Why Does E=mc2? (And Why Should We Care?), by Brian Cox and Jeff Forshaw, Da Capo, 242pp, £12.99, ISBN: £12.99, 978-0306817588
E=mc2 is without a doubt the most famous equation in the world. While most people can repeat it, few grasp its significance and even fewer understand the physics behind it. This book aims not only to explain this physics but also its importance today. As well as answering both questions posed in its title in a surprisingly clear and comprehensible manner, the book brings home to the reader the beauty of the workings of the universe described in mathematical equations, each tying in with the other in a breathtaking tapestry.
In 1960 the Nobel-prizewinning theoretical physicist Eugene Wigner focused on the strangeness of this phenomenon in his essay “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, where he states that “it is not at all natural that laws of Nature exist, much less that man is able to discover them”. But laws of nature do exist and we have been able to discover many of them, though what is perhaps most remarkable of all is that we are able to express these laws so well using the language of mathematics. The simplicity of Einstein’s equation E=mc2 is just one example of this.When most of us picture Albert Einstein, probably the most famous scientist of all time, we think of an elderly man with unruly white hair sticking his tongue out at the camera. However, the special theory of relativity and its poster child, the equation E=mc2, along with much of the rest of Einstein’s most important work – such as the photoelectric effect, which is the basis of solar energy, and his work on the statistical mechanics of particles in a fluid or gas – was produced by him when he was a relatively young man. In 1905, when he published four seminal papers, he was a twenty-six-year-old technical expert, third class, in the Swiss patent office, a position he took up having failed to secure a teaching post at either university or secondary school level. Shortly before he published his paper on special relativity in 1905 he had been turned down for promotion to technical expert second class, a grade he finally attained a year later. He secured a “proper” academic position only in 1908. While it may at first appear that Einstein was a lone maverick scientist working in isolation he was actually in regular contact with other scientists whom he met socially and with whom he discussed physics, often late into the night. While he worked in the patent office by day, physics was relegated to his spare time, but he later spoke of this as the happiest period of his life because it gave him time to think about physics. Scientific views in this period were changing. Physics throughout the nineteenth century was marked by an ever increasing emphasis on the use of mathematics to describe the physical world. Research focused mainly on the so-called imponderables: light, heat, electricity and magnetism. Each was studied individually, but as the century progressed it became apparent that they were related. As the German-American academic JB Stallo put it in 1880, these phenomena appeared to be “different manifestations of the same fundamental energy”. It was becoming evident that there might be more to the workings of the universe than the theories of Newton allowed.Isaac Newton’s Philosophiae Naturalis Principia Mathematica (often referred to as his Principia), published in 1687, is considered to be one of the most influential scientific texts ever written. In it Newton described universal gravitation, in which he demonstrated how the motion of objects here on earth and of the celestial bodies are governed by the same natural laws, the conservation of momentum and angular momentum, and three laws of motion that define the relationship between force and movement and which became fundamental to mechanics. In the post-Newtonian world view it was believed that everything was made up of small parts that obeyed Newton’s laws and that this was the full theory that could explain everything in the universe. There were a few minor questions that remained to be answered – such as what the little parts are made of and how they actually stick together – but it was believed that once a few details were filled in there would be a unified theory of everything. However, as the nineteenth century progressed, more and more phenomena were observed that appeared to contradict Newton’s laws. This led to modern physics, in which it is no longer believed that Newton’s theory lies at the heart of everything but that it is, in fact, just an approximation of nature. It appears to be correct for many of the phenomena around us but as we approach the speed of light it becomes less and less accurate.
But to get to where physics stood at the beginning of the twentieth century, when Einstein developed his theories of relativity, explanations of much earlier theories and experiments are first required. Going as far back as ancient Greece and Aristotle’s view of the world – that the earth is stationary at the centre of the universe and the sun, moon, stars and other planets orbit around it attached to fifty-five concentric crystalline spheres stacked inside one another like Russian dolls – Cox and Forshaw dismantle the notion of absolute space following in the steps of Galileo Galilei. Galileo’s principle of relativity states that there is no such thing as absolute motion; we can only say that we are stationary or moving relative to another object and we have no absolute position in space, only positions relative to other objects. While Galileo was correct in his elimination of the concept of absolute space he believed that time was immutable: that it would be possible to place a clock on the surface of the earth, another on the surface of the sun and another orbiting the Milky Way and, provided they all started off telling the same time (and were of good quality), they would continue to tell the same time forever and ever. Here Galileo was wrong. Time is not absolute. This counterintuitive concept is quite difficult to swallow, but what is truly incredible is that the experiments that first undid the notion of absolute time were carried out on a laboratory desktop with relatively basic apparatus and are the same kinds of experiments that led to a deeper understanding of electricity and magnetism and the foundations of much modern technology. The majority of these experiments were carried out by a man considered by many to be the greatest experimental scientist of all time, Michael Faraday.
