There Was a Young Lady Named Bright
Whose Speed Was Far Faster Than Light.
She Set Out One day
In a Relative Way
And Returned on the Previous Night.
— A. H. Reginald Buller [1]
6.1 A Vietnamese Fairy Tale: The Land of Bliss
Over five centuries ago, a young mandarin, named Tu Thuc, was chief of the Tien Du district in what is now Vietnam. He was a devoted scholar and possessed many books on all subjects, except for any on the Land of Bliss, which is known to be a fairy place and a land of eternal youth and pleasure.
Once when travelling during the Flower Festival, Tu Thuc came across a disturbance near an ancient pagoda covered in beautiful blossoms. A young maiden of startling beauty had just grasped a branch to better admire a sacred moon flower, when the branch had broken off in her hand. The priests arrested her and imposed a heavy fine, which she was unable to pay. Although Tu Thuc had insufficient money with him to cover the fine, he offered the priests his expensive brocaded coat, which they accepted as payment. Tu Thuc was praised by the local community for his generosity.
Some years later, tired from the duties of his office, Tu Thuc resigned and set out to explore his native land and write poetry on events and scenes that inspired him. High in the mountains, he sat down to rest and composed a poem on the Land of Bliss (see Appendix 6.2). As he completed it, a cavern appeared before him in a mountain, and it seemed almost to beckon him. He entered and passed into a land of crystalline springs, lotus leaves and wonderful birds with bright colourful feathers. A choir of angelic voices sang gently in the background.
A group of lovely maidens presented him to their queen, who smiled and welcomed him. He was astonished to see at her side the young maid he had befriended years before. “This is my daughter, Giang Huong,” she said. “We have never forgotten the help you gave her when she was in distress.” Tu Thuc felt his heart go out to the demure fairy princess. “You are obviously a kind and generous man,” continued the queen, “and I offer you my daughter’s hand in marriage.”
The wedding was celebrated that day, with great pomp and festivities. Tu Thuc had never been so happy. However, as time passed, he began to feel nostalgia for his old village. Finally, he asked the Queen if he might be allowed to visit his friends and relatives for just a few weeks. He promised to return, for his love for his young wife was as strong as ever.
The queen reluctantly acceded and suggested to Tu Thuc that he lie down on the ground and close his eyes. He did so and fell asleep immediately. When he awoke, he looked around but could not recognize where he was. He stopped a passing old man and said to him. “I am Tu Thuc, and I am seeking my village in Tien Du.”
The old man shook his head. “There was once a Tu Thuc, who was chief of the Tien Du district, but that was a hundred years ago, long before I was born. He set out on a long journey, and never returned. Some say he perished, and others that he found a secret way into heaven.”
Tu Thuc realized that during the time he had been in the Land of Bliss, a hundred years had passed by on earth. There was nothing left for him anymore in the village of his birth. All his friends were dead, and the customs of the young people appeared strange to him. He set out once more in the direction of the mountains and was never seen again. Whether he found his way back to the Land of Bliss, or became lost and died in the rugged highlands, is not known.
6.2 What is Time?
The above tale is a traditional Vietnamese fairy story. In the manner of such stories, there are several variants which differ in their details and endings. It has now been adapted into “The Vietnamese Legend, Giang Huong – the Musical”, which has been staged in the Opera House of Ho Chi Minh City.
A feature of the story is that in Fairyland, time flows more slowly than on earth. This is a characteristic shared with other tales. “Rip Van Winkle” by Washington Irving is one of the best known, where Rip falls asleep and wakes to find twenty years have passed him by. In “The king of Elfland’s Daughter” by Irish writer, Lord Dunsany, there is much in common with the Vietnamese story.
However, these are only stories, where the writer can create whatever fantasy he or she desires. In the real world, here on earth, time has no surprises, but passes at the same rate for everybody, wherever and whenever they are.
Or does it?
Before we can talk of time, and whether it races or crawls by, we need to be clear exactly what we are talking about. The Oxford English Dictionary offers the following definition for time: “the indefinite continued progress of existence and events in the past, present and future regarded as a whole.” This may be all right, so far as it goes, provided that one does not probe too deeply into what is meant by past, present and future. For instance, past is defined as “gone by in time and no longer existing”, and the circularity of the definition becomes immediately apparent.
