WHEN NEWTON AND EINSTEIN AGREE

Einstein's general theory of relativity and Newton's law of gravitation offer radically different interpretations of the phenomenon of gravity. Yet, in practical terms, the difference between their predictions seem to be very small. In Chapter 5 we saw two examples of observations in the solar system: the precession of the orbit of Mercury and the bending of light rays from a distant star by the Sun. In both cases the differences in the predictions of Newton and Einstein are very small and are measurable only with very patient and sophisticated astronomical observations. Is it just a coincidence that these two approaches give almost the same answer?

A mathematical analysis of Einstein's equations tells us that the agreement between the two approaches is not coincidental. It can be shown that, in all phenomena of weak gravitational effects and where the gravitating bodies are moving slowly compared to light, the two theories must almost agree. In our discussion of the escape speed in Chapter 3, we saw how to measure the relative strength of gravity. We use the criterion of the escape speed in the present context to understand the difference between ‘weak’ and ‘strong’ gravity. The rule is simple: compare the escape speed V with the speed of light c. If the ratio V/c is very small compared to 1, the gravitational effects are weak. If the ratio is comparable to 1, say between 0.1 and 1, the gravitational effects are strong (see Figure 6–1). Referring back to Table 3–2, we see that the gravitational effects are weak in all cases except on the surface of neutron stars.

THE PROBLEMS OF LARGE-SCALE STRUCTURE

More than seven decades have elapsed since Friedmann proposed his mathematical models that describe the expanding Universe. As we saw in Chapter 9, these models lead to the conclusion that the Universe was created some 10–15 billion years ago in a big explosion (the so-called big bang) after which it has been expanding but more and more slowly because of brakes applied by gravity. This model also tells us that the Universe was very hot to begin with, and dominated by radiation, but with expansion it has cooled down and the temperature of the radiation background today is 2.73 kelvin (see Figure 9–10) as measured by the COBE satellite and other groundbased detectors. And one other set of relics of the hot era, namely the light nuclei like deuterium, helium, etc., are found in the right amount all over the Universe. Thus, we concluded the last chapter with a fair degree of confidence in the big–bang scenario.

However, over the last quarter of a century astronomical observations have become more sophisticated and the views of the largescale structure of the Universe they present go well beyond the simplified assumptions of a ‘homogeneous and isotropic Universe’. We shall see, for example, in Figure 10–1 how galaxies are distributed over the sky in depth. The dots in the figure represent galaxies and their distribution is clearly not smooth, as a homogeneous Universe would have us believe.

WAS THERE REALLY A BIG BANG?

The big-bang cosmology described in the last two chapters has a large following amongst the astronomical community. The models of Friedmann are able to account for the observed expansion of the Universe, for the smooth background of microwave radiation, and for the abundance of light nuclei that cannot be generated inside stars. Are these not good enough reasons for believing in the overall picture?

Playing the devil's advocate in this chapter, let me voice a few dissenting views. First, a scientific theory, howsoever successful it may be, must always be vulnerable to checks of facts and conceptual consistency. Even a well established theory like Newton's had to give way to Einstein's when it was found wanting under these checks (see Chapter 5). The formidable facade of big-bang cosmology is likewise developing cracks that can no longer be plastered over.

The first crack has actually been there right from the beginning and may have been noticed by the reader of Chapter 9. He or she may ask the questions, ‘What preceeded the big bang? How did the matter and radiation in the Universe originate in the first place? Does it not contradict the law of conservation of matter and energy?’

These questions cannot be answered within the framework of Einstein's general theory of relativity, which was used to construct the Friedmann models. The moment of ‘big bang’ is a singular epoch, according to the theory, just as the end of a collapsing object, described in Chapter 7, is in a singularity.

Our discussion of gravity began with the falling apple and has taken us from ocean tides to the planets, comets, and satellites of the solar system, to the different stages in the evolution of a star, to the curved spacetime of general relativity, to the illusions of gravitational lensing, to the weird effects associated with black holes and white holes, and finally to the large-scale structure of the Universe itself. None of the other basic forces of physics has such a wide range of applications. Although gravity is by far the weakest of the four known basic forces, its effects are the most dramatic.

Indeed, it would be an amusing exercise to speculate on the state of the world if there were no gravity at all! Would atoms and molecules be affected? As far as we know, the presence or absence of gravity does not play a crucial role in the existence and stability of the microworld. The strong, weak, and electromagnetic forces are the main forces at this level. Even at the macroscopic level of the objects we see around us in our daily lives, gravity does not appear to play a crucial role in their constitution or equilibrium. After all, even astronauts have demonstrated that they can live in simulated conditions of weightlessness. Neither the astronauts nor their spacecraft come apart in such circumstances. The basic binding force at this level is the force of electricity and magnetism.

But we can go no further in dispensing with gravity. If we eliminate gravity on a bigger scale, disasters lie in store.

WHY DID THE APPLE FALL?

Apples have played a prominent role in many legends, myths, and fairytales. It was the forbidden apple that became the source of temptation to Eve and ultimately brought God's displeasure upon Adam. It was the apple of discord that led to the launching of a thousand ships and the long Trojan War. It was a poisoned apple that nearly killed Snow White, and so on.

