The Universe Within

by Neil Turok

  As a young scientist, I was amazed to see the confidence that these theorists placed in their little equations when describing a realm so entirely remote from human experience. There was no direct evidence for anything they were discussing: the exponential blow-up of the universe during inflation, the scalar fields and their potential energy which they hoped would drive it, and — what they were most excited about at the meeting — the vacuum quantum fluctuations that they hoped inflation would stretch and amplify into the seeds of galaxies. Of course, they drew their confidence from physics’ many previous successes in explaining how the universe worked with mathematical ideas and reasoning.

  But there seemed to me a big difference. Maxwell and Einstein and their successors had been guided by a profound belief that nature works in simple and elegant ways. Their theories had been extremely conservative, in the sense of introducing little or no arbitrariness in their new physical laws. Getting inflation to work was far more problematic. The connection to grand unified theory sounded promising, but the Higgs fields, which were introduced in order to separate the different particles and forces, would typically not support the kind of inflation needed: they would either hold the universe stuck in an exponential blow-up forever or they would end the inflation too fast, leaving the universe curved and lumpy. Working models of inflation required a fine tuning of their parameters and strong assumptions about the initial conditions. Inflationary models looked to me more like contrivances than fundamental explanations of nature.

  At the same time, the attention theorists were now giving to cosmology was enormously energizing to the field. Although the inflationary models were artificial, their predictions gave observers a definite target to aim at. Over the next three decades, the inflationary proposal, along with other ideas linking fundamental physics to cosmology, helped drive a vast expansion of observational efforts directed at the biggest and most basic questions about the universe.

  THE STORY OF MODERN cosmology begins with Einstein’s unification of space, time, energy, and gravity, which closely echoed Maxwell’s unification of electricity, magnetism, and light. When Einstein visited London, a journalist asked him if he had stood on the shoulders of Newton. Einstein replied, “That statement is not quite right; I stood on Maxwell’s shoulders.” 59 Just as happened with Maxwell’s theory, many spectacular predictions would follow from Einstein’s. Maxwell’s equations had anticipated radio waves, microwaves, X-rays, gamma rays — the full spectrum of electromagnetic radiation. Einstein’s equations were even richer, describing not only the fine details of the solar system but everything from black holes and gravitational waves to the expansion and evolution of the cosmos. His discoveries brought in their wake an entirely new conception of the universe as a dynamic arena. Einstein’s theory was more complicated than Maxwell’s, and it would take time to see all of its implications.

  The most spectacular outcome of Maxwell’s unified theory of electricity and magnetism had been its prediction of the speed of light. This prediction raised a paradox so deep and far-reaching in its implications that it took physicists decades to resolve. The paradox may be summarized in the simplest of questions: the speed of light relative to what? According to Newton, and to everyday intuition, if you see something moving away and chase after it, it will recede more slowly. If you move fast enough, you can catch up with it or overtake it. An absolute speed is meaningless.

  In every argument, there are hidden assumptions. The more deeply they are buried, the longer it takes to reveal them. Newton had assumed that time is absolute: all observers could synchronize their clocks and, no matter how they moved around, their clocks would always agree. Newton had also assumed an absolute notion of space. Different observers might occupy different positions and move at different velocities, but again they would always agree on the relative positions of objects and the distances between them.

  It took Einstein to realize that these two seemingly reasonable assumptions — of absolute time and space — were incompatible with Maxwell’s theory of light. The only way to ensure that everyone would agree on the speed of light was to have them each experience different versions of space and time. This does not mean that the measurements of space and time are arbitrary. On the contrary, there are definite relations between the measurements made by different observers.

  The relations between the measurements of space and time made by different observers are known as “Lorentz transformations,” after the Dutch physicist Hendrik Lorentz, who inferred them from Maxwell’s theory. In creating his theory of relativity, Einstein translated Lorentz’s discovery into physical terms, showing that Lorentz’s transformations take you from the positions and times measured by one observer to those measured by another. For example, the time between the ticks of a clock or the distance between the ends of a ruler depends on who makes the observation. For an observer moving past them, a clock goes more slowly and a ruler aligned with the observer’s motion appears shorter than for someone who sees them at rest. These phenomena are known as “time dilation” and “Lorentz contraction,” and they become extremely important when observers move relative to one another at speeds close to the speed of light.

  The Lorentz transformations mix up the space and time coordinates. Such a mixing is impossible in Newton’s theory, because space and time are entirely different quantities. One is measured in metres, the other in seconds. But once you have a fundamental speed, the speed of light, you can measure both times and distances in the same units: seconds and light seconds, for example. This makes it possible for space and time to mix under transformations. And because of this mixing, they can be viewed as describing a single fundamental entity, called “spacetime.”

