Big Bang

One of the most amazing outcomes of Einstein’s special theory of relativity is that our familiar notion of time is fundamentally wrong. Scientists and non-scientists had always pictured time as the progression of some kind of universal clock that ticked relentlessly, a cosmic heartbeat, a benchmark against which all other clocks could be set. Time would therefore be the same for everybody, because we would all live by the same universal clock: the same pendulum would swing at the same rate today and tomorrow, in London or in Sydney, for you and for me. Time was assumed to be absolute, regular and universal. No, said Einstein: time is flexible, stretchable and personal, so your time may be different from my time. In particular, a clock moving relative to you ticks more slowly than a static clock alongside you. So if you were on a moving train and I was standing on a station platform looking at your watch as you whizzed by, then I would perceive your watch to be running more slowly than my own watch.

This seems impossible, but for Einstein it was logically unavoidable. What follows in the next few paragraphs is a brief explanation of why time is personal to the observer and depends on the travelling speed of the clock being observed. Although there is a small amount of mathematics, the formulas are quite simple, and if you can follow the logic then you will understand exactly why special relativity forces us to change our view of the world. However, if you do skip the mathematics or get stuck, then don’t worry, because the most important points will be summarised when the mathematics is complete.

To understand the impact of the special theory of relativity on the concept of time, let us consider an inventor, Alice, and her very unusual clock. All clocks require a ticker, something with a regular beat that can be used to count time, such as a swinging pendulum in a grandfather clock or a constant dripping in a water-clock. In Alice’s clock, the ticker is a pulse of light that is reflected between two parallel mirrors 1.8 metres apart, as shown in Figure 22(a). The reflections are ideal for keeping time, because the speed of light is constant and so the clock will be highly accurate. The speed of light is 300,000,000 m/s (which can be written as 3 × 10

m/s), so if one tick is defined as the time for the light pulse to travel from one mirror to the other and back again, then Alice sees that the time between ticks is

Alice takes her clock inside a train carriage, which moves at a constant velocity down a straight track. She sees that the duration for each tick remains the same—remember, everything should remain the same because Galileo’s principle of relativity says that it should be impossible for her to tell whether she is stationary or moving by studying objects that are travelling with her.

Meanwhile, Alice’s friend Bob is standing on a station platform as her train whizzes past at 80% of the speed of light, which is 2.4 X 10

m/s (this is an express train in the most extreme sense of the word). Bob can see Alice and her clock through a large window in her carriage, and from his point of view the light pulse traces out an angled path, as shown in Figure 22(b). He sees the light pulse as following its usual up-and-down motion, but for him it is also moving sideways, along with the train.

In other words, in between leaving the lower mirror and arriving at the upper mirror, the clock has moved forward, so the light has to follow a longer diagonal path. In fact, from Bob’s perspective, the train has moved forward 2.4 metres by the time the pulse has reached the upper mirror, which leads to a diagonal path length of 3.0 metres, so the light pulse has to cover 6.0 metres (up and down) between ticks. Because, according to Einstein, the speed of light is constant for any observer, for Bob the time between ticks must be longer because the light pulse travels at the same speed but has farther to travel. Bob’s perception of the time between ticks is easy to calculate:

Figure 22 The following scenario demonstrates one of the main consequences of Einstein’s special theory of relativity. Alice is inside her railway carriage with her mirror-clock, which ’ticks’ regularly as the light pulse is reflected between the two mirrors. Diagram (a) shows the situation from Alice’s perspective. The carriage is moving at 80% of the speed of light, but the clock is not moving relative to Alice, so she sees it behaving quite normally and ticking at the same rate as it always has.

Diagram (b) shows the same situation (Alice and her clock) from Bob’s perspective. The carriage is moving at 80% of the speed of light, so Bob sees the light pulse follow a diagonal path. Because the speed of light is constant for any observer, Bob perceives that it takes longer for the light pulse to follow the longer diagonal path, so he thinks that Alice’s clock is ticking more slowly than Alice herself perceives the ticking.

It is at this point that the reality of time begins to look extremely bizarre and slightly disturbing. Alice and Bob meet up and compare notes. Bob says that he saw Alice’s mirror-clock ticking once every 2.0 x 10

s, whereas Alice maintains that her clock was ticking once every 1.2 × 10

s. As far as Alice is concerned, her clock was running perfectly normally. Alice and Bob may have been staring at the same clock, but they perceived the ticking of time to be passing at different rates.

Einstein devised a formula that described how time changes for Bob compared to Alice under every circumstance:

It says that the time intervals observed by Bob are different from those observed by Alice, depending on Alice’s velocity (v

) relative to Bob and the speed of light (c). If we insert the numbers appropriate to the case described above, then we can see how the formula works:

Einstein once quipped: ‘Put your hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour, and it seems like a minute. That’s relativity.’ But the theory of special relativity was no joke. Einstein’s mathematical formula described exactly how any observer would genuinely perceive time to slow down when looking at a moving clock, a phenomenon known as time dilation. This seems so utterly perverse that it raises four immediate questions:

1. Why don’t we ever notice this peculiar effect?

The extent of the time dilation depends on the speed of the clock or object in question compared with the speed of light. In the above example the time dilation is significant because Alice’s carriage is travelling at 80% of the speed of light, which is 240,000,000 m/s. However, if the carriage were travelling at a more reasonable speed of 100 m/s (360km/h), then Bob’s perception of Alice’s clock would be almost the same as her own. Plugging the appropriate numbers into Einstein’s equation would show that the difference in their perception of time would be just one part in a trillion. In other words, it is impossible for humans to detect the everyday effects of time dilation.

