suggested that it might contain more of the relevant material. The latest publications included are those that reached my hands in September 1994.

2. I looked through the subject indexes to all volumes of Physics Abstracts, beginning with the 1915 volume, and studied the sections on cosmology, general relativity, gravitation, gravitational collapse and spacetime configurations. Whenever any keyword of title or abstract suggested that the paper might be relevant for the review, I added the reference to the list of papers to look up. The last index so surveyed was Part I of the 1994 volume.

3. While reading or looking through the papers, I added every reference that seemed relevant to the list.

Stage 3 produced more than 1000 references in addition to about 1000 found in stage 2 (but about two-thirds of the total number of papers were discarded according to the criteria listed in Section 1.1).