St Augustine (354–430 CE) once famously said: “What then is time? If no one asks me, I know what it is. If I wish to explain it to him who asks, I do not know.” Since then, philosophers have debated the subjectivity of time, and particularly duration which is a measure of elapsed time. The difficulty arises because the only time one can actually experience is the present, and an estimate of elapsed time requires the memory of a past event, which is subjective. Time appears to pass more quickly if we are having fun.
Physicists have long tried to put the measurement of time on a quantitative basis. Galileo, in his experiments on the mechanics of falling bodies relied on his own pulse, and a primitive water clock, to measure the elapsed time. He later studied pendulums, which led to the development in 1656 of a pendulum clock by Christiaan Huygens. The construction of the chronometer in the Eighteenth Century by John Harrison enabled the accurate determination of longitude by ships at sea, and revolutionized the art of navigation. Today, the most accurate clocks use the period of electromagnetic radiation emitted from the Caesium-133 atom as the standard unit of time.
Before the advent of train travel and timetables, village life was leisurely, and it mattered little if the clocks in one village were running slightly ahead or behind those of another village further down the road. Now, with data exchanged around the globe at the speed of light, much more importance is attached to time standards. Surely, with today’s incredibly accurate clocks, we would notice if time in one place were passing at a different rate from time somewhere else.
Or would we? We shall see.
It has long been the custom of physicists to designate the position of an object in the three-dimensional (3D) space in which we live by three coordinates. We begin by choosing a reference point in space, from which all other dimensions are measured. This point is known as the origin, and is designated by the letter O. It can be anywhere, but it is more practical to choose an origin that is convenient for the problem we are studying. Choosing an origin in another star system, such as Alpha Centauri1 is not appropriate for calculating the location of planets in our own solar system.
Coordinates of point P in a 3D Cartesian coordinate system
So far, so good. The three spatial coordinates are all that is required to define the position of a point in 3D space, provided the point is stationary in time. If the point is moving, then at each time t the point will have different values of x, y, and z. To describe such an evolving system, physicists introduce a fourth axis, called the T (time) axis, assumed to be perpendicular to the other three. An arbitrary time is selected for the origin of the T axis. The coordinate t is then the elapsed time since t = 0, which occurs at the origin.
This is the stage at which our intuition (or common sense) warns us that we have departed from reality. Space, as we know it, has only three dimensions, so how can we talk about a fourth axis perpendicular to X, Y, and Z? Nevertheless let us consider this four-dimensional (4D) space–time as just a mathematical device that enables us to formulate physical problems, e.g. the calculation of the trajectories of moving objects, in simpler mathematics. So now, in the same way that x, y and z identify a point in space, the four coordinates x, y, z, and t identify a point (which we shall call an event) in four-dimensional (4D) space–time.
As we have seen, there is an arbitrariness to the values of these four coordinates because of the arbitrary choice of the origin O. Peter might choose the location and time of his birth for his choice of O, while Mary might choose something altogether different. A clear requirement of any physical theory is that it must predict the results of experiments and observations, regardless of the assumptions made. Clearly the path of the moon around the earth does not depend on where or when Peter was born. Indeed, if our theory is good, we should be able to choose any origin we like, and still accurately predict the results of the measurements we are making. Let us explore this a little further in the next Section.
6.3 Walking on a Train
We all feel that we can distinguish when we are in motion from when we are at rest. However, the work of Galileo and Newton showed that making such a decision is actually impossible, except when our velocity changes. At the moment, sitting in front of a computer screen, and reaching for a cup of delicious Ethiopian Harar coffee, I do not have any feeling of motion. However, I live on the surface of a planet that is rotating at a speed of roughly 460 m per second (at the equator). Also the earth is revolving around the sun at about 30 km per second, and the whole solar system is plummeting towards the constellation of Leo at 371 km per second. Would anyone call this “being at rest”?
The reason that I have no perception of these enormous speeds is explained by Newton’s first law of motion:every body continues in a state of rest or uniform motion unless acted upon by an external force. I, and my coffee cup, and my computer, continue on our journey towards Leo, maintaining the same relative separation from each other unless some force acts upon one of us, e.g. if I inadvertently knock the cup and spill the coffee. Suppose I move to the study I have set up in my private jet aircraft.2 Once again, after the plane has taken off and reached cruising altitude, I will have no sensation of motion, unless we encounter turbulence, which is an external force likely to send my coffee mug sliding across the desk.
