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Having acquainted ourselves with the trials and tribulations of working in non-Euclidean spacetimes we are now prepared for the next step, that of describing physics in such curved spacetimes. For we recall from Chapter 2 that the Einstein programme for general relativity consists of replacing the Newtonian perception of gravitation as a force by the notion that its effect makes the geometry of spacetime ‘suitably non-Euclidean’. What we mean by ‘suitably’ will be clear in the next two chapters. But given that the geometry is non-Euclidean we first need to know how the rest of physics is described in it.
For example, how do we describe the motion of a particle under a non-gravitational force? How do we write Maxwell's equations? What is the role of energy-momentum tensors? Can a dynamical action principle be written in curved spacetime? Such questions need our attention before we turn to the basic issue of how gravity actually leads to curved spacetime.
To this end we will introduce a concept that Einstein took as a basic principle in formulating general relativity. It is known as the principle of equivalence.
The principle of equivalence
Let us go back to the purely mathematical result embodied in the relations shown in Section 4.6 and attempt to describe their physical meaning. These relations tell us that special (locally inertial) coordinates that behave like the coordinates (t, x, y, z) of special relativity exist in the neighbourhood of any point P in spacetime.
The special theory of relativity reviewed in the last chapter marked a major advance in physics. The basic assumption that the fundamental laws of physics are invariant for all inertial observers looks at first sight a reasonable premise. However, as we saw in Chapter 1, its application to Maxwell's equations of electromagnetic theory led to a drastic revision of how such observers make and relate their measurements of space and time. One consequence was that the Newtonian notions of absolute space and absolute time had to be abandoned and replaced by a unified entity of spacetime. The Galilean transformation relating the space and time measurements of two inertial observers had to be replaced by the Lorentz transformation. Strange and non-intuitive though the consequences of this transformation were, as we saw in Chapter 1, several experiments confirmed them.
In spite of these successes, Einstein felt that the special theory addressed limited issues. For example, what was the nature of physical laws when viewed not in the inertial frames of reference, but in an accelerated one? Was there some more general principle that, when applied to these laws, preserved their form? Intuitively Einstein felt that some such situation must prevail. But that required a formalism more general than that provided by the Lorentz transformation.
On another matter, of the two classical theories of physics known in the first decade of the twentieth century, the electromagnetic theory had played a major role in the genesis of special relativity.
General relativity is now an essential part of undergraduate and graduate courses in physics, astrophysics and applied mathematics. This simple, user-friendly introduction to relativity is ideal for a first course in the subject. Beginning with a comprehensive but simple review of special relativity, the book creates 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 students appreciate the underlying concepts. There are also 180 exercises which test and develop students' understanding of the subject. The textbook presents all the necessary information and discussion for an elementary approach to relativity. Password-protected solutions to the exercises are available to instructors at www.cambridge.org/9780521735612.
We begin this chapter by introducing the idea of a field in physics (to be distinguished from the ‘field’ in algebra that mathematicians talk about). The idea was popularized by Michael Faraday in the context of the electric and magnetic fields. Figure 4.1 shows what happens when iron filings are sprinkled in the vicinity of a bar magnet. The filings get distributed in a pattern somewhat like that in this figure. Faraday called these curves lines of force. If we imagine a magnetic pole placed anywhere on one of these lines, it will move along that line, being guided by the magnetic force on it. The lines of force therefore represent the ‘magnetic field’ B both in strength and direction at any point in the vicinity of the magnet. In short the magnet generates a ‘field’ of B vectors all around it, representing the force exerted by it on another magnetic pole.
We generalize the concept of a vector field by defining a vector function of spacetime variables, so that at each point a vector is defined. This idea may further be generalized by having tensor fields as functions of spacetime coordinates. Thus we could argue that equations of physics involve fields related by partial differential equations. If we additionally require that these equations do not change their form under changes of coordinates (thereby being the same for all observers), then they should be represented by tensor fields.
Do Einstein's equations permit the existence of gravitational waves? As in the case of electromagnetism, where Maxwell's field equations led to the important deduction of electromagnetic waves carrying energy and momentum with the speed of light, one expects the relativistic equations to imply the existence of gravitational waves that do the same. However, several issues intervene to make the answer to our question nontrivial.
The first problem is posed by the non-linearity of the Einstein field equations. In the wave motion discussed in electromagnetic theory, acoustics, elastic media, etc. the basic equations are linear and a superposition principle holds. There is no corresponding situation in general relativity. Secondly, there is no corresponding vector or tensor in relativity that plays the role of the Poynting vector in the transport of electromagnetic energy.
A third difficulty arises from the general covariance of the field equations. With the facility available to use any coordinate system as per convenience, it is not clear whether a particular ‘wavelike’ solution is a real physical effect or a pure coordinate effect. Thus one has to be on guard against solutions that describe coordinate waves that may travel ‘with the speed of thought’.
Even during Einstein's lifetime, the above question did not receive an unequivocal answer.
1905 is often described as Einstein's annus mirabilis: a wonderful year in which he came up with three remarkable ideas. These were the Brownian 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.
It is perhaps a remarkable circumstance that, ever since the initiation of modern science with the works of Galileo, Kepler and Newton, there has emerged a feeling towards the end of each century that the end of physics is near: that is, most in-depth fundamental discoveries have been made and only detailed ‘scratching at the surface’ remains. This feeling emerged towards the end of the eighteenth century, when Newtonian laws of motion and gravitation, the studies in optics and acoustics, etc. had provided explanations of most observed phenomena. The nineteenth century saw the development of thermodynamics, the growth in understanding of electrodynamics, wave motion, etc., none of which had been expected in the previous century. 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., relativity and quantum theory.
