4 The intangible Universe

4.1 Electromagnetism and fields

When Newton wrote about 'The System of the World' in Part 3 of Principia, the only forces he could discuss in any detail were the contact forces that arose when one object touched another, and gravity, which acted at a distance. Even so, Newton thought that there were other forces at work in the world, and hoped they might eventually be brought within his overall scheme just as gravity had been. In fact, Newton wrote:

'I wish we could derive the rest of the phenomena of Nature by the same kind of reasoning from mechanical principles, for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of the bodies, by some causes hitherto unknown, are either mutually impelled towards one another, and cohere in regular figures, or are repelled and recede from one another.'
Isaac Newton (1686), Principia.

Amongst the phenomena familiar to Newton, but which he could not treat mathematically, were those of electricity and magnetism, both of which had been known since antiquity

Figure 1.16 Examples of electric and magnetic forces
figure 1.16a
Examples of electric and magnetic forces. The ancient Greeks were aware that when samples of amber, which they called
figure 1.16b (electron)
were rubbed with wool or fur they acquired the ability to attract light objects such as feathers. They were also aware that the substance we now call lodestone, which could be found in northern Greece in the area known as Magnesia, had the ability to attract pieces of iron.
figure 1.16b
One of the key concepts that Newton lacked, but which eventually proved to be crucial to the quantification of both electricity and magnetism was that of electric charge. This was originally viewed as something like a fluid that could be passed from one object to another, but is now seen, rather like mass, as a fundamental attribute of matter.

Just as Newton had been able to make gravity an effective part of the mechanistic world-view by declaring that the gravitational force between two point-like bodies was proportional to the product of their masses and the inverse square of their separation, so the French scientist Charles Coulomb (1736 - 1806) was able to do the same for electricity by showing that the electrical force between two point-like bodies was proportional to the product of their charges and the inverse square of their separation. In terms of symbols, this can be expressed as:
equation 24aequation 24b
However, electrical charge can be positive or negative, and the electrical forces can be attractive or repulsive in accordance with the famous dictum

'like charges repel; unlike charges attract'

Forces between magnets could be treated in a similar way by using north and south magnetic poles in place of positive and negative charges.

The incorporation of electrical and magnetic forces into the mechanistic world-view appeared to be a triumphant vindication of Newton's foresight. But it was really only the beginning of a story, not the end of one. Subsequent investigations were to show that an electric current - a flow of charge - could produce a magnetic force. This showed that the apparently separate subjects of electricity and magnetism were actually different aspects of a single subject: electromagnetism.
Two of the other books in the Physical World series, Static fields and potentials and Dynamic fields and waves give a thorough discussion of electromagnetism
It was within this unified subject that a new physical concept was to arise, that of a field. The field concept was destined to play an enormously important role in reshaping the physicistís view of the world. It would initially augment the mechanistic world-view, then around 1900, come to rival it, and ultimately, after 1926, play an important part in its downfall.

The field theory of electromagnetism was mainly the creation of two men, Michael Faraday and James Clerk Maxwell. They are, in a sense, the Galileo and the Newton of field theory.
figure 1.20, the fundamental laws of electromagnetismFigure 1.20
Maxwell's equations, the fundamental laws of electromagnetism.

The problem that led Faraday to introduce the concept of a field was an old one; how could one body exert a force on another that was separated from it by empty space? Scientists and philosophers of earlier ages had devised essentially two possible answers.

The simpler but less appealing possibility was that it just happened - that action at a distance was part of the fundamental reality of Nature and, as such, needed no further explanation.

The other possibility was that the notion of empty space was a delusion, that the Universe was actually full of matter, albeit a very subtle and unusual form of matter, and that force was transmitted from one place to another by direct contact between parts of that matter. There were several different proposals concerning the exact nature of this 'subtle matter' that could transmit forces, but it was generally referred to as ether, and theories that made use of it were therefore called ether theories.

Newton's law of gravitation was taken to be an example of action at a distance. The law described the force that one body would exert on another some distance away without any regard to what was in between and without any hint of a mechanism for transmitting the force.
the magnetic force on an electric current was not simply attractive or repulsive, it could cause rotation
Newton was aware that this was a feature of his 'System of the World' that many would find unattractive, but he also realized that he had no evidence on which to base a detailed explanation of gravitational forces. He contented himself with describing gravitational forces mathematically, and said in the Principia, that he would 'form no hypotheses' as to their cause.