Can we all, as young children, remember sitting in a train at a railway station, waiting impatiently for our journey to begin. Looking out of a window, we notice that at last we have begun to edge forwards and are finally underway. We sit back in our seat, turn towards our parents to tell them, and are astonished to see that the station platform on the other side of the train remains absolutely still. What we had observed was another train pulling into the platform on the track next to us and have mistaken its movement for our own.
Technically we can summarize the above considerations by stating that any system moving at constant velocity (which we call an inertial frame of reference) is equivalent to any other, in the sense that any experiment performed in one gives the same result if it is performed in another. Consequently, it is impossible to decide which frame of reference is moving, and with what velocity. Of course, if the speed of the train is not constant (i.e. it is accelerating or braking) we feel the effect of the acceleration or deceleration, but not of the speed. For example, if the train suddenly stops, we are thrown forwards and might even fall, if we happen to be on our feet at the time.
Suppose that Peter is standing on a platform, and Mary is on a train departing slowly from the station. Peter may claim that Mary is moving away from him at 10 kph, which he determines to be the speed of the train, as measured in his frame of reference. Mary, on the other hand, may claim equally validly that Peter is moving away from her at 10 kph in the opposite direction, and that she is the one at rest. If Peter throws a ball towards Mary at 20 kph, as measured by him, Mary will see it go past her at 10 kph. This is all quite straightforward and is explained readily by Newton’s mechanics.
Whether we decide to set up a physics experiment in Peter’s frame of reference on the platform, or Mary’s on the train, is unimportant. We will get the same results. Indeed, we can readily transform our equations of motion from Peter’s to Mary’s frames of reference (or vice versa) by what is called a Galilean Transformation.
So, what is all the fuss about? It is surely just “common sense.” Having just had our notion of common sense trashed in the previous Chapter, we should by now be inured to what is to come. To lead us in gently, let us first discuss two important results that were disturbing physicists, as the Nineteenth Century, with appropriate fireworks and popping of corks, passed over into the Twentieth.
6.4 Harbingers of a Scientific Revolution
An air of complacency pervaded physics at this time. Classical Mechanics, as formulated by Newton and extended by Lagrange and Hamilton, accounted successfully for terrestrial and celestial phenomena involving moving bodies and their collisions. Maxwell had completed a seminal work unifying electricity and magnetism into the single field of Electromagnetism. Some physicists believed that all important work in physics had been completed, and all that remained was to fill in the details.
However, a close inspection revealed that Maxwell’s equations of electromagnetism are not invariant under a Galilean Transformation. This may sound erudite, but it is just physicists’ jargon, which we shall now explain. The Galilean Transformation, which we encountered in the last Section, is the method we use to transform from equations in one frame of reference (e.g. Peter’s on the platform) to those in another (e.g. Mary’s on the moving train). What was discovered was that the predictions of Maxwell’s theory depend on the frame of reference used, something that we have maintained is unphysical. In other words, an experiment on electromagnetism would be able to reveal our absolute velocity, contrary to what was affirmed in the previous Section. So, who is right: Galileo or Maxwell? Or are they both wrong?
In the last Chapter, we saw that electromagnetic radiation can propagate in a vacuum. This is in contradistinction to sound, which requires a material such as air to maintain the wave motion, and sea waves, which of course require water. In the 19th Century it was thought that an as yet undiscovered, but all pervading, medium must exist to enable the propagation of light waves. This was called the quintessence3 or ether (aka aether). According to the physics of Newton and Galileo, the velocity of light on earth with respect to the ether should be given by the same Galilean equations discussed in the previous Section for the motion of a person on a train (i.e. the velocity of light as measured on earth should be equal to the velocity of light through the ether plus or minus the speed of the earth along its direction of motion).