  The unification of space and time in Einstein’s theory, which he called “special relativity,” allowed him to infer relationships between quantities which, according to Newton, were not related. One of these relations became the most famous equation in physics.

  IN 1905, THE SAME year that he introduced his theory of special relativity, Einstein wrote an astonishing little three-page paper that had no references and a modest-sounding title: “Does the Inertia of a Body Depend Upon Its Energy-Content?” This paper announced Einstein’s iconic formula, E = mc2.

  Einstein’s formula related three things: energy, mass, and the speed of light. Until Einstein, these quantities were believed to be utterly distinct.

  Energy, at the time, was the most abstract of them: you cannot point at something and say, “That is energy,” because energy does not exist as a physical object. All you can say is that an object possesses energy. Nevertheless, energy is a very powerful idea, because under normal circumstances (not involving the expansion of space), while it can be converted from one form into another, it is never created or destroyed. In technical parlance, we say energy is conserved.

  The concept of mass first arose in Newton’s theory of forces and motion, as a measure of an object’s inertia: how much push is required to accelerate the object. Newton’s second law of motion tells you the force you need to exert to create a certain acceleration: force equals mass times acceleration.

  So how does energy equal the mass of an object times the speed of light squared? Einstein’s argument was simple. Light carries energy. And objects like atoms or molecules can absorb and emit light. So Einstein just looked at the process of light emission from an atom, from the points of view of two different observers.

  The first observer sees the atom at rest emit a burst of electromagnetic waves. From energy conservation, it follows that the atom must have had more energy before it emitted the light than it had afterward. Now let’s look at the same situation from the point of view of a second observer, moving relative to the first. The second observer sees the atom moving, both before and after the emission. According to the second observer, the atom has some energy of motion, or kinetic energy. The second observer also sees a slightly more energetic burst of radiation compa
red to the first, just because she is in motion. This extra energy can be calculated from Maxwell’s theory, using a Lorentz transformation.

  Now Einstein just wrote down the equations for energy conservation. The total energy before the emission must equal the energy after it, according to both observers. From these two equations it follows that the atom’s kinetic energy after the emission, as seen by the second observer, must equal the atom’s kinetic energy before the emission plus the extra energy in the burst of radiation. This equation relates the energy in the burst of radiation to the mass of the atom before and after the emission. And the equation implies that the atom’s mass changes by the energy it emits divided by the square of the speed of light. If the atom loses all of its mass in this process, and just decays completely into the burst of radiation, the same relation applies. The amount of radiation energy released must be equal to the original mass times the speed of light squared, or E = mc2.

  Einstein put it this way: “Classical physics introduced two substances: matter and energy. The first had weight, but the second was weightless. In classical physics we had two conservation laws: one for matter, the other for energy. We have already asked whether modern physics still holds this view of two substances and the two conservation laws. The answer is: No. According to the theory of relativity, there is no essential distinction between mass and energy. Energy has mass and mass represents energy. Instead of two conservation laws we have only one, that of mass-energy.”60 E = mc2 is a unification. It tells us that mass and energy are two facets of the same thing.

  What Einstein’s magical little formula tells us is that we are surrounded by vast stores of energy. For example, that sachet of sugar you are about to stir into your coffee has a mass energy equivalent to a hundred kilotons of TNT — enough to level New York. And of course, his discovery prefigured the development of nuclear physics, which eventually led to nuclear energy and the nuclear bomb.

  In Newton’s theory, there was no limit to the speed of an object. But in Einstein’s theory, nothing travels faster than light. The reason is fundamental: if something did travel faster than light, then according to Lorentz’s transformations, some observers would see it going backward in time. And that would create all sorts of causality paradoxes.

  IN DEVELOPING THE THEORY of relativity, the next question facing Einstein, which echoed concerns raised by Michael Faraday more than half a century earlier, was whether the force of gravity could really travel faster than light. According to Newton, the gravitational force of attraction exerted by one mass on any other mass acts instantaneously — that is, it is felt immediately, right across the universe. As a concrete example, the tides in Earth’s oceans are caused by the gravitational attraction of the moon. As the moon orbits Earth, the masses of water in the oceans follow. According to Newton, the moon’s gravity is felt instantly. But moonlight takes just over a second to travel from the moon to Earth. Faraday and Einstein both felt it unlikely that the influence of gravity travelled any faster.

  In constructing a theory of gravity consistent with relativity, one of the key clues guiding Einstein was something that Galileo had noticed: all objects fall in the same way under gravity, whatever their mass. An object in free fall behaves as if there is no gravity, as we know from the weightlessness that astronauts experience in space: an astronaut and her space capsule fall together. This behaviour suggested to Einstein that gravity was not the property of an object, but was instead a property of spacetime.