Faraday, like others, was willing to accept this situation as far as a purely attractive force like gravity was concerned, or even for a force that could be attractive or repulsive like Coulomb's, but Faraday's own invention of the electric motor showed that the magnetic force on an electric current was not simply attractive or repulsive, it could cause rotation (see Figure 1.18). Faraday felt that for a wire to rotate around a magnet there had to be something, produced by the magnet but present at the location of the wire, that pushed the wire to one side rather than another. It was this agency, filling the space around the magnet, that Faraday eventually came to call a magnetic field.

Faraday's views about the nature of the magnetic field changed over time; for complex reasons, he spoke about his field as being different from an ether. Whatever his precise views, Faraday was convinced that fields held the key to understanding magnetic and electrical phenomena. He certainly felt that the curved pattern of lines revealed by sprinkling iron filings onto a sheet of paper placed over a magnet
figure 1.21, magnetic field linesFigure 1.21 Magnetic field lines, as revealed by sprinkling iron filings onto a sheet of stiff paper placed on top of a magnet.
showed the presence of a magnetic field. Like a collection of miniature compass needles, the filings showed the fieldís strength and direction in each region of space. However, he also realized that in order to provide convincing evidence of the reality of the field something more was needed, such as a demonstration that a disturbance at one point in the field would take a finite time to propagate through the field and have visible effects elsewhere. Faraday tried to observe such delays, but failed. Nevertheless, his belief in the physical reality of fields guided his experiments and lead him on to new discoveries.

When Maxwell started to work on electromagnetism he studied Faraday's experimental researches and, unlike most of his contemporaries, was impressed by the notion of a field. However, Maxwell had his own reasons for believing in an ether. In particular, he believed that an ether was necessary to account for the propagation of light, which was generally regarded as a kind of wavelike disturbance and was therefore thought to require a medium just as ocean waves require water. Maxwell therefore decided to combine Faraday's field ideas with the ether concept. He set out to treat electricity and magnetism in terms of fields that were themselves interpreted as manifestations of pressure, tension and motion within the ether.

In many ways, Maxwell was extraordinarily successful. He did formulate a mechanical model of the electromagnetic field
figure 1.22, a part of Maxwell's mechanical model of the electromagnetic fieldFigure 1.22 A part of Maxwell's mechanical model of the electromagnetic field. The model has been described as 'the most ingenious but least credible ever invented'.
and used it as a guide in writing down his now famous equations. Amongst many other things, Maxwell's equations implied that light is a wave phenomenon in which electric and magnetic fields oscillate in space and time. In an astonishing demonstration of the power of these ideas, Maxwell took the fundamental constants of electricity and magnetism, entered them in his equations, and derived an accurate value for the speed of light. In this way, the subjects of electricity, magnetism and optics, which had seemed quite distinct at the beginning of the nineteenth century, were unified into a single branch of physics. The equations even led Maxwell to predict the existence of a wider family of electromagnetic waves, most with wavelengths beyond the range of human sight. In 1888 Heinrich Hertz completed a series of experiments which confirmed the existence of electromagnetic waves with wavelengths much greater than those of visible light. These were the radio waves which, within a few decades would transform both communication and entertainment. In 1895 Wilhelm Röntgen discovered X-rays, which proved to be electromagnetic waves with wavelengths much smaller than those of visible light. Yet, in spite of these successes, Maxwell's mechanical model of the electromagnetic field remained unconvincing. From about 1865, Maxwell himself drew a clear distinction between his equations, which described the behaviour of electric and magnetic fields, and the underlying ether mechanism that was supposed to account for them. Maxwell firmly believed that he had discovered the correct equations, but did not try to defend the model that had led to them.

If Maxwell had succeeded in accounting for the electromagnetic field in terms of motion in the ether, the mechanical world-view would have reigned supreme; but it was not to be. As investigations continued, particularly after Maxwell's untimely death, it became increasingly clear that it would be impossible to find a convincing mechanical basis for the electromagnetic field. On the other hand it also became clear that Maxwell's field theory of electromagnetism, as embodied in his equations, was stunningly successful.