As we mentioned earlier, we are always in motion with respect to the ether, so the time taken for a ray of light to travel a certain distance in our direction of motion, and return, is expected to be different from the time taken to travel the same distance at right angles to our direction of motion, and return. It is analogous to a person swimming in a moving stream of water. The time taken to swim 100 m upstream and then return downstream to the starting point is different from the time taken to swim 100 m across the stream, and return. The huge magnitude of the velocity of light (300,000 kms per second) compared with our planetary speeds means that this time difference is very small, but we should still be able to measure it, if Newtonian physics is correct.
In a series of brilliantly conceived experiments, beginning in 1881, Albert A. Michelson and Edward W. Morley split a beam of light into two rays, sent these off on perpendicular paths, bounced them back from mirrors and then let the returning two rays produce an interference pattern on a screen. By this approach they were able to show that the velocity of light in the two perpendicular directions was actually the same. This did not seem possible. It was analogous to Mary, when measuring the velocity of the ball thrown by Peter in the last Section, obtaining a velocity of 20 kph, even though she was located on a moving train. It made no sense.
In 1892, the Dutch physicist, Hendrick Lorentz, remarked that the negative results of the Michelson-Morley experiments could be accounted for by replacing the Galilean Transformations by another, more complicated set of transformations, now known as the Lorentz Transformations. This explanation was, however, totally ad hoc, and as such only acceptable as a working proposition. In this regard, it was similar to Max Planck’s ad hoc quantum explanation of black body radiation that we discussed in the last Chapter. The physics that underlay the Lorentz Transformations was supplied by the same young genius who had sorted out what lay behind Planck’s mysterious quanta, in between carrying out his duties as a patent clerk in Berne, Switzerland.
6.5 Here Comes Einstein
In the annus mirabilis of 1905, the 26 year-old Albert Einstein published six papers, three of which would revolutionise the field of Physics. One of them [2] reintroduced the Lorentz Transformations as a consequence of a very daring assumption, i.e. that the speed of light c had to be exactly the same for all inertial frameworks, thus becoming a universal constant and rendering the ether superfluous.
Fame is a Fickle Food
Upon a shifting plate [3].
Not all the kudos in science finishes up where it rightly belongs, and a discussion of the pros and cons of the many examples would fill a lengthy monograph. However, in this case the reason for Einstein’s fame is because his conjecture provides a full justification for the Lorentz transformations and leads us, as we shall see, to some astonishing conclusions.
What does it mean when we say that the speed of light is constant in all inertial frames of reference? Let us imagine that, in our example from the previous Sections, Peter, instead of throwing a ball towards Mary, directs a ray of light past her. He measures the speed of the light as it leaves him and obtains a value of c kph. Mary then measures the speed of the light as it passes her, and, because the train is moving at 10 kph, using Galilean transformations we would expect her to obtain a value of c-10 kph. However, she doesn’t. She also obtains a value of c kph.
We would surely expect something of this kind if the speed of light were infinite. Adding or subtracting finite numbers from infinity yields infinity.4 However, although very large, c is not infinite.5 The error of 10 kph, compared with the value of c, amounts to roughly 1 part in 100 million, and in most circumstances it could be completely disregarded. However, when, instead of balls, subatomic particles are shot out in nuclear reactions, they may be moving at speeds comparable with c. In this case the error obtained by using Galilean transformations would be quite large.
Indeed the consequences of Special Relativity, as the new field springing out of Einstein’s conjecture was called by Einstein himself, are astonishing and manifold, resulting in some very bizarre effects, rivalling those we encountered with QM in the last Chapter. They made Einstein an instant (and sometimes controversial) celebrity, even among the general public. In the next Sections, we shall discuss briefly some of these effects: notably time dilation, Lorentz contraction, the puzzle of simultaneity and mass/energy equivalence.
6.6 Time Dilation
In the real world, as distinct from the world of fiction and fairy tales, time has always flowed inexorably from the past to the present, and at the same rate for everybody, no matter where they are or what is their state of motion. Or so it was believed, Giang Huong and Tu Thuc notwithstanding. However, the Special Theory of Relativity put an end to this complacency.
