The Universe Within

by Neil Turok

  When I was ten, we moved to London, England, just in time to see the Apollo 11 lunar landing and watch Neil Armstrong step onto the moon. Who could ever forget the picture of Earth as a gorgeous blue marble floating above the moon’s horizon? We were swept up in the moment and filled with optimism for the future.

  It was the end of the sixties, and space was suddenly the coolest thing around. It’s hard to convey the sense of the excitement, how the space program bound together people from all walks of life and every political opinion. It symbolized a certain spirit — ambitious and aglow with the crazy idea of using technology to fling a climbing rope up to the cosmos.

  Equally as enthralling as the moon landing was the drama of Apollo 13, only one year later. Imagine you’re 320,000 kilometres from home, out in the void of deep space, and you hear a loud bang. “Houston, we have a problem . . .” One of two main oxygen tanks had exploded, leaking precious oxygen into space over the next two hours. The three astronauts crowded into the only lifeboat they had: the little lunar explorer capsule, which had nowhere near the fuel they needed to get back to Earth. The drama was incredible. There were daily bulletins on TV. All over the world, people were biting their nails. How could the astronauts possibly survive?

  NASA’s engineers came up with a fantastic solution. They used the moon’s gravity to pull them towards it, then slingshot the little pod around its dark side and back to Earth. A few days later, there the astronauts were, their hot little tin can dive-bombing into the Pacific, where they were fished out and then, incredibly, waving to us from the TV, gaunt, unshaven, but alive. Everyone survived. It was pure magic.

  The trajectory for this manoeuvre was computed using the equations discovered by the founder of the field we now call theoretical physics, and also one of the most capable mathematicians of all time: Isaac Newton.

  Newton, like Galileo, was an outsider. He came from an ordinary background but possessed an extraordinary mind. He was deeply religious but highly secretive about his beliefs. And understandably so, since, for example, he passionately rejected the idea of the Holy Trinity while spending the duration of his scientific career at Trinity College in Cambridge. Newton seems also to have been motivated to a large degree by mysticism — he wrote far more on interpretations of the Bible and on the occult than he ever did on science. The famous economist John Maynard Keynes studied Newton’s private papers, a box of which he had acquired at auction, and came to this conclusion: “Newton was not the first of the age of reason. He was the last of the magicians, the last of the Babylonians and Sumerians, the last great mind which looked out on the visible and intellectual world with the same eyes as those who began to build our intellectual inheritance rather less than 10,000 years ago.”20

  Newton spent most of his early scientific years on alchemy, researching transmutation (turning base elements into gold) and trying to find the elixir of life. None of these efforts were successful; he seems to have succeeded only in poisoning himself with mercury. This poisoning may have contributed to a nervous breakdown he is believed to have suffered around the age of fifty-one, after which he largely gave up doing serious science.

  Newton’s mathematical researches were his magic that worked. He searched for mathematical formulae that would describe the motion of objects on Earth and the planets in space. He found spectacularly simple and successful answers. In the late sixteenth century, a series of very accurate measurements of the motions of celestial bodies were made by the astronomer Tycho Brahe from the world’s greatest observatory of the time, Uraniborg in Denmark. Brahe’s protégé, German mathematician and astronomer Johannes Kepler, had successfully modelled the data with some ingenious empirical rules. It fell to Newton to develop Galileo’s insights into a complete mathematical theory.

  BEFORE GALILEO, COPERNICUS HAD pioneered the idea that the Earth was not the centre of the universe. The prevailing wisdom, tracing back to Aristotle and Ptolemy, held that the sun, moon, and planets moved around the Earth carried on a great interlocking system of celestial spheres, which could be carefully arranged to fit the observations. Aristotle claimed that it was just in the Earth’s nature not to move. Earthly bodies followed earthly laws, and celestial bodies obeyed celestial laws.