Nowadays, with the scaffolding stripped away, we can recognize that the true achievement of Faraday and Maxwell was in establishing the importance of fields, arguably the most radical concept in physics since the time of Newton. We now know that there are many types of field: magnetic fields, electric fields, gravitational fields and so on. Each of these fields has a particular value at each point in space. The key idea is that a particle passing through a given point will experience forces that depend on the fields at that point, or in its immediate vicinity. This means that forces are determined locally - there is no action at a distance. When two particles interact they do so because one particle creates a field in the space around itself and the other particle then responds to this field. What is more, as Faraday anticipated and Maxwellís equations established, the fields have dynamics of their own, allowing disturbances of electric and magnetic fields to spread out as waves. Crucially, this means that fields should be thought of as part of the fabric of the world - more intangible than matter, but just as real. The electromagnetic field on Earth is incredibly complex. While you are reading this, electromagnetic waves from all the channels that your radio and television could possibly receive are passing straight through your head. Added to this are signals from power lines, domestic appliances, cars and the Big Bang; tiny electromagnetic signals are even reaching you from the brains of those around you. One of the attractions of physics is its ability to reveal a much richer world than is immediately apparent to our senses. And much stranger things are yet to come.

4.2 Relativity,space,time and gravity

Throughout the development of mechanics and electromagnetism, the role of space and time had been clear and simple. Space and time were simply the arena within which the drama of physics was played out. Speaking metaphorically, the principal 'actors' were matter and ether/fields; space and time provided the setting but didn't get involved in the action. All that changed with the advent of the theory of relativity.

The theory was developed in two parts. The first part is called the special theory of relativity, or, occasionally, the restricted theory, and was introduced in 1905. The second part is called the general theory, and dates from about 1916. Both parts were devised by the same man, Albert Einstein
The special theory of relativity is discussed in Dynamic fields and waves, book 6 of the Physical World series
The origins of the special theory of relativity can be traced back a long way. In 1632, Galileo wrote:

'Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drop falls into the vessel beneath; and, in throwing something to your friend, you need throw no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though there is no doubt that when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still.'
Galileo Galilei (1632), Dialogue Concerning the Two Chief Systems of the World.

In other words, any phenomenon you care to study occurs in just the same way in a steadily moving ship as in a stationary ship. The underlying physical laws and fundamental constants must therefore be exactly the same for all uniformly moving (or stationary) observers. This fact, which dozing train passengers may accept with gratitude, is the central idea of the theory of special relativity. Indeed, it is called the principle of relativity. This leaves one obvious question: how did Einstein gain both fame and notoriety for promoting an idea that was nearly three hundred years old?

The answer is that a lot of physics had been discovered between the time of Galileo and that of Einstein. Most notably Maxwell's theory of electromagnetism had achieved the feat of predicting the speed of light using fundamental constants of electromagnetism, constants that could be measured using simple laboratory equipment such as batteries, coils and meters. Now, if the principle of relativity were extended to cover Maxwell's theory, the fundamental constants of electromagnetism would be the same for all uniformly moving observers and a very strange conclusion would follow: all uniformly moving observers would measure the same speed of light. Someone running towards a torch would measure the same speed of light as someone running away from the torch. Who would give credence to such a possibility?

Einstein had the courage, self-confidence and determination to reassert the principle of relativity and accept the consequences. He realized that, if the speed of light were to remain the same for all uniformly moving observers, space and time would have to have unexpected properties, leading to a number of startling conclusions, including the following:

Moving clocks run slow. If I move steadily past you, you will find that my wrist watch is ticking slower than yours. Our biological clocks are also ticking, and you will also find that I am ageing less rapidly than you.

Moving rods contract. If an observer on a platform measures the length of a passing railway carriage, he or she will measure a shorter length than that measured by a passenger who is sitting inside the carriage.

Simultaneity is relative. Suppose you find two bells in different church towers striking at exactly the same time (i.e. simultaneously). If I move steadily past you, I will find that they strike at different times (i.e. not simultaneously). It is even possible for you to find that some event A happens before some other event B and for me to find that they occur in the opposite order.

The speed of light in a vacuum is a fundamental speed limit. It is impossible to accelerate any material object up to this speed.

If these consequences seem absurd, please suspend your disbelief. It took the genius of Einstein to realize that there was nothing illogical or contradictory in these statements, but that they describe the world as it is. Admittedly we donít notice these effects in everyday life but that is because we move slowly: relativistic effects only become significant at speeds comparable with the speed of light (2.998 times 108 metres per second). But not everything moves slowly. The electrons in the tube of a TV set are one example, found in most homes, where relativistic effects are significant.

One of the first people to embrace Einstein's ideas was his former teacher, Hermann Minkowski (1864-1909). He realized that although different observers experience the same events, they will describe them differently because they disagree about the nature of space and the nature of time. On the other hand space and time taken together form a more robust entity:

'Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.'
Hermann Minkowski, Space and Time in A. Einstein et al. (1952), The Principle of Relativity, New York, Dover Publications.

