Principles of Cosmology and Gravitation

Michael V Berry

• Publisher: CRC Press
• Year: 2017
• eBook ISBN: 9781351421904
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Astronomy & Astrophysics

General relativity and quantum mechanics have become the two central pillars of theoretical physics. Moreover, general relativity has important applications in astrophysics and high-energy particle physics. Covering the fundamentals of the subject, Principles of Cosmology and Gravitation describes the universe as revealed by observations and presents a theoretical framework to enable important cosmological formulae to be derived and numerical calculations performed. Avoiding elaborate formal discussions, the book presents a practical approach that focuses on the general theory of relativity. It examines different evolutionary models and the gravitational effects of massive bodies. The book also includes a large number of worked examples and problems, half with solutions.

1	Introduction

It is customary to start with definitions, which are often all too glib and rob a subject of its richness. Nevertheless it can help to fix our ideas if we do have a definition, providing we do not take it too seriously. According to Chambers’s Dictionary, cosmology is ‘the science of the universe as a whole’. Again, it is customary to divide a subject into neatly-separated sections, even though this can obscure the richness of their interconnections. Nevertheless, it does help to distinguish three main aspects of cosmology.

Firstly, we have cosmography: cataloguing the objects in the universe and charting their positions and motions. Unlike geographers, we are restricted to one vantage-point – the Earth – where we sit and receive electromagnetic radiation. All our information about the universe is contained in the directional distribution of this radiation (a star here, a galaxy there) and in its spectral composition (light, X-ray, radio, etc.). By comparison, we have learned very little from the analysis of cosmic rays and meteorites (objects falling in from space), or from our first toddling steps outside the Earth.

Secondly, there is theoretical cosmology, where we search for a frame-work within which to comprehend the information from cosmography. Even here the tidy scheme breaks down, because it is not possible to discuss even the simplest observations without a theoretical framework – for example, ‘the distance of an object’ can have at least five different meanings, depending on how it is measured. Theoretical cosmology employs the physical laws established on and near the Earth, and makes the outrageous extrapolation that they apply throughout the universe. But physics, even extrapolated, is not enough; to escape from the prison of our single vantage-point we need something more: a ‘cosmological principle’. This is essentially philosophical in nature; it does not follow from the laws of physics. In simple terms, the cosmological principle says: ‘There is nothing special, cosmologically, about the Earth; therefore our large-scale observations are the same as those which would be made by observers anywhere else in the universe’. How fickle our intellect is! To mediaeval man it was completely natural to consider the Earth as being at the centre of the universe, yet here we are, a mere few centuries later, elevating anti-anthropocentrism to the level of a basic principle. The cosmological principle is immensely powerful: it enables us to select from all the complicated solutions of the equations of physics those which have certain simple symmetries.

What kind of physics does cosmology require? I am afraid it is a pretty rich stew, whose basic ingredient is a theory of gravitation, since that is the dominant force on the cosmic scale. The best description of gravity that we have is Einstein’s ‘general theory of relativity’, and this will form the core of our account of cosmology. To flavour the stew there will be a bit of electromagnetism, some thermodynamics, and even a dash of particle physics. The proof of any stew lies in the eating, which in this case means comparison with observation. We shall see that theoretical cosmology based on general relativity is capable of explaining the observations. However, these are not yet precise enough or extensive enough to indicate which of a range of ‘universe models’ applies to our actual world.

The third aspect of cosmology is cosmogony; this is the study of the origin (or perhaps the infinitely distant past) of the universe. Here our arrogance will be extreme, for we shall extrapolate the laws of physics to the most distant times as well as places. We shall find ourselves interpreting the most modern radio-astronomical observations as giving detailed information about conditions in the chaos of a ‘big bang’ ten thousand million years ago. Distinguishing the distant past from the distant future involves the nature of time itself, especially its reversibility, and this leads to connections – still mysterious – between cosmology and laboratory physics.

2	Cosmography

2.1 What the universe contains

On the very largest scale, the universe consists of clusters of galaxies. About 10¹¹ galaxies can be seen in the largest optical telescopes. From a cosmological point of view galaxies are the ‘atoms’ of the universe, and their distribution, motion and origin must be determined and explained. However, galaxies are of course complicated gravitationally-bound structures, each consisting of up to 10¹¹ stars as well as gas clouds. These are often distributed throughout a disc-shaped region with a central core and spiral arms; however, many other forms are possible (see the frontispiece), and these are beautifully illustrated in the Hubble atlas of galaxies (see the bibliography). Each of the component stars is a nuclear powerhouse whose detailed behaviour is the concern of astrophysics. Our Sun is a typical star, situated about halfway out towards the edge of the disc of our galaxy. The projection of the plane of our galaxy can be seen in the sky as the faint white stripe called the ‘Milky Way’. The Earth is one of a few planets gravitationally bound to the Sun. It is not known what fraction of stars have planetary systems, because no planets have been observed outside our solar system (although several have been inferred) and because the mechanism of planet formation is not understood. However, even if the mechanism relies on an improbable event like the near-collision of two stars (although this is now thought unlikely) the vast numbers of stars make it virtually certain that there are many planets in the universe not too dissimilar from our own. Therefore we can hardly claim our vantage-point to be exceptional; on the contrary, it is likely to be typical.

What about distances? It used to be common to illustrate the immensity and emptiness of space with imaginary scale models. For example, if the Sun were a watermelon in Piccadilly Circus, the Earth would be a grape pip a hundred metres away, and the nearest star would be another watermelon in Australia. Our imagination rapidly loses its grip on such models, and in this preliminary survey of cosmic distances we instead make use of light time. The speed of light in vacuo is

Table 1

Mean Earth–Moon distance ≡ rℂ = 3.84 × 10^8m = 1.28 light-seconds

Mean Earth–Sun distance ≡ r⨀ = 1.496 × 10^11 m = 8.3 light-minutes 9.46 × 10^15 m = 1 light-year = 0.307 pc 3.26 light years = 1 pc

Distance to nearest star ≈ 4 light years ≈ 1.2 pc

Diameter of our galaxy ≈ 10^5 light-years ≈ 3 × 10^4 pc

Distance of nearest large galaxy ≈ 2 × 10^6 light-years ≈ 6 × 10^5 pc

Distance of farthest galaxy seen optically ~ 3 × 10^9 light-years ≈ 10^9 pc

$▶c=2.998\times {10}^8\text{m}{\text{s}}^{\text{–1}}.$

(2.1.1)

Light travels seven times round the Earth in one second, so that the circumference of the Earth is about ‘one-seventh of a light-second’. section 2.2.1).

There may well be other matter in the universe besides the galaxies we see, for instance galaxies that have ceased to radiate, black holes of all sizes (see section 5.6), and intergalactic dust and gas, but firm experimental evidence for this is lacking. However, a whole range of exotic astronomical objects has been discovered, most puzzling being the quasistellar objects (QSO’s); these appear to be as distant and as bright (both optically and at radio frequencies) as galaxies, but they are very compact – at most a few thousand light-yea...

Table of contents

• Cover
• Title Page
• Copyright Page
• Table of Contents
• Preface to the 1989 reprint
• Preface to the first edition
• 1 Introduction
• 2 Cosmography
• 3 Physical basis of general relativity
• 4 Curved spacetime and the physical mathematics of general relativity
• 5 General relativity near massive objects
• 6 Cosmic kinematics
• 7 Cosmic dynamics
• 8 In the beginning
• Appendix A: Labelling astronomical objects
• Appendix B: Theorema egregium
• Problems
• Solutions to odd-numbered problems
• Useful numbers
• Bibliography
• Index