  What then is gravity? Gravity is replaced, in Einstein’s theory, by the bending of space and time caused by matter. Earth, for example, distorts the space­time around it, like a bowling ball sitting in the centre of a trampoline. If you roll marbles inwards, the curved surface of the trampoline will cause them to orbit the bowling ball, just as the moon orbits Earth. As the physicist John Wheeler would later put it, “Matter tells spacetime how to curve, and spacetime tells matter how to move.”61

  After ten years of trying, in 1916 Einstein finally discovered his famous equation — now called Einstein’s equation — according to which the curvature of space­time is determined by the matter contained within it. He used the mathematical description of curved space invented by the German mathematician Bernhard Riemann in the 1850s. Before Riemann, a curved surface, such as a sphere, had always been thought of as embedded within higher dimensions. But Riemann showed how to define the key concepts in geometry, like straight lines and angles, intrinsically within the curved surface, without referring to anything outside it. This discovery was very important, because it allowed one to imagine that the universe was curved, without it having to be embedded inside anything else.

  Einstein’s new theory, which he called “general relativity,” brought our view of the universe much closer to that of the ancient Greeks: the universe as a vital, dynamic entity with a delicate balance between its elements — space, time, and matter. Einstein altered our view of the cosmos, from the inert stage I had envisaged as a child to a changeable arena that could curve or expand.

  In welcoming Einstein to London, the celebrated playwright George Bernard Shaw told a jokey story about how a young professor — Albert Einstein — had demolished the Newtonian picture of the world. Upon learning that Newton’s gravity was no more, people asked him: “But what about the straight line? If there is no gravitation, why do not the heavenly bodies travel in a straight line right out of the universe?” And, Shaw continues, “The professor said, ‘Why should they? That is not the way the world is made. The world is not a British rectilinear world. It is a curvilinear world, and the heavenly bodies go in curves because that is the natural way for them to go.’ And at last the whole Newtonian universe crumbled up and vanished, and it was succeeded by the Einsteinian universe.”62

  In the early days Max Born described Einstein’s theory of general relativity thus: “The theory appeared to me then, and it still does, the greatest feat of human thinking about nature, the most amazing combination of philosophical penetration, physical intuition, and mathematical skill. But its connections with experience were slender. It appealed to me like a great work of art to be admired from a distance.”63 Today, Born’s statement is no longer true. Einstein’s younger successors applied his theory to the cosmos and found it to work like a charm. Today, general relativity is cosmology’s workhorse, and almost every observation we make of the universe — whether from the Hubble Telescope or giant radio arrays or X-ray or microwave satellites — relies on Einstein’s theory for its interpretation.

  General relativity is not an easy subject. As an undergraduate, I trudged my way through a famously massive textbook on the subject called Gravitation, which weighs 2.5 kilograms. It was a quixotic attempt — while the subject is conceptually simple, its equations are notoriously difficult. After six months of trying, I decided instead to take a course. That made it all so much easier. Physics, like many other things, is best learned in person. Seeing someone else do it makes you feel you can do it too.

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  THE DISCOVERY OF GENERAL relativity, and its implication that spacetime was not rigid, raised a question: what is the universe doing on the very largest scales, and how is it affected by all the matter and energy within it? Like everyone else at the time, when Einstein started to think about cosmology, he assumed the universe was static and eternal. But immediately a paradox arose. Ordinary matter attracts other ordinary matter under gravity, and a static universe would just collapse under its own weight. So Einstein came up with a fix. He introduced another, simpler form of energy that he called the “cosmological term.” Its main properties are that it is absolutely uniform in spacetime, and it looks exactly the same to any observer. The best way to visualize the cosmological term is as a kind of perfectly elastic, stretchy substance, like a giant sponge filling space. It has a “tension,” or negative pressure, meaning that as you stretch it out it stores up energy just like an elastic band. But no matter how much you str
etch it, its properties do not change — you just get more of it.

  At first, a negative pressure sounds like exactly what you don’t want holding up the universe. It would suck things inward and cause a collapse. However, as we described earlier, the expansion of the universe is not like ordinary physics. It is not an explosion: it is the expansion of space. And it turns out that the effect of a negative pressure, in the Einstein equations, is exactly the opposite of what you might expect. Its gravity is repulsive and causes the size of the universe to blow up. (This effect of repulsive gravity is the same one that Guth used in his theory of inflation.)

  So Einstein made his mathematical model of the universe stand still by balancing the attractive gravity of the ordinary matter against the repulsive gravity of his cosmological term. The model was a failure. As noticed by the English astrophysicist Arthur Eddington, the arrangement is unstable. If the universe decreased a little in size, the density of the ordinary matter would rise and its attraction would grow, causing the universe to collapse. Likewise, if the universe grew a little in size, the matter would be diluted and the cosmological term’s repulsion would win, blowing the universe up.