Faraday (1791-1867), the son of a Surrey blacksmith, had only a basic formal education and at the age of fourteen became apprenticed to a bookbinder and bookseller. During his seven years’ apprenticeship he had the opportunity to read a wide range of books and developed a fervent interest in science, and electricity in particular. Towards the end of his apprenticeship, at the age of twenty, he attended a series of lectures given by the eminent scientists Humphrey Davy, of the Royal Academy and the Royal Society, and John Tatum, of the City Philosophical Society. He took extremely detailed notes and afterwards posted Davy a three-hundred-page book based on the lectures. Davy was so impressed that when he damaged his eyesight in an experiment he decided to employ Faraday as a secretary: he went on to become an assistant at the Royal Institution and has been called Davy’s greatest scientific discovery. While his family background meant that Faraday was never really accepted as a gentleman in class-based British society, his time at the Royal Institution provided him with the opportunity to meet the European scientific elite and gave him access to a host of stimulating ideas.
Michael Faraday carried out his meticulous experiments, which ultimately led to the mathematical expression of electromagnetism, with equipment as simple and inexpensive as a few coils of wire, magnets and a compass, long before Thomas Edison had invented a reliable lightbulb. He noted that if you send an electric pulse through a wire in the vicinity of a compass the compass needle (which is a magnet) will deflect in time with the electric pulse. Faraday realised that he was seeing a connection between electricity and magnetism – two phenomena that had previously appeared unrelated. He found, through careful experimentation and observation, that electric currents make magnetic fields and moving magnets generate electric currents. These two phenomena, known today as electromagnetic induction, are the basis for generating electricity in all modern power stations and all the electric motors we use every day, from power tools to disc drives. His experiments produced a large amount of data, which was published. However, until a mathematical expression of the relationship between electricity and magnetism was developed the usefulness of the experimental data was limited.
Three years before Faraday’s death, James Clerk Maxwell (1831-1879), a Scottish theoretical physicist, wrote down a series of equations that expressed the observations of Faraday and many other experimental scientists on electricity and magnetism. These equations provided the key to a deeper understanding of nature, which the experimental data alone could not. Maxwell’s equations express the relationship between electric charges and currents and the electric and magnetic fields they create. These equations give us a simple and unified picture of electricity and magnetism that allowed all of Faraday’s and others’ desktop experiments to be predicted and understood and have enabled us to create far more complex electrical and electronic technologies. In order to be mathematically consistent, Maxwell was forced to add to his equations an extra piece that was not required by the experimental data. He called this extra piece a displacement current. The displacement current meant that these equations could be recast in the form of wave equations – equations that describe the motion of waves. Maxwell had unintentionally predicted the existence of electromagnetic waves. These equations also indicated that the waves move forward at a fixed speed, defined by the ratio of the strengths of the electric and magnetic fields, two easily measurable quantities. The speed of electromagnetic waves as predicted by Maxwell’s equations and Faraday’s experiments is 299,792,458 metres per second. This value is the same as c, or the speed of light, which was already known at the time. Maxwell had inadvertently stumbled across an explanation of light itself as an electromagnetic wave.
The mathematical beauty of Maxwell’s equations impressed Einstein so deeply that he decided to take seriously the idea that the speed of light is constant for all observers. Einstein carried out a thought experiment to see what consequences might arise from this. He considered a clock that consisted of two mirrors between which a beam of light bounces, each bounce being equivalent to one tick, and placed this clock on a high speed train. He then considered the time it would take for one tick to occur according to an observer on the train compared with an observer standing on a station platform as the train whizzed past. Einstein came to the astonishing conclusion that if the speed of light is the same for all observers, one tick of the clock will appear longer for the person on the station platform than for someone on the train moving at the same speed as the clock. This is because the mirrors themselves will move, relative to the stationary observer, in between the bounces of light. The somewhat disturbing implication of this is that time moves forward at different rates depending on how we are moving relative to someone else, doing away with Galileo’s concept of absolute time.
The fact that neither space nor time is universal or absolute leads to another conclusion: that space and time are not in fact separate. They are both part of the four-dimensional space-time continuum. In 1908, Hermann Minkowski, a friend and tutor of Einstein, wrote an obituary of the separate concepts of space and time: “From henceforth, space by itself, and time by itself, have vanished into the merest shadows and only a kind of blend of the two exists in its own right.” So we are all moving through the space-time continuum; and while we can reverse our direction in space we can travel in only one direction along the time axis. In fact, we are all hurtling through space-time at the same cosmic speed limit – c or approximately 300,000 kilometres per second.