In 1915 Einstein put the finishing touches to the general theory of relativity. The Schwarzschild solution described in Chapter 9 was the first physically significant solution of the field equations of general relativity. It showed how spacetime is curved around a spherically symmetric distribution of matter. The problem solved by Schwarzschild was basically a local problem, in the sense that the deviations of spacetime geometry from the Minkowski geometry of special relativity gradually diminish to zero as we move further and further away from the gravitating sphere. This result can be easily verified from the Schwarzschild line element by letting the radial coordinate go to infinity. In technical jargon a spacetime satisfying this property is called asymptotically flat. In general any spacetime geometry generated by a local distribution of matter is expected to have this property. Even from Newtonian gravity we expect an analogous result: that the gravitational field of a local distribution of matter will die away at a large distance from the distribution. Can the Universe be approximated by a local distribution of matter?
Einstein rightly felt that the answer to the above question would be in the negative. Rather, he expected the Universe to be filled with matter, howsoever far we are able to probe it. A Schwarzschild-type solution cannot therefore provide the correct spacetime geometry of such a distribution of matter. Since we can never get away from gravitating matter, the concept of asymptotic flatness must break down.
In 1978 I wrote an introductory textbook on general relativity and cosmology, based on my lectures delivered to university audiences. 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 1978 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.
Thus I offer no apology to a critic who may find the book lacking in some of his/her favourite topics. 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.
The work covered in Chapter 14 did not tell us two important items of information about the Universe: (1) the rate at which it expands as given by the function S(t); and (2) whether its spatial sections t = constant are open or closed as indicated by the parameter k. To find answers to these questions, it is necessary to go beyond the Weyl postulate and the cosmological principle. We require a dynamical theory that tells us how the scale factor and curvature are determined by the matter/radiation contents of the universe.
A comparison of Newton's law of gravitation with the general theory of relativity shows the latter as enjoying advantages both on the theoretical and on the observational front. General relativity gets round the criticism of Newtonian gravity of violating the light-speed limit. It allows for the permanence of gravitation by identifying its effect with the curvature of spacetime. Observationally it performs better vis-à-vis the Solar-System tests and explains the shrinking of binaries through gravitational radiation. It therefore generates greater confidence than Newton's approach does, especially for use in cosmology, where strong gravitational fields are likely to be involved and where distances are so large that the assumption of instantaneous action at a distance would be misleading. Hence we will adopt general relativity as the underlying theory for constructing models of the Universe.
We will now undertake that exercise by constructing the models which Friedmann in 1922–4 and Lemaître in 1927 came up with before Hubble's results became known.
Figure 5.1 repeats the previous example of non-Euclidean geometry on the surface of a sphere which we discussed in Section 2.2 of Chapter 2. We have the triangle ABC of Figure 2.3 whose three angles are each 90°. Consider what happens to a vector (shown by a dotted arrow) as it is parallely transported along the three sides of this triangle. As shown in Figure 5.1, this vector is originally perpendicular to AB when it starts its journey at A. When it reaches B it lies along CB; it keeps pointing along this line as it moves from B to C. At C it is again perpendicular to AC. So, as it moves along CA from C to A, it maintains this perpendicularity, with the result that when it arrives at A it is pointing along AB. In other words, one circuit around this triangle has resulted in a change of direction of the vector by 90°, although at each stage it was being moved parallel to itself!
A similar experiment with a triangle drawn on a flat piece of paper will tell us that there is no resulting change in the direction of the vector when it moves parallel to itself around the triangle. So our spherical triangle behaves differently from the flat Euclidean triangle.
The phenomenon illustrated in Figure 5.1 can also be described as follows.
The general theory of relativity, like any other physical theory, must submit itself for experimental verification. It started with a disadvantage in that it was competing with a well-established paradigm, viz. the Newtonian laws of motion and gravitation. Any test that could be designed for testing general relativity had at the same time to show ways of distinguishing its predictions from those of the Newtonian framework. Here the situation has been different from the case of special relativity. Many laboratory tests have been designed (see some in the opening chapter) to study the dynamics of fast-moving particles. For, in this case, the crucial factor γ, distinguishing relativity from Newtonian dynamics, is significantly different from unity. For really testing general relativity we need situations of strong gravitational fields that cannot be arranged in a terrestrial laboratory. The differences from Newtonian predictions can and do exist in relatively weak fields, however, provided that we look at astronomical situations. Therefore astronomical tests have figured prominently in establishing the general theory.
In the early days Einstein proposed three tests, which are known as the classical tests of general relativity. More tests emerged in later years, although their number is still small. In this chapter we will disuss both classes of tests. All except one require an astronomical setting.
To place matters in proper perspective, let us see how ‘strong’ or ‘weak’ the Earth's gravitational field is at its maximum, i.e., on the surface of the Earth.
We found in the previous chapter that, if a massive star runs out of nuclear fuel, it would lose its equilibrium and begin to shrink. Even when nuclear fuel is available to the star, it may be insufficient to meet the demands for the star's equilibrium. In the early 1930s the young astrophysicist Subrahmanyan Chandrasekhar had encountered a somewhat similar situation when discussing the state of stars like the Sun, after they run out of their nuclear fuel. He found that the star can still sustain equilibrium if its internal matter can attain the degenerate state. Degeneracy can arise if the density of matter is so high that all available energy levels of atoms are filled up, up to some low energy. In such a situation further compression of matter is not possible and gravity is held at bay. This is an excellent example of a macroscopic effect of quantum mechanics: a star as massive as the Sun feels an effect whose origin is in quantum mechanics. We cannot describe it in detail since that would take us farther away from our present interest.
The early work on degenerate matter by R. H. Fowler had shown that every star on sufficient compression attains degeneracy, thereby ensuring that the star would rest in peace in a state of very high density and small radius. It was felt that white dwarf stars are precisely the stars which are in this state.