2. Is this difference in time real?

Yes, it is very real. There are numerous pieces of sophisticated hi-tech gadgetry that have to take into account time dilation in order to work properly. The Global Positioning System (GPS), which relies on satellites to pinpoint locations for devices such as car navigation systems, can function accurately only because it takes into account the effects of special relativity. These effects are significant because the GPS satellites travel at very high speeds and they make use of high-precision timings.

3. Does Einstein’s special theory of relativity apply only to clocks relying on light pulses?

The theory applies to all clocks and, indeed, to all phenomena. This is because light actually determines the interactions that take place at the atomic level. Therefore all the atomic interactions taking place in the carriage slow down from Bob’s point of view. He cannot view these individual atomic interactions, but he can view the combined effect of this atomic slowing-down. As well as seeing Alice’s mirror-clock ticking more slowly, Bob would see her waving to him more slowly as she passed by; she would blink and think more slowly, and even her heartbeat would slow down. Everything would be similarly affected by the same degree of time dilation.

4. Why can’t Alice use the slowing of her clock and her own movements to prove that she is moving?

All the peculiar effects described above are as observed by Bob from outside the moving train. As far as Alice is concerned, everything inside the train is perfectly normal, because neither her clock nor anything else in her carriage is moving relative to herself. Zero relative motion means zero time dilation. We should not be surprised that there is no time dilation, because if Alice noticed any change in her immediate surroundings as a result of her carriage’s motion, it would contravene Galileo’s principle of relativity. However, if Alice looked at Bob as she whizzed past him, it would appear to her that it was Bob and his environment that was undergoing time dilation, because he is moving relative to her.

The special theory of relativity impacts on other aspects of physics in equally staggering ways. Einstein showed that as Alice approaches, Bob perceives that she contracts along her direction of motion. In other words, if Alice is 2 m tall and 25 cm from front to back, and she is facing the front of the train as it approaches Bob, then he will see her as still 2 m tall but only 15 cm from front to back. She appears to be thinner. This is nothing as trivial as a perspective-based illusion, but is in fact a reality in Bob’s view of distance and space. It is a consequence of the same sort of reasoning that showed that Bob observes Alice’s clock ticking more slowly.

So, as well as assaulting traditional notions of time, special relativity was forcing physicists to reconsider their rock-solid notion of space. Instead of time and space being constant and universal, they were flexible and personal. It is not surprising that Einstein himself, as he developed his theory, sometimes found it difficult to trust his own logic and conclusions. ‘The argument is amusing and seductive,’ he said, ‘but for all I know, the Lord might be laughing over it and leading me around by the nose.’

Nevertheless, Einstein overcame his doubts and continued to pursue the logic of his equations. After his research was published, scholars were forced to acknowledge that a lone patent clerk had made one of the most important discoveries in the history of physics. Max Planck, the father of quantum theory, said of Einstein: ‘If [relativity] should prove to be correct, as I expect it will, he will be considered the Copernicus of the twentieth century.’

Einstein’s predictions of time dilation and length contraction were all confirmed by experiments in due course. His special theory of relativity alone would have been enough to make him one of the most brilliant physicists of the twentieth century, providing as it did a radical overhaul of Victorian physics, but Einstein’s stature was set to reach even greater heights.

Soon after publishing his 1905 papers, he set to work on a programme of research that was even more ambitious. To put it into context, Einstein once called his special theory of relativity ‘child’s play’ compared with what came after it. The rewards, however, would be well worth the effort. His next great discovery would reveal how the universe behaved on the grandest scale and provide cosmologists with the tools they needed to address the most fundamental questions imaginable.

The Gravity Battle: Newton v. Einstein

Einstein’s ideas were so iconoclastic that it took time for mainstream scientists to welcome this deskbound civil servant into their community. Although he published his special theory of relativity in 1905, it was not until 1908 that he received his first junior academic post at Berne University. Between 1905 and 1908, Einstein continued to work at the patent office in Berne, where he was promoted to ‘technical expert, second class’ and given the time to push ahead with his effort to extend the power and remit of his theory of relativity.

The special theory of relativity is labelled special because it applies only to special situations, namely those in which objects are moving at constant velocity. In other words, it could deal with Bob observing Alice’s train travelling at a fixed speed on a straight track, but not with a train that was speeding up or slowing down. Consequently, Einstein attempted to reformulate his theory so that it would cope with situations involving acceleration and deceleration. This grand extension of special relativity would soon become known as general relativity, because it would apply to more general situations.