What is so special about light is that it uses up all of its 300,000 kilometres per second travelling through space, whereas the rest of us use a large proportion of it travelling through time too. In fact, no particle that possesses mass can actually reach or exceed the speed of c through space while all massless particles (like light) always travel at the speed of c through space and don’t use up any of their space-time speed quota travelling through time. We can take different paths through space-time and arrive at the same destination having got there using a path that travelled more through space or through time. This leads to such wonderfully unsettling puzzles as the Twins Paradox. If a set of twins is born here on earth and one heads off in a spaceship travelling at high speeds to the far reaches of the galaxy then turns around and comes back to earth again, she will find that her twin has aged much more rapidly than she has and while she is still young her earth-bound twin is an old lady.
In reaching the conclusion that E really does equal mc2 the writers recall Newton’s law of the conservation of momentum. The momentum of an object depends on its mass, which is represented by m in Einstein’s equation, and its speed; it can be thought of as the difference between being hit by a tennis ball travelling at forty miles per hour and being hit by a train travelling at the same speed. Both objects are moving at the same speed but because of its greater mass the train will do a lot more damage. Conservation of momentum means that in, for example, a collision between two billiard balls the total momentum before the collision must be the same as the total momentum after the collision even though the speeds of the two balls may change. This is a description of momentum in space, but in Einstein’s relativity we cannot consider space alone: we must have an expression for momentum in four-dimensional space-time.
Cox and Forshaw describe momentum in four dimensions using diagrams of arrows and some basic algebra that is explained quite clearly and simply. The explanation is reasonably easy to follow, though it requires considerable concentration on the part of the reader. At low speeds the space-time momentum is the same as the momentum in space alone and this is how Newton’s laws were completely right in the world that we are used to living in, that of large things moving slowly. The closer an object’s speed gets to the speed of light the more their space and space-time momenta differ. Another law of conservation also comes into play, the conservation of energy, a law that was discovered during the Industrial Revolution by a number of engineers and experimenters who found that while energy can be converted from one form to another it cannot be destroyed. This is the principle behind the steam engine, in which heat energy is converted into movement. By showing that the energy of a moving object is equal to mc2 + ½mv2 and that the kinetic energy of the object (its energy due to its motion) is ½mv2, where v is the velocity of the object, we come to the conclusion that the first part of mc2 + ½mv2 must be an inherent energy due to the mass of the particle itself. In other words, there is an amount of energy stored in matter simply by virtue of the fact that it has mass and this energy can be expressed as E = mc2. The reader can’t help but feel a tingle of excitement when this most famous of equations is derived.
The next portion of the book deals with the second part of the title: Why should we care? In some ways this is a more difficult question than the first. But the truth is that the conversion of mass to energy takes place around us all the time. It is fundamental to how we heat our homes, how the sun heats the earth; it is even part of the reason that any elements other than hydrogen exist. After burning a log of wood the mass of the ash left behind after the fire plus the mass of the gas that floated away will be slightly less than the mass of the log before burning it. This is because some of the mass of the original log was converted into energy, mainly in the form of heat. A small amount of mass can produce a very large amount of energy; one millionth of a gram would be enough to heat a house for a period of eight hours if it were entirely converted to heat energy. This sounds very exciting and if we were able to efficiently convert mass to energy we would have solved the world’s energy crisis. Unfortunately, we have not, as yet, been able to convert anything like one hundred per cent of an object’s matter into energy. The conversion of mass to energy is, however, the basis for nuclear power and its rather unpleasant cousins, nuclear weapons. It is also the way in which our biggest energy source, the sun, generates energy, constantly converting mass to energy and sending it out throughout the solar system.
At the heart of the atom lies the nucleus, made up of positively charged protons and uncharged neutrons. The negatively charged electrons are in a cloud around the outside of the atom. If two nuclei are brought very close together they may fuse into one larger nucleus – or the nucleus of a heavier element – a process known as nuclear fusion. As long as the two original nuclei are lighter than iron, the mass of the final nucleus will be slightly less than the total mass of the two original nuclei, the rest of this mass being released as energy. Because the nucleus is positively charged and like charges repel each other it is very difficult to bring two nuclei close enough together for fusion to take place. However, if the nuclei are moving at high enough speeds they can overcome their mutual repulsion and fuse. This means that they need to be at an extremely high temperature (temperature is just a measure of how fast the atoms or molecules in a material are moving). A temperature of about 10 million degrees centigrade is required for nuclear fusion – not a temperature we reach very easily here on earth so fusion doesn’t take place around us very often. Fortunately for us, these kinds of temperatures are reached in the hearts of stars – like our sun.