Let us remove Mary from the comfort of her railway carriage, train her as an astronaut, and place her on a high-velocity rocket ship, which can attain speeds comparable to (but never equalling or exceeding) that of light. When Peter looks into the spaceship moving away from him, he notices an amazing thing: everything and everyone on board appears to be in a state of slow motion. Even the cabin clock has slowed down. Comparing it with his own earthbound timepiece, Peter observes the second hand of Mary’s clock completing only one revolution of the dial in the time it takes his own clock to complete two. Time is actually moving more slowly in Mary’s frame of reference than in his own. She is indeed in a sort of Land of Bliss. This effect, which is known as Relativistic Time Dilation, is a direct consequence of the Lorentz transformations. However, even without using these transformations, the effect is easily deduced from Einstein’s assumption that the speed of light is the same for all observers in inertial frameworks.
Ray of light passing between two points in spaceship: a when ship is stationary; and b when ship is moving to the right with velocity v
Now what does Mary think when she peers out of the spaceship rear window and sees Peter vanishing off into the distance? To her, everything in Peter’s frame of reference is moving more slowly than in hers. She sees the second hand on his clock crawling around the dial, and suddenly realises that time itself is changing more slowly in Peter’s surroundings than in her own. When she returns from her voyage, Peter will still be a young man, and she will be older.
Of course, a paradox is immediately evident: both Peter and Mary see the other person ageing less rapidly than themselves. They can’t both be right. This contradiction is called the twin paradox, as it is usually formulated in terms of two identical twins travelling apart on separate space ships.
In mathematics and physics, a paradox is normally an indicator that we have made a mistake in our logic. Is Special Relativity wrong? No, but we have indeed violated the conditions under which it applies. Special Relativity deals with inertial frames of reference only, which means that Peter and Mary must be moving apart at a constant velocity for its provisions to apply. In that case, they can never get back together to compare their clocks or lament their wrinkles. The only way this can happen is if Mary turns the spaceship around and brings it back home. In doing so, she undergoes an acceleration, positive and/or negative, and thereby violates the conditions of Special Relativity. The paradox is therefore resolved. However, time dilation will still occur, and as a consequence of the acceleration she has experienced, Mary will age less than Peter.
Time dilation is extremely small, except for velocities v close to c. The energy required to accelerate a rocket ship increases rapidly as its speed approaches that of light, for reasons that will become apparent later in this Chapter. The highest speed attained by a rocket was achieved in 2006, when an Atlas V Rocket, carrying the New Horizons probe to Pluto, was launched from Cape Canaveral in Florida, USA. Its maximum speed, relative to the launch pad was 58,536 kph (about 16.3 kms per second), which yields a v/c ratio of 0.0000542. Accordingly, its relativistic time dilation effect was of the order of one in a billion, which is very small, but significant over a long journey.
The first direct test of time dilation for normal objects and velocities was carried out by Hafele and Keating [4] in 1971. They flew four caesium atom-beam clocks around the world in commercial jet airliners, both eastwards and westwards, and compared the clocks afterwards with reference clocks that had remained behind in the U.S. Naval Observatory. The clocks lost 59 ± 10 ns during the eastward flight and gained 273 ± 7 ns during the westward flight, compared with the land-based clocks. The difference between the eastward and westward results is explained by taking into account the rotation of the earth. These figures agree well with the predictions of Einstein’s theory of Relativity.
It should be noted that the altitude of the airliner varied throughout the flight, but was generally around 9000 m. At this altitude, the gravitational force experienced by the clocks in the aircraft was less than that experienced by the ground-based clock. As we shall see in the next Chapter, gravitational fields produce another form of time dilation, which must be included in the analysis of the results of the Hafele-Keating experiment.
A routine application of Einstein’s theories may be found in a device that most of us use regularly. Because of the difference in altitude and speed between GPS satellite transmitters and the receivers in the navigators of our vehicles and mobile phones, corrections for time dilation—due to the relative motion of the satellites and our phones, and to the different gravitational fields that they experience—must be included in the software that is used to calculate GPS positions.
