Fundamentals Page 4

  Now let us look within. There, too, we will discover abundance. We will find again that there’s plenty of space we can use, and much more that we can admire.

  Different kinds of microscopes open our eyes to the riches contained in small things. Microscopy is a vast subject, full of ingenious and useful ideas. But here I will only briefly sketch four basic techniques, which reveal different levels of the deep structure of matter.

  The simplest and most familiar microscopes exploit the ability of glass and some other transparent substances to bend light. By sculpting glass lenses and deploying them strategically, one can bend incoming light rays so as to spread out the angles with which they arrive at an observer’s retina or a camera’s plate. That makes the incoming image appear larger. This trick provides a wonderfully powerful and flexible way to explore the world down to distances of about a millionth of a meter, or a little less. (A meter is about 40 inches.) Using it, we can see the cells from which plants, animals, and humans are made. And we can glimpse the bacterial hordes, which both help and plague them.

  When we push this light-bending technique to try to resolve even smaller objects, we run up against a fundamental problem. The technique is based on manipulating the paths of light rays. But the idea that light is composed of rays is only approximately right, since light travels in waves. Using waves to pick out details that are smaller than the size of the waves themselves is like trying to pick up a marble while wearing boxing gloves. The wavelengths involved in visible light are approximately half of one-millionth of a meter, so microscopes based on imaging visible light get fuzzy below that distance.

  X-rays have wavelengths a hundred or a thousand times smaller than visible light, so they allow, in principle, access to much shorter distances. But there is no equivalent for x-rays of what glass supplies for visible light, namely a material that we can sculpt to make lenses and manipulate rays. Without lenses, the classical methods for magnifying images can’t get off the ground.

  Fortunately, there’s a radically different approach that works. It is called x-ray diffraction. In x-ray diffraction, we dispense with lenses. We shine an x-ray beam on the object of interest, let the object bend and scatter the beam, and record what comes out. (To avoid confusion, let me note that these are quite different from the more familiar, simpler sort of x-ray images that doctors and dentists use. Those are much cruder projections, basically x-ray shadows. X-ray diffraction uses much more carefully controlled beams and smaller target samples.) The “pictures” that an x-ray diffraction camera takes don’t look like that object at all, but they contain a lot of information about its shape, in a coded form.

  A long and fascinating saga, strewn with Nobel Prizes, hinges on that qualification “a lot of.” Unfortunately, x-ray diffraction patterns don’t supply enough information to let you reconstruct the object purely by mathematical calculation. They’re like corrupted files of digital images.

  To address that problem, several generations of scientists constructed an interpretive ladder, which allows us to climb from simple objects to more complicated ones. The first objects to be deciphered from their x-ray diffraction patterns were simple crystals, starting with table salt. In that example, people had a good idea, based on chemistry, of what the answer should look like, namely a regular array of equal numbers of two kinds of atoms, sodium and chlorine. They also had reason to expect, based on the observed form of large salt crystals, that the array should be cubical. They did not know, however, the distance separating the atoms. Fortunately, you can calculate what the x-ray diffraction pattern looks like for model crystals with any possible value for the distance. By finding a match to the observed pattern, you both validate the model and determine the interatomic distance.

  As scientists geared up to study more complicated materials, they used a kind of bootstrap procedure. At each stage, they used previously validated models to build up more elaborate models as candidates to describe materials with more elaborate spatial structures. Then they compared x-ray diffraction patterns calculated using the candidates to the ones they observed. Through a combination of inspired guesswork and heroic labors, success was sometimes achieved. With each new success structural features emerged, which could be fed as input into the next generation of models.

  Historical highlights from this line of work include the extraordinary chemist Dorothy Crowfoot Hodgkin’s determination of the three-dimensional structures of cholesterol (1937), penicillin (1946), vitamin B12 (1956), and insulin (1969), and the determination of the three-dimensional structure of DNA (1953)—the famous double helix—by Francis Crick and James Watson, who decoded x-ray diffraction pictures taken by Maurice Wilkins and Rosalind Franklin.

  Today’s much more advanced computers, using programs that incorporate the successful work of the past, allow chemists and biologists to solve more complicated x-ray diffraction problems routinely. In this way, they’ve determined the structure of tens of thousands of proteins and other important biomolecules. The art of scientific image manufacturing remains a vital frontier of biology and medicine.

  To me, the interpretive ladder is both a beautiful example of and a metaphor for how we construct our models of the world more generally. In natural vision, we must turn the two-dimensional patterns that arrive at our retinas into a useful rendition of the three-dimensional world of objects in space. Abstractly, it is an impossible problem—there simply is not enough information. To compensate, we add assumptions about how the world works. We exploit abrupt changes in patterns of color, shadows, and motion to identify objects, their properties, their motion, and their distances.

  Babies, or blind people suddenly given vision, have to learn how to see. They learn by experience to work with what they’ve got, building up from simple cases to construct a world that makes sense. Learning to “see” an object in its x-ray diffraction pattern has been a collective effort to accomplish something very similar; that is, to find a bag of tricks that lets us make sense of the world.

  Our third technique, scanning microscopy, is refreshingly direct. One holds a needle with a tiny tip close to a surface of interest and “scans” by moving the tip parallel to the surface. If one does this while applying an electric field, then electric currents flow from the surface into the needle. The nearer the tip is to the surface, the larger the current. In this way, one can read out the topography of the surface with subatomic resolution. In images that reflect this data, one sees individual atoms towering up like mountains above a flat landscape.