For physicists, however, the most important apple legend concerns the apple that fell in an orchard in Woolsthorpe in Lincolnshire, England, in the year 1666. This particular apple was seen by Isaac Newton, who ‘fell into a profound meditation upon the cause which draws all bodies in a line which, if prolonged, would pass very nearly through the centre of the earth.’

The quotation is from Voltaire's Philosophie de Newton, published in 1738, which contains the oldest known account of the apple story. This story does not appear in Newton's early biographies, nor is it mentioned in his own account of how he thought of universal gravitation. Most probably it is a legend.

It is interesting to consider how rare it is to see an apple actually fall from a tree. An apple may spend a few weeks of its life on the tree, and after its fall it may lie on the ground for a few days. But how long does it take to fall from the tree to the ground? For a drop of, say, 3 metres, the answer is about three-quarters of a second.

THE MASS OF THE EARTH

Although with Newton's pioneering discoveries, gravity was the first basic force of nature to be described and studied quantitatively, it is the weakest of all known basic forces of nature. The other basic forces are the forces of electricity and magnetism and the forces of ‘strong’ and ‘weak’ interaction which act on subatomic particles. It is a measure of the success achieved to date that physicists are able to explain all observed natural and laboratory phenomena in terms of these four basic forces. As we shall see in later chapters, many physicists hope that one day they will be able to bring all the basic forces under the umbrella of one unified force.

Although atomic physicists consider gravity to be the weakest of the four known basic forces of nature, for astronomers gravity is the most dominant force in the celestial environment. How do we assess the strength of gravity in any given situation? We will try to answer this question with a few examples in this chapter.

All of us on the Earth are conscious of gravity. The feeling of weight that we have results from the gravitational pull the Earth exerts on us. Newton's inverse-square law of gravitation described in Chapter 2 tells us how strong this force is on any given body on the Earth's surface. Let m be the mass of the body and M the mass of the Earth. The distance between the body and the Earth is denoted by d.

THE COSMIC ENERGY PROBLEM

We here on Earth are constantly reminded by experts that with advancing technology our energy needs are growing, and that we need to worry about stocks of oil, coal, nuclear fuel, etc. that are needed to generate energy to meet these demands all over the world. How long will the supplies last? Can we extend that period by conserving energy? If so, how? These questions are being debated by experts and lay people alike.

Astronomers face the ‘energy problem’ in their investigations of cosmic sources of radiation. The age-old problem, where the Sun gets its energy to shine so brightly and steadily, has been solved. In Chapter 4 we saw that the key to solar energy lies in the nuclear fusion going on in the central core of the Sun.

But in the 1950s new problems with far greater magnitude began to confront the astronomers. The radio astronomers began to find sources of radio emission whose total energy reservoir exceeded that of the Sun by several billion. Where did the source of this energy lie? The problem was exacerbated in the early 1960s with the discovery of quasi-stellar sources, commonly called quasars. Initially mistaken for stars, quasars turned out to be far more energetic, and far more dramatic in spending their energy.

A typical quasar radiates in visible light as much as a galaxy of hundred billion stars. It also radiates in X-rays and possibly other wavebands.

IS NEWTON's LAW PERFECT?

We left Chapter 2 with the impression that Newton's law of gravitation gave a successful account of the diverse nature of phenomena in which gravity is believed to play a leading role. Not only is this law able to account for motions of such celestial bodies as planets, comets, and satellites, it also helps us in understanding the complex problem of the structure and evolution of the Sun and other stars. Modern scientists use the same law in determining the rocket thrusts, spacecraft trajectories, and the timing of space encounters. That a good scientific law should be basically simple but universal in application is epitomized in Newton's law of gravitation. What more could one ask for?

Yet science by nature is perfectionist. The laws and theories of science are accepted as long as they are able to fulfil its primary purpose of explaining natural phenomena. Any law of science, despite a history of past successes, is inevitably discarded if it fails in even one particular instance. To the scientist, such an event brings mixed feelings. Disappointment and confusion that an old, well established idea has to be given up or modified are coupled with excitement and expectation that nature is about to reveal a new mystery.

Newton's law of gravitation was no exception to this rule. By the beginning of the present century, cracks were beginning to appear in the impressive facade of physics erected on the Newtonian ideas of motion and gravitation.

THE STATIC UNIVERSE IN NEWTON'S THEORY

Back in the 1690s, Isaac Newton attempted an ambitious application of his law of gravitation. Newton wanted to describe, with the help of his theory of gravity, the largest physical system that can be imagined – the Universe. How did Newton fare in this attempt?

In a letter to Richard Bentley dated 10 December 1692, Newton expressed his difficulties in the following words:

Figure 9–1, which illustrates a finite and uniform distribution of matter in the form of a sphere initially at rest, helps explain Newton's difficulty. Will the sphere stay at rest forever? The matter in the sphere has its own force of gravity, which tends to pull the different parts of the sphere toward one another, with the result that the sphere as a whole contracts. We have encountered this force of selfgravity in stars (Chapter 4) and in the phenomenon of black-hole formation (Chapter 7).