  Newton’s point of view was quite different: his law of gravitation was the first step on a path towards “unification,” a single, neat set of mathematical laws describing all of physical reality. It was the most far-reaching idea, that exactly the same laws should apply everywhere — on Earth, in the solar system, right across the cosmos. Newton’s law of gravitation states that the gravitational force of attraction between two objects depends only on their masses and how far apart they are. The more massive the object, the more strongly it attracts and is attracted. The farther apart two objects are, the weaker the force of attraction between them.

  In order to work out the consequences of this law of gravity, Newton had to develop a theory of forces and motion. It required a whole new type of mathematics, called “calculus.” Calculus is the study of continuous processes, such as the motion of an object whose position is given as a function of time. The velocity measures the rate of change of the object’s position, and the acceleration tells you the rate of change of the velocity. Both are calculated over infinitesimally small times, so calculus implicitly rests on a notion of infinitely small quantities. Once he had developed it, Newton’s theory had applications well beyond gravity or the solar system. It predicts how any collection of objects will move when any set of forces acts upon it.

  In describing the motion of objects, Newton’s starting point was an idealization. How would an object behave if it were released in empty space, with nothing else around it? To be specific, picture a hockey puck floating all alone in an absolute void that stretches to infinity. Let’s ignore gravity, or any other forces. What would you expect the puck to do? If it was all alone, and there was nothing nearby to measure its position from, how could you tell if it was moving?

  Now imagine a second hockey puck, also floating freely in the void. Picture two tiny people, each of them standing on one of the pucks and seeing the other puck some way off. What do they see? And how will each puck move?

  Newton’s answer was simple. According to the view from either puck, the other puck will move in a straight line and at a constant speed, forever. If you imagine more and more pucks, with none more special than any other, then according to every puck’s viewpoint, every other puck will move in the same way. This was Newton’s first law of motion: in the absence of forces, the velocity of any object remains constant.

  Let us come back down to earth, to a perfectly smooth, slippery ice rink. The world’s greatest Zamboni has just gone over it. Imagine a puck sliding along the ice in a perfectly straight line. But now you skate alongside it and push it with your stick. Push on its side and the trajectory will curve; push behind it and you can speed it up. Newton’s second law describes both effects in one equation: force equals mass times acceleration.

  Finally, when you push on anything — the puck, another person, or the side of the rink — it pushes back at you equally hard. This is described by Newton’s third law, which says that for every force there is always an equal and opposite force.

  Newton’s three laws are simple but incredibly powerful. They describe everything known about motion prior to the twentieth century. In combination with his law of gravitation, they explain how the force due to the sun’s gravity pulls the planets inwards — just as a string pulls on a whirling stone — and bends the motion of the planets into orbit around it. According to Newton’s third law, just as the string pulls in the stone or the sun pulls the Earth around it, the stone pulls the string out and the Earth pulls back on the sun, causing the sun’s position to wobble back and forth slightly as the Earth goes around it. The same effect is now used to search for planets in orbit around other stars: the slight wobble in a distant star’s position causes a tiny modulat
ion in the colour of the light we receive, which can be detected. More familiar is the effect of the moon’s gravitational pull on the water in Earth’s oceans, which is responsible for the tides.

  Implicit within these laws was the idea, dating back to Galileo, that it is only the relative positions and motions of objects that really matter. Galileo pointed out that a person travelling in the hold of a ship, which is sailing steadily along, simply cannot tell from watching anything inside the ship — for example, a fly buzzing around — whether the ship is moving. Today we experience the same thing when we sit in an aircraft moving at 1,000 kilometres per hour and yet everything feels just as if we are at home in our living room.

  In our ice-rink world, we can see the same effect. Imagine two pucks that happen to be sliding along the ice exactly parallel to each other and moving at the same speed. From either puck’s point of view, the other is not moving. However, from a third puck’s point of view, both would be moving in straight lines at the same velocity. In this ice-rink world, all that really matters are the relative positions and motions of the objects. Because Newton’s laws never mention a velocity, the point of view of any puck moving at any constant velocity is equally valid. All such observers agree on forces and accelerations, and they would all agree that Newton’s laws are valid.