The union of space and time of which Minkowski spoke is now generally referred to as space-time. It represents a kind of melding together of space and time, and since space is three-dimensional, and time is one-dimensional, space-time is four-dimensional. Any particular observer, such as you or I, will divide space-time into space and time, but the way in which that division is made may differ from one observer to another and will crucially depend on the relative motion of the observers.

A very rough attempt at representing diagrammatically this change of attitude towards space and time is shown in

Figure 1.23
figure 1.23a, The pre-Einsteinian view of space and time(a) The pre-Einsteinian view of space and time. Not only are space and time separate and distinct they are also absolute. All observers agree on what constitutes space and what constitutes time, and they also agree about what it means to speak of 'the whole of space at a particular time'.
figure 1.23b, The post-Einsteinian view in which space and time are seen as aspects of a unified space-time(b) The post-Einsteinian view in which space and time are seen as aspects of a unified space-time. Different observers in uniform, relative motion will each slice space-time into space and time, but they will do so in different ways. Each observer knows what it means to speak of 'the whole of space at a particular time', but different observers no longer necessarily agree about what constitutes space and what constitutes time.
Before Einstein introduced special relativity, the phrase 'the whole of space at a particular time' was thought to have exactly the same meaning for all observers. After Einstein's work it was felt that each observer would understand what the phrase meant, but that different observers would disagree about what constituted the whole of space at a particular time. All observers would agree on what constituted space-time, but the way in which it was sliced up into space and time would differ from one observer to another, depending on their relative motion. No observer had the true view; they were all equally valid even though they might be different.

In retrospect, special relativity can be seen as part of a gradual process in which the laws of physics attained universal significance. The earliest attempts to understand the physical world placed Man and the Earth firmly at the centre of creation. Certain laws applied on Earth, but different laws applied in the heavens. Copernicus overturned this Earth-centred view and Newton proposed laws that claimed to apply at all places, and at all times. Special relativity continues this process by insisting that physical laws should not depend on the observer's state of motion - at least so long as that motion is uniform. It is therefore not surprising that Einstein was led to ask if physical laws could be expressed in the same way for all observers, even those who were moving non-uniformly. This was the aim of his general theory of relativity.

Einstein realized that many of the effects of non-uniform motion are similar to the effects of gravity. (Perhaps you have experienced the sensation of feeling heavier in a lift that is accelerating upwards.) With unerring instinct he treated this as a vital clue: any theory of general relativity would also have to be a theory of gravity. After more than ten years of struggle, the new theory was ready. According to general relativity, a large concentration of mass, such as the Earth, significantly distorts space-time in its vicinity. Bodies moving through a region of distorted space-time move differently from the way they would have moved in an undistorted space-time.

For example, meteors coming close to the Earth are attracted to it and deviate from uniform, steady motion in a straight line. Newton would have had no hesitation in saying that these deviations are due to gravitational forces. In Einstein's view, however, there is no force. The meteors move in the simplest way imaginable, but through a distorted space-time, and it is this distortion, generated by the presence of the Earth, that provides the attraction. This is the essence of general relativity, though the mathematics required to spell it out properly is quite formidable, even for a physicist.

The central ideas of general relativity have been neatly summarized by the American physicist John Archibald Wheeler. In a now famous phrase Wheeler said:

'Matter tells space how to curve.

Space tells matter how to move.'

Purists might quibble over whether Wheeler should have said 'space-time' rather than 'space', but as a two-line summary of general relativity this is hard to beat.
Figure 1.24
A highly schematic diagram showing space-time curvature near the Sun and indicating the way in which this can lead to the bending of starlight as it grazes the edge of the Sun. (The bending has been hugely exaggerated for the sake of clarity.) The observation of this effect in 1919, during a total eclipse of the Sun, did much to make Einstein an international celebrity.

If you tried to summarize Newtonian gravitation in the same way all you could say is: 'Matter tells matter how to move'; the contrast is clear.

General relativity is a field theory of gravity. At its heart are a set of equations called the Einstein field equations. To this extent general relativity is similar to Maxwell's field theory of electromagnetism. But general relativity is a very unusual field theory. Whereas electric and magnetic fields exist in space and time, the gravitational field essentially is space and time. Einstein was well aware of the contrast between gravity and electromagnetism, and spent a good deal of the later part of his life trying to formulate a unified field theory in which gravity and electromagnetism would be combined into a single 'geometric' field theory. In this quest he was ultimately unsuccessful, but general relativity remains a monumental achievement.