This volume describes the advances in the quantum theory of fields that have led to an understanding of the electroweak and strong interactions of the elementary particles. These interactions have all turned out to be governed by principles of gauge invariance, so we start here in Chapters 15-17 with gauge theories, generalizing the familiar gauge invariance of electrodynamics to non-Abelian Lie groups.

Some of the most dramatic aspects of gauge theories appear at high energy, and are best studied by the methods of the renormalization group. These methods are introduced in Chapter 18, and applied to quantum chromodynamics, the modern non-Abelian gauge theory of strong interactions, and also to critical phenomena in condensed matter physics.

Chapter 19 deals with general spontaneously broken global symmetries, and their application to the broken approximate SU(2) × SU(2) and SU(3) × SU(3) symmetries of quantum chromodynamics. Both the renormalization group method and broken symmetries find some of their most interesting applications in the context of operator product expansions, discussed in Chapter 20.

The key to the understanding of the electroweak interactions is the spontaneous breaking of gauge symmetries, which are explored in Chapter 21 and applied to superconductivity as well as to the electroweak interactions. Quite apart from spontaneous symmetry breaking is the possibility of symmetry breaking by quantum-mechanical effects known as anomalies. Anomalies and various of their physical implications are presented in Chapter 22.

It is often useful to consider quantum field theories in the presence of a classical external field. One reason is that in many physical situations, there really is an external field present, such as a classical electromagnetic or gravitational field, or a scalar field with a non-vanishing vacuum expectation value. (As we shall see in Chapter 19, such scalar fields can play an important role in the spontaneous breakdown of symmetries of the Lagrangian.) But even where there is no actual external field present in a problem, some calculations are greatly facilitated by considering physical amplitudes in the presence of a fictitious external field. This chapter will show that it is possible to take all multiloop effects into account by summing ‘tree’ graphs whose vertices and propagators are taken from a quantum effective action, which is nothing but the one-particle-irreducible connected vacuum-vacuum amplitude in the presence of an external field. It will turn out in the next chapter that this provides an especially handy way both of completing the proof of the renormalizabilty of non-Abelian gauge theories begun in Chapter 15, and of calculating the charge renormalization factors that we need in order to establish the crucial property of asymptotic freedom in quantum chromodynamics.

The Quantum Effective Action

Consider a quantum field theory with action I[ϕ], and suppose we ‘turn on’ a set of classical currents Jr(x) coupled to the fields ϕr(x) of the theory.

The quantum field theories that have proved successful in describing the real world are all non-Abelian gauge theories, theories based on principles of gauge invariance more general than the simple U(1) gauge invariance of quantum electrodynamics. These theories share with electrodynamics the attractive feature, outlined at the end of Section 8.1, that the existence and some of the properties of the gauge fields follow from a principle of invariance under local gauge transformations. In electrodynamics, fields ψn(x) of charge en undergo the gauge transformation ψn(x) → exp(ienΛ(x))ψn(x) with arbitrary Λ(x). Since ∂μψn(x) does not transform like ψn(x), we must introduce a field Aμ(x) with the gauge transformation property Aμ(x) → Aμ(x)+∂μΛ(x), and use it to construct a gauge-covariant derivative ∂μψn(x)–ienAμ(x)ψn(x), which transforms just like ψn(x) and can therefore be used with ψn(x) to construct a gauge-invariant Lagrangian. In a similar way, the existence and some of the properties of the gravitational field gμn(x) in general relativity follow from a symmetry principle, under general coordinate transformations. Given these distinguished precedents, it was natural that local gauge invariance should be extended to invariance under local non-Abelian gauge transformations.

In the original 1954 work of Yang and Mills, the non-Abelian gauge group was taken to be the SU(2) group of isotopic spin rotations, and the vector fields analogous to the photon field were interpreted as the fields of strongly-interacting vector mesons of isotopic spin unity.

Much of the physics of this century has been built on principles of symmetry: first the space time symmetries of Einstein's 1905 special theory of relativity, and then internal symmetries, such as the approximate SU(2) isospin symmetry of the 1930s. It was therefore exciting when in the 1960s it was discovered that there are more internal symmetries than could be guessed by inspection of the spectrum of elementary particles. There are exact or approximate symmetries of the underlying theory that are ‘spontaneously broken,’ in the sense that they are not realized as symmetry transformations of the physical states of the theory, and in particular do not leave the vacuum state invariant. The breakthrough was the discovery of a broken approximate global SU(2) × SU(2) symmetry of the strong interactions, which will be discussed in detail in Section 19.3. This was soon followed by the discovery of an exact but spontaneously broken local SU(2) × U(1) symmetry of the weak and electromagnetic interactions, which will be taken up along with more general broken local symmetries in Chapter 21. In this chapter we shall begin with a general discussion of broken global symmetries, and then move on to physical examples.

Degenerate Vacua

We do not have to look far for examples of spontaneous symmetry breaking. Consider a chair. The equations governing the atoms of the chair are rotationally symmetric, but a solution of these equations, the actual chair, has a definite orientation in space.