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The textbook begins with a comprehensive, but simple, review of special relativity, creating a framework from which to launch the ideas of general relativity.
After describing the basic theory, it moves on to describe important applications to astrophysics, black-hole physics, and cosmology. Several worked examples, and numerous figures and images, help stu- dents appreciate the underlying concepts. The textbook presents all the necessary information and discus- sion for an elementary approach to relativity.
Password-protected solu- tions to the exercises are available to instructors at www. He is author of An Introduc- tion to Cosmology, now in its third edition Cambridge University Press, , and has been active in teaching and researching cosmology, theoret- ical astrophysics, gravitation and relativity for nearly five decades.
An Introduction to Relativity Jayant V. Narlikar This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. Includes bibliographical references and index.
ISBN hardback 1. General relativity Physics I. N The book was well received and had been in use for about 15—20 years until it went out of print. The present book has been written in response to requests from students as well as teachers of relativity who have missed the earlier text. An Introduction to Relativity is therefore a fresh rewrite of the text, updated and perhaps a little enlarged.
As I did for the earlier text, I have adopted a simple style, keeping in view a mathematics or physics undergraduate as the prospective reader. The topics covered are what I consider as essential features of the theory of relativity that a beginner ought to know. A more advanced text would be more exhaustive. I have come across texts whose formal and rigorous style or enormous size have been off-putting to a student wishing to know the A, B, C of the subject.
I am sure the readers of this book will be in a position to read and appreciate those topics after they have completed this preliminary introduction. Cambridge University Press published my book An Introduction to Cosmology, which was written with a similar view and has been well received. Although the present book contains chapters on cosmology, they are necessarily brief and highlight the role of general relativity. The reader may find it useful to treat the cosmology volume as a companion volume.
It is a pleasure to acknowledge the encouragement received from Simon Mitton for writing this book. I also thank Vince Higgs, Lindsay Barnes, Laura Clark and their colleagues at Cambridge University Press for their advice and assistance in preparing the manuscript for publica- tion. I do hope that teach- ers and students of relativity will appreciate this rather unpretentious offering!
Jayant V. These were the Brow- nian motion in fluids, the photoelectric effect and the special theory of relativity.
Each of these was of a basic nature and also had a wide impact on physics. In this chapter we will be concerned with special relativity, which was arguably the most fundamental of the above three ideas. This feeling emerged towards the end of the eighteenth century, when Newto- nian laws of motion and gravitation, the studies in optics and acoustics, etc.
The nine- teenth century saw the development of thermodynamics, the growth in understanding of electrodynamics, wave motion, etc. So the feeling again grew that the end of physics was nigh. As we know, the twentieth century saw the emergence of two theories, fundamental but totally unexpected by the stalwarts of the nineteenth century, viz.
Finally, the success of the attempt to unify electromagnetism with the weak interaction led many twentieth-century physicists to announce that the end of physics was not far off.
That hope has not materialized even though the twenty-first century has begun. Although Newton had wrongly resisted the notion that light travels as a wave, during the nineteenth century the concept of light travelling as a wave had become experimentally established through such phenomena as interference, diffraction and polarization.
However, this understanding raised the next question: in what medium do these waves travel? For, conditioned by the mechanistic thinking of the Newtonian paradigm, physicists needed a medium whose distur- bance would lead to the wave phenomenon. Water waves travel in water, sound waves propagate in a fluid, elastic waves move through an elastic substance Indeed, many unsuccessful attempts were made to detect it.
The most important such experiment was conducted by Michelson and Morley. Michelson and E. Morley in can be understood by invoking the example of a person rowing a boat in a river. Figure 1. A boatman who can row his boat at speed c in still water is trying to row along and across the river in different directions.
In Figure 1. Likewise see Figure 1. What is his speed when he rows across the river in the perpendicular direction as shown in Figure 1. Clearly he must row in an oblique direction so that his velocity has a component v in a direction opposite to the current. Suppose now that he does this experiment of rowing down the river a distance d and back the same distance and then rows the same distance perpendicular to the current and back.
Light from a source S is made to pass through an inclined glass plate cum mirror P. Part of the light from the source passes through the transparent part of the plate and, travelling a distance d1 , falls on a plane mirror A, where it is reflected back. It then passes on to plate P and, getting reflected by the mirror part, it moves towards the viewing telescope.
A second ray from the source first gets reflected by the mirror part of the plate P and then, after travelling a distance d2 , gets reflected again at the second mirror B. From there it passes through P and gets into the viewing telescope.
Now consider the apparatus set up so that the first path length d1 is in the E—W direction. In the actual experiment the apparatus was turned by a right angle so that the E—W and N—S directions of the arms were interchanged.
The experiment was repeated several times. In the case that the Earth was at rest relative to the aether at the time of the experiment, six months later its velocity would be maximum relative to the aether. But an experment performed six months later also gave a null result. The Michelson—Morley experiment generated a lot of discussion. Did it imply that there was no medium like aether present after all?
Physicists not prepared to accept this radical conclusion came up with novel ideas to explain the null result. Their conclusion as summarized by J. Larmour in a contemporary pre-relativity text on electromagnetic theory reads as follows:. It is clear that a factor of this kind would resolve the problem posed by the Michelson—Morley experiment. For, by reducing the length trav- elled in the E—W direction by the above factor, we arrive at the same time of travel for both directions and hence a null result.
Lorentz went further to give an elaborate physical theory to explain why the Fitzgerald contraction takes place.
The Michelson—Morley experiment was explained much more ele- gantly when Einstein proposed his special theory of relativity. We will return to this point after decribing what ideas led Einstein to propose the theory.
As we will see, the Michelson—Morley experiment played no role whatsoever in leading him to relativity. An elegant conclusion derived from them was that the electromagnetic fields propagated in space with the speed of light, which we shall henceforth denote by c. It was how this fundamental speed should transform, when seen by two observers in uniform relative motion, that led to the conceptual problems.
The Newtonian dynamics, with all its successes on the Earth and in the Cosmos, relied on what is known as the Galilean transformation of space and time as measured by two inertial observers. Let us clarify this notion further. Laws of physics were expected to be invariant relative to such frames of reference. Indeed, we may state a general expectation that the basic laws of physics should turn out to be invariant under the Galilean transformations. This may be called the principle of relativity.
Paving the way to a mechanistic philosophy, Newtonian dynamics nurtured the belief that the basic laws of physics will turn out to be mechanics-based and as such the Galilean transformation would play a key role in them. But then we run foul of the principle of relativity: that the basic laws of physics are invariant under Galilean transformations. This was the problem Einstein worried about and to exacerbate it he took up the imaginary example of an observer travelling with the speed of the wave.
What would such an observer see? Let us look at the equations from a Galilean standpoint first. Indeed, if we want the equations to be invariant for all inertial observers, then we need, for example, the speed of light to be invariant for them, as seen from the above example of the wave equation. Can we think of some other transformation that will guarantee the above invariances? In particular, let us ask this question: what is the simplest modifi- cation we can make to the Galilean transformation in order to preserve 1 In this book, as a rule, we will choose units such that the speed of light is unity when measured in them.
Then from 1. A more elaborate algebra will also show that the Maxwell equations are also invariant under the above transformation.
An Introduction to Relativity
Jayant V. He is author of An Introduction to Cosmology, now in its third edition Cambridge University Press, , and has been active in teaching and researching cosmology, theoretical astrophysics, gravitation and relativity for nearly five decades. An Introduction to Relativity. General relativity is now an essential part of undergraduate and graduate courses in physics, astrophysics and applied mathematics.
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