When physicists explore the domain of cosmic rays, they encounter particles that are moving at speeds approaching that of light, when time dilation effects are much more pronounced. The muon is such a particle. In Chap. 9 we shall discuss muons, and other subatomic particles, in some detail. Here we need only note that the muon’s average lifetime, when it is at rest, is 2.2 millionths of a second. In this time, if we disregard the time dilation of Relativity, even if the muon were travelling at the velocity of light, it would only travel 660 m. However, time dilation cannot be disregarded: thanks to it, physics students in their laboratories today routinely observe the arrivals of thousands of muons from the upper atmosphere, i.e. from distances of 15 km to more than 100 km.6
6.7 Lorentz Contraction
Suppose now that we manage to hitch a ride on a muon on its way from the upper atmosphere to the earth’s surface below. We have done the calculations and are aware that with the muon’s average lifetime of 2.2 µs, we will only have travelled an average of about 660 m before the muon decays beneath us. And yet when we were back on earth, we had observed most of the muons arriving safely, having travelled much further. It might seem that the outcome depends on which frame of reference we choose for our calculations. Doesn’t this violate Einstein’s edict that the results of experiments should be independent of the frame of reference?
No, because the Lorentz transformations contain another surprise: not only does the rate of passage of time vary with the speed, but so also does the length. An observer travelling at a certain velocity relative to an object will notice a decrease in the length of the object in the direction of motion.
There is a reciprocity between time dilation, and what is now called the Lorentz contraction of distance. Observer A in the laboratory on earth observes a long-lived (due to time dilation) muon travelling several kilometres to the earth; Observer B on the muon sees a short-lived muon travelling a much shorter distance (due to Lorentz contraction) to the earth’s surface. Thus both observers see the muon arriving safely at the laboratory before the muon decays.
These concepts have now been taught in physics classes for the best part of a century. However, when first published by Einstein, they puzzled not only lay people, but specialists as well. Shortly after the publication of Einstein’s paper, it was asserted by Croatian mathematician and theoretical physicist, Vladimir Varicak, trying to come to grips with the new theory, that the length contraction is real according to Lorentz, while it is only apparent, or subjective, according to Einstein. Einstein “clarified” the situation, as follows:
“The author unjustifiably states a difference between Lorentz's view and mine concerning the physical facts. The question as to whether length contraction really exists or not is misleading. It doesn't “really” exist, in so far as it doesn't exist for a co-moving observer, though it “really” exists, i.e. in such a way that it could be demonstrated in principle by physical means by a non co-moving observer” [5].
6.8 Simultaneity
Mary in the moving carriage fires a flash, the effects of which are also observed by Peter on the platform
What follows from her point of view is straightforward: light from the flash travels at speed c, as measured by her, and reaches both ends of the carriage at exactly the same time. Nothing strange there. Now let us consider what Peter sees. The light travelling towards the rear of the train will reach the back end of the carriage before the light travelling forwards reaches the front end. Why? The reason is that the back end of the carriage is moving towards the flash, and thus the light has a shorter distance to travel, compared with the light travelling towards the front end, which is moving away. So, the two events (when the light impacts on each end of the carriage) appear to be simultaneous to Mary, but not to Peter, who sees light arrive at the rear end first.
The implications arising from the dependence of the simultaneity of events on the state of motion of the observer are profound. Will someone watching a murder from a speeding carriage see the victim die before the trigger of the revolver is pulled? Relativity protects us from such absurdities. Events that are causally connected, i.e. where one event is a direct consequence of the other, cannot have their order reversed by Lorentz transformations. All observers, irrespective of their motion, see these events in the correct sequence.
However, we saw in the last Chapter when discussing Entanglement, that Quantum Mechanics plays free and easy with the concept of simultaneity. When an observation specifies the state of one of a pair of entangled particles, the state of the other is simultaneously specified, no matter how far away that particle is located. There is a disagreement here with the Theory of Relativity, and it cannot be swept under the carpet. We shall discuss this point further in Chap. 12, Part 3.
Let us now move on to the final topic in our brave new world. It is so momentous, not just to physics, but to the history and future of humankind, that we break the rule we have set ourselves for this book: we allow ourselves just one mathematical formula.
6.9 The Only Formula in This Book
Before Einstein, there existed two separate conservation laws in chemistry and physics; i.e. the Law of Conservation of Mass, and the Law of Conservation of Energy. In chemical reactions, the total mass of the reactants had to be equal to the total mass of the reaction products. In physics, it had long been known that energy can be converted from one form to another (e.g. potential energy to kinetic energy to heat), but never created or destroyed. Since Einstein, these two quantities, mass and energy, can be considered as different forms of a single entity. In nuclear reactions, mass can be converted to energy, and vice versa. The same is also true in chemical reactions, but the energies involved are so small that any change in mass of the reactants is unobservable.