Hydrogen is the lightest element in the periodic table of elements – it contains just one proton and one electron – and it is the most abundant element in the universe. Stars like the sun form when hydrogen atoms start to clump together under the force of gravity. As the clump grows more atoms are attracted and the force of gravity increases. The atoms in the star are constantly falling inwards more and more quickly as the star grows – this means that the temperature rises. When the temperature reaches about ten thousand degrees the electrons are ripped away from their atoms and protons and electrons whizz around in an ion gas known as a plasma. The star continues to heat up until finally it reaches ten million degrees and fusion begins to take place. Massive amounts of energy are generated and heavier elements are created. Every second our own sun turns six hundred million tons of hydrogen to helium (the second element in the periodic table), losing four million tons of mass with each second that passes. The sun will eventually run out of hydrogen to convert to helium but this won’t happen for a few billion years and so is not a pressing concern just yet for the inhabitants of earth.
Nuclear power plants here on earth rely on another means for the generation of energy: nuclear fission. Nuclear fission is the process by which the nuclei of heavy elements spontaneously split up into lighter ones. Some elements, like uranium and plutonium, can exist in different forms, known as isotopes – they contain the same number of protons and electrons but a different number of neutrons. Some of these isotopes decay by nuclear fission into lighter elements, the combined mass of which is less than the mass of the original nucleus, the excess mass being converted into energy. In a nuclear power plant unstable forms of uranium or plutonium are created, which then decay and release energy. Nuclear fission is a process that also happens naturally on earth, deep inside the earth as well as in some rocks on the earth’s surface. The most stable naturally occurring form of uranium contains ninety-two protons and 146 neutrons and has a half-life of around 4.5 billion years. This means that after 4.5 billion years half the atoms will have split up into lighter elements, the heaviest of which in this case is lead. There also exists a less stable isotope of uranium that contains 143 neutrons, which decays into a different form of lead, and has a half-life of 704 million years. The decay of uranium into lead can be used to discover the age of some very old rocks that don’t naturally contain lead but do contain uranium. As the uranium decays lead starts to appear in the rocks. By counting the lead atoms present in a sample the age of the rock can be calculated. The energy that is generated when uranium atoms split up inside the earth also plays an important role in keeping the earth warm. Without it earth would be a much colder planet and humans might never have evolved.
The final two chapters of the book go beyond the title. The first delves into what is currently our best explanation for the laws of nature, the Standard Model of particle physics, and goes some way to explain the origins of mass as well as the physics behind and the motivation for CERN’s large hadron collider. Finally, Einstein’s theory of general relativity, in which the space-time continuum warps under the force of gravity, is tackled. Einstein did not publish his generalised theory of relativity until about ten years after E=mc2, partly because of the complexity of the mathematics involved. This means that the theory is tremendously difficult to explain to the lay person and the authors are required to omit a great deal. However, the reader gains a sense of the weird world in which the earth travels in a straight line but the gravitational field of the sun bends space-time so that the earth appears to move in a circle.
Einstein’s contributions to science and physics cannot be overstated. His work changed the way scientists view the world and opened the door to modern physics. However, as with all scientists, his achievements built on the work of those who came before him. Thousands of years of human endeavour have brought us to our current understanding of the universe. As Isaac Newton put it: “If I have seen further than others, it is by standing upon the shoulders of giants.” Meaningful scientific advancement is impossible without an understanding and appreciation of what has already been done and Einstein understood this as well as anyone, keeping portraits of Isaac Newton, Michael Faraday and James Clerk Maxwell on the wall of his study. Future breakthroughs may reveal that Einstein and his theories of relativity are not entirely correct and like Newton’s theories merely an approximation of nature. To quote Newton again: “I do not know what I may appear to the world, but to myself I seem to have been only a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.” It is very likely that most of that vast ocean is still undiscovered.
Anyone interested in learning more about Einstein’s theories and the workings of the universe would do well to read this book. The writers’ belief that absolutely no experience of mathematics is required by the reader may be a little over-optimistic and means that explorations of the fundamentals of relativity are mixed up with explanations of basic algebra. However, the book is very clear and is an enjoyable and enlightening read in which two very clever physicists (one of whom previously achieved fame as the keyboard player in the pop group D:Ream) offer an accessible path to some seemingly forbidding physics.
Eva McGuire is a graduate of Trinity College Dublin where she studied physics. She is currently studying for a PhD in biophysics at Imperial College London.