  The idea that the same laws of motion apply for any observer moving at a constant velocity was very important. It explained how it can be that we are moving rapidly through space around the sun without feeling any effect. Our orbital speed is huge — around 30 kilometres per second — but, as Galileo realized, it is imperceptible to us because everything around us on the surface of the Earth is travelling right alongside us, with exactly the same enormous velocity. Today we know that the sun is moving, at an even more fantastical speed of 250 kilometres per second, around our galaxy, and that our galaxy, the Milky Way, is moving at a yet greater speed of 600 kilometres per second through the universe. We are actually space travellers, but because Newton’s laws do not care about our velocity, we don’t feel a thing!

  Newton’s law of gravity describes with exquisite precision the invisible, inexorable tie that binds the seat of your pants to your chair, holds the Earth and the planets in orbit around the sun, holds the stars in their spherical shape and keeps them in their galaxies. At the same time, it explains how Earth’s gravity affects everything from baseballs to satellites. That exactly the same laws should apply in the unearthly and hitherto divine realm of the stars as in the imperfect human world around us was a conceptual and indeed a spiritual break with the past. As Stephen Hawking has said, Newton unified the heavens and the Earth.

  Newton’s laws are as useful as ever. They are still the first rules that every engineer learns. They govern how vehicles move, on Earth or in space. They allow us to build everything from machines and bridges to planes and pipelines — not just by crafting, eyeballing, and adjusting, but by design. Although Newton discovered his laws by thinking about the motion of planets, they enabled the development of a vast number of technologies here on Earth, from bridge building to the steam engine. His notion of force was the key to all of it. It explained how we, through controlling and governing forces, could harness nature to our purposes.

  More than three centuries after he published his findings in Mathematical Principles of Natural Philosophy, known as the Principia, Newton’s universal laws of motion and gravitation are still the foundation for much of engineering and architecture. His discoveries underpinned the Industrial Revolution that transformed the organization of human society.

  The universe that Newton’s laws describe is sometimes called the “classical” or “clockwork” universe. If you know the exact position and velocity of every object at one time, then in principle Newton’s laws predict exactly where every object was or will be at any time in the past or future, no matter how remote. This classical universe is completely deterministic, and it is straightforward and intuitive. But as we shall see, in this respect it is utterly misleading. Before we get to that part of the story, we must discuss another outsider who, two hundred years later, would make a discovery even greater than Newton’s.

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  THE STORY OF THE discovery of the nature of light begins, appropriately enough, with the great flowering of intellectual thought known as the Scottish Enlightenment. At the turn of the eighteenth century, after a dark and brutal period of domination by the monarchy and the Catholic Church, England was preoccupied with building the British Empire in Africa, the Americas, and Asia, giving Scotland the space to establish a unique identity. Scotland emerged with a powerful national spirit, determined to set its own course and to create a model society. Scotland’s parliament founded a unique public school system with five hundred schools, which, by the end of the eighteenth century, had made their country more literate and numerate than any other in the world. Four universities were founded — in Glasgow, St. Andrews, Edinburgh, and Aberdeen — and they were far more affordable than Oxford or Cambridge, the only universities in England. The Scottish universities became centres of public education as well as academic study.

  Edinburgh became the leading literary centre in Europe and home to luminaries such as David Hume and the political philosopher Adam Smith. According to Arthur Herman, author of How the Scots Invented the Modern World, it “was a place where all ideas were created equal, where brains rather than social rank took pride of place, and where serious issues could be debated . . . Edinburgh was like a giant think tank or artists’ colony, except that unlike most modern think tanks, this one was not cut off from everyday life. It was in the thick of it.”21

  Scottish academia likewise followed a distinct course, emphasizing foundational principles and encouraging students to think for themselves, explore, and invent. There was a lively debate, for example, over the meaning of basic concepts in algebra and geometry, and their relation to the real world.22

  This focus on the fundamentals was remarkably fruitful. As just one instance, English mathematician and Presbyterian minister Reverend Thomas Bayes, whose famous “Bayes theorem” was forgotten for two hundred years but now forms the basis for much of modern data analysis, attended Edinburgh University at the same time as Hume. Fast on the heels of Scotland’s academic flowering came the great Scottish engineers, such as James Watt, inventor of the steam engine, and Robert Stevenson, who built the Bell Rock Lighthouse, off the coast of Angus, Scotland.