Einstein’s formula is remarkable from a philosophical point of view because it unexpectedly relates together three variables which nobody had ever before dared to assume were associated, i.e. mass, energy and the speed of light. It may be interesting to some readers to recall that a similar kind of unexpected association between apparently unconnected quantities can be found in mathematics. We refer to Euler’s formula, which establishes a fundamental relationship between the hitherto unrelated fields of trigonometric functions, exponentials and complex numbers.7 The physicist, Richard Feynman, called Euler’s equation “our jewel” and “the most remarkable formula in mathematics". It may be a subject of philosophical speculation to investigate the underlying reasons for these (and a few other) unexpected connections.
A formal demonstration of the mass/energy relationship is beyond the scope of our book, since it involves a lengthy mathematical treatment. (A somewhat simpler justification of it can be found in [6].) What is more instructive for our purposes here is to examine some of its consequences.
We begin by considering the problem of an accelerating rocket ship. In classical physics, as the rocket consumes fuel and ejects the combustion products, it receives a propulsive force which drives it forwards. As long as this force remains, the rocket will continue increasing its velocity, and thereby its kinetic energy. According to Einstein, however, some of this kinetic energy will be transformed into mass, thus making the rocket ship more difficult to accelerate further. The closer the rocket’s velocity gets to the speed of light, the more pronounced this effect becomes. If the rocket’s speed were ever to reach c, its mass would be infinite. This is mathematics’ way of telling us that it is not possible ever to accelerate a rocket, or any other material object, to speeds greater than or equal to c. The velocity of light is a universal limiting speed that nothing can exceed.
One may raise the objection that photons are particles, and surely, since they are particles of light, they travel at speed c. Yes, they do, but they are a special type of particle, possessing no rest mass. The rest mass of a particle is, as the name implies, the mass of the particle when it is stationary. When it is in motion, its total mass is greater than its rest mass because of the contribution from the conversion of kinetic energy into mass, as we have discussed in the previous paragraph. Photons, having no rest mass, must spend their entire existence zipping hither and thither at speed c. They can never slow down.
This upper limit on the speed at which material objects, e.g. astronauts, can travel has long been the bane of science fiction writers, as it puts restrictions on how far one can travel in a human lifetime. Given that the closest stars, located in the Alpha Centauri system, require 4.3 years of travel at speed c to reach them, it is clear that by far the vast majority of the cosmos is beyond the reach of humankind. The invention of Hyperspace, a region outside the realm of Relativity, has become an accepted method in the domain of science fiction to overcome the limits imposed by Einstein’s theory. In the real world, we can only sit and peer through our telescopes in frustration or find something else nearer to home to sate our curiosity.
Let us now put our wistfulness aside and return to the application of Einstein’s formula to more earthly pursuits. Its importance and huge impact on the history of the twentieth century is due to the magnitude of the coefficient c2 in our lonely equation. To give a simple example, if one kilogram of matter collides with one kilogram of antimatter (we will discuss antimatter in Chaps. 8 and 9), they annihilate each other: the two masses disappear and as much as fifty billion kilowatt-hours (kwh) of energy are produced.
To put this number into perspective, we recall that in the oil industry, the unit toe (tonne of oil equivalent) is defined as the amount of energy released by burning one tonne of crude oil, and is approximately equal to 11,600 kwh. Hence the annihilation of 1 kg of matter and 1 kg of antimatter would yield the same energy as burning 4.3 million tonnes of crude: i.e. almost enough to provide for the entire energy needs of the United States for one day (about 6 million toes).
In reality, however, kilograms of antimatter are difficult to find, and the practical transformation of mass into energy relies on one of the two processes of nuclear fission and nuclear fusion. The result is either nuclear energy, released under controlled conditions and deployable for practical purposes, or a violent burst of a huge quantity of destructive energy. The latter is the atomic bomb, if nuclear fission is employed, or the H-bomb, with the potential for an even more devastating effect, if nuclear fusion is adopted.
Given the consequences, should we blame Einstein (or science) for having incautiously opened such a Pandora’s Box? Or rather humans, for their stockpiling of a terrifying arsenal of weapons of mass destruction? This question has been debated for the past seventy years, and we shall not discuss it further here.