When Einstein made his first breakthrough in building general relativity in 1907, he called it ‘the happiest thought of my life’. What followed, however, was eight years of torment. He told a friend how his obsession with general relativity was forcing him to neglect every other aspect of his life:‘I cannot find the time to write because I am occupied with truly great things. Day and night I rack my brain in an effort to penetrate more deeply into the things that I gradually discovered in the past two years and that represent an unprecedented advance in the fundamental problems of physics.’

In speaking of ‘truly great things’ and ‘fundamental problems’, Einstein was referring to the fact that the general theory of relativity seemed to be leading him towards an entirely new theory of gravity. If Einstein was right, then physicists would be forced to question the work of Isaac Newton, one of the icons of physics.

Newton was born in tragic circumstances on Christmas Day 1642, his father having died just three months earlier. While Isaac was still an infant, his mother married a sixty-three-year-old rector, Barnabas Smith, who refused to accept Isaac into his home. It fell to Isaac’s grandparents to bring him up, and as each year passed he developed a growing hatred towards the mother and stepfather who had abandoned him. Indeed, as an undergraduate, he compiled a list of childhood sins that included the admission of ‘threatening my father and mother Smith to burne them and the house over them’.

Not surprisingly, Newton grew into an embittered, isolated and sometimes cruel man. For example, when he was appointed Warden of the Royal Mint in 1696, he implemented a harsh regime of capturing counterfeiters, making sure that those convicted were hung, drawn and quartered. Forgery had brought Britain to the brink of economic collapse, and Newton judged that his punishments were necessary. In addition to brutality, Newton also used his brains to save the nation’s currency. One of his most important innovations at the Mint was to introduce milled edges on coins to combat the practice of clipping, whereby counterfeiters would shave off the edges of coins and use the clippings to make new coins.

In recognition of Newton’s contribution, the British £2 coin issued in 1997 had the phrase STANDING ON THE SHOULDERS OF GIANTS around its milled edge. These words are taken from a letter that Newton sent to fellow scientist Robert Hooke, in which he wrote: ‘If I have seen further it is by standing on the shoulders of giants.’ This appears to be a statement of modesty, an admission that Newton’s own ideas were built upon those of illustrious predecessors such as Galileo and Pythagoras. In fact, the phrase was a veiled and spiteful reference to Hooke’s crooked back and severe stoop. In other words, Newton was pointing out that Hooke was neither a physical giant, nor, by implication, an intellectual giant.

Whatever his personal failings, Newton made an unparalleled contribution to seventeenth-century science. He laid the foundations for a new scientific era with a research blitz that lasted barely eighteen months, culminating in 1666, which is today known as Newton’s annus mirabilis. The term was originally the title of a John Dryden poem about other more sensational events that took place in 1666, namely London’s survival after the Great Fire and the victory of the British fleet over the Dutch. Scientists, however, judge Newton’s discoveries to be the true miracles of 1666. His annus mirabilis included major breakthroughs in calculus, optics and, most famously, gravity.

In essence, Newton’s law of gravity states that every object in the universe attracts every other object. More exactly, Newton defined the force of attraction between any two objects as

The force (F) between the two objects depends on the masses of the objects (m

and m

)—the bigger the masses, the bigger the force. Also, the force is inversely proportional to the square of the distance between the objects (r

), which means that the force gets smaller as the objects move farther apart. The gravitational constant (G) is always equal to 6.67 × 10

Nm

kg

, and reflects the strength of gravity compared with other forces such as magnetism.

The power of this formula is that it encapsulates everything that Copernicus, Kepler and Galileo had been trying to explain about the Solar System. For example, the fact that an apple falls towards the ground is not because it wants to get to the centre of the universe, but simply because the Earth and the apple both have mass, and so are naturally attracted towards each other by the force of gravity. The apple accelerates towards the Earth, and at the same time the Earth even accelerates up towards the apple, although the effect on the Earth is imperceptible because it is much more massive than the apple. Similarly, Newton’s gravity equation can be used to explain how the Earth orbits the Sun because both bodies have a mass and therefore there is a mutual attraction between them. Again, the Earth orbits the Sun and not vice versa because the Earth is much less massive than the Sun. In fact, Newton’s gravity formula can even be used to predict that moons and planets will follow elliptical paths, which is exactly what Kepler demonstrated after analysing Tycho Brahe’s observations.

For centuries after his death, Newton’s law of gravity ruled the cosmos. Scientists assumed that the problem of gravity had been solved and used Newton’s formula to explain everything from the flight of an arrow to the trajectory of a comet. Newton himself, however, suspected that his understanding of the universe was incomplete: ‘I do not know what I may appear to the world, but to myself I seem to have been only a little boy playing on the seashore, and diverting myself now and then in finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay undiscovered before me.’

And it was Albert Einstein who first realised that there might be more to gravity than Newton had imagined. After his own annus mirabilis in 1905, when Einstein published several historic papers, he concentrated on expanding his special theory of relativity into a general theory. This involved a radically different interpretation of gravity based on a fundamentally different vision of how planets, moons and apples attract one another.