  As the Western world entered the nineteenth century, the Industrial Revolution permeated and remade every aspect of life. The power of steam engines revolutionized the economy. Distances shrank with trains, steamships, and other conveyances; people moved en masse into cities to work in factories that made everything from textiles to pots and pans and that in so doing redefined notions of both work and economic value. A new breed of “natural philosophers” — mainly gentleman hobbyists — set out to understand the world in ways that had never before been possible. The effect of the Scottish Enlightenment was felt at the highest levels of science. Having spawned philosophers, writers, engineers, and inventors, Scotland now produced great mathematicians and physicists. One particular young genius would expose nature’s inner workings to a degree that outshone even Newton.

  Newton’s physics explains a great many things, from the ebb and flow of the tides, caused by the moon’s gravitational attraction, to the orbits of planets, the flow of fluids, the trajectories of cannonballs, and the stability of bridges — everything involving motion, forces, and gravity. However, Newtonian physics could never predict or explain the transmission or reception of radio waves, the telephone, electricity, motors, dynamos, or light bulbs. The understanding of all this, and a great deal more, we owe to the experimental work of Michael Faraday, born in 1791, and its theoretical elaboration by James Clerk Maxwell, born four decades later.

  One can see the pair, Faraday and Maxwell, as the experimental yin and the theoretical yang of physics. To
gether, they typify the golden age of Victorian science. The well-born, well-educated Maxwell (he was heir to a small Scottish estate) fits a definite type: a gentleman scientist who, largely freed from the pressures of earning a living, pursued science as an ardent, passionate hobbyist.

  James Clerk Maxwell was a bright and curious child, born in southern Scotland. Having the run of his family’s estate in Glenlair (click to see photo), he was interested in everything natural and man-made. “What’s the go o’ that?” he asked, again and again, picking up insects or plants or following the course of a stream or a bell-wire in the house. Joining a private school — the Edinburgh Academy — at age ten, he was known as “Dafty” and bullied, in part for his strange clothes, designed by his father who, though a lawyer by profession, was scientifically minded. By fourteen, with his father’s encouragement, Maxwell had become a keen mathematician, preparing a paper describing a new way to draw ovals, which was read to the Royal Society of Edinburgh by a local professor.

  The Scottish educational system was particularly strong in mathematics. Rather than learning mathematics by rote as what one professor contemptuously termed a “mechanical knack,” students worked through the fundamentals from first principles and axioms. When James Clerk Maxwell found his first great friend, Peter Guthrie Tait, as a schoolkid, they amused themselves by trading “props,” or “propositions” — questions they’d make up to try to outwit one another. It became their bond, and decades later, when they were both eminent physicists, Maxwell continued to send his old friend questions that stumped him and whose answers helped him piece together the puzzle of electromagnetism.

  Maxwell, Tait, and William Thomson — later Lord Kelvin — who was educated at Glasgow University, formed a Scottish triumvirate, with all three becoming leading physicists of their time. Tait and Thomson co-authored the Treatise on Natural Philosophy, the most important physics textbook of the nineteenth century. Tait founded the mathematical theory of knots and Lord Kelvin made major contributions to many fields, including the theory of heat, where his name is now attached to the absolute scale of temperature. Alexander Graham Bell, another great Scottish inventor, followed Maxwell to university in Edinburgh before emigrating to Canada and developing the telephone.