  Finally, let’s discuss how scientists probe the smallest distances. The first experiment to get a look inside atoms was done by Hans Geiger and Ernest Marsden in 1913, with Ernest Rutherford guiding the effort. In their experiment, Geiger and Marsden pointed a beam of alpha particles at a gold foil. Some of the alpha particles were deflected by the foil. Geiger and Marsden counted how many got deflected through different angles. Before doing the experiment, they expected that few, if any, particles would be deflected by large angles. The alpha particles have a lot of inertia, so only close encounters with much heavier objects can change their course significantly. If the mass of the gold foil were spread out evenly, then large deflections wouldn’t happen.

  What they observed was quite different from their expectations. There were, in fact, significant numbers of large-angle deflections. Occasionally, alpha particles even reversed direction, returning back the way they came. Rutherford later recalled his reaction to the news:

  It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you. On consideration, I realized that this scattering backward must be the result of a single collision, and when I made calculations I saw that it was impossible to get anything of that order of magnitude unless you took a system in which the greater part of the mass of the atom was concentrated in a minute nucleus. It was then tha
t I had the idea of an atom with a minute massive centre, carrying a charge.

  Rutherford’s detailed analysis of the Geiger-Marsden observations gave birth to the modern picture of atoms. He showed that to account for the data one must assume that most of the mass and all of the positive electric charge in an atom is concentrated in a tiny nucleus. Further refinements made those conclusions quantitative. An atomic nucleus contains more than 99 percent of the atom’s mass. Yet a nucleus extends less than one-hundred-thousandth of its atom’s radius and—being nearly spherical—occupies less than one part in a million of one part in a billion of its volume. Those are literally astronomical numbers. The way a nucleus is dwarfed by its atom parallels how the Sun is dwarfed by its surrounding interstellar space.

  The Geiger-Marsden experiment established a paradigm for exploring the subatomic world that has dominated experimental work on fundamental interactions ever since. By bombarding targets with particles of ever-higher energy, and studying the patterns of their deflections, we learn about the targets’ interiors. Here, too, we construct an interpretive ladder, using our understanding of what’s revealed at each stage to design and interpret new experiments that probe deeper.

  THE FUTURE OF SPACE

  Beyond the Horizon

  We can’t see beyond the distance that light has traveled since the time of the big bang. That defines our cosmic horizon. But with each passing day the big bang recedes farther into the past. Space that was beyond our horizon yesterday comes within it today, and is newly opened to view.

  Of course, since adding one more day, or even thousands of years, increases the age of the universe by only a small fraction, the fractional growth in the visible universe is hardly noticeable on human time scales. But it is entertaining to consider what kind of universe our distant descendants might perceive and to exercise our minds by thinking about what might be happening beyond the horizon. As Tennyson has Ulysses say:

  . . . all experience is an arch wherethrough

  Gleams that untraveled world whose margin fades

  For ever and forever when I move.

  How dull it is to pause, to make an end. . . .

  The expanding cosmic horizon poses many questions. For instance, as the horizon expands, might the entire universe come within it? If space is finite, that will eventually happen. Famously, finite space need not have an edge. A sphere—that is, the surface of a ball—is an example of a space that is finite yet has no boundary. The surfaces of ordinary balls are two-dimensional. Though they are challenging to visualize, for mathematicians it is child’s play to define three-dimensional spaces that, like ordinary spheres, are finite yet have no boundary. Such spaces provide candidate shapes for a finite universe.

  The visible universe is remarkably uniform. It contains the same kinds of materials, obeying the same laws, organized in the same ways, evenly distributed throughout. Another question raised by the expanding horizon is whether or not that “universal” pattern holds up for the parts we can’t see yet.

  Or is the universe really a “multiverse,” home to many different patterns or laws? The most straightforward way to answer this question would be to observe outlandish things happening far away. Were that to happen, we could establish the multiverse experimentally. A sad but perfectly logical possibility is that other facts about the fundamental laws and cosmology will suggest that we live in a multiverse, but will also suggest that the “different” parts will become visible only in the very distant future, when the horizon expands to contain them. I call this possibility sad, because to me using an idea to say something concrete about the world we experience brings it to another level. It’s where the magic is. Also, testing keeps you honest.

  Particles of Space?

  Euclid assumed that one could continue to measure distance more and more finely, without limit, using the same conceptual tools. He didn’t know about atoms, elementary particles, or quantum mechanics. Now we know better. When we divide matter into very small parts, things change a lot! A placid drop of water, which appears continuous and at rest, breaks up into atoms and even more basic units, which jitter and jive to the tune of quantum mechanics.

  When we come to measure subatomic distances, we must use tools that are very different from the sorts of rigid rulers Euclid had in mind. Scalable versions of those instruments simply don’t exist. Yet Euclid’s geometry lives on, triumphant, within our fundamental equations. Within those equations, elementary particles (and the fields that support them) occupy a seamless continuum, equivalent in all its parts, measured in lengths and angles, obedient to Pythagoras’s theorem, just as Euclid assumed. It’s uncanny that Nature has let us get away with it. So far . . .

  . . . but probably not forever. According to Einstein’s general theory of relativity, space is a kind of material. It is a dynamic entity, which can bend and move. In our later discussions, many other reasons to consider space as a material will emerge as well. According to the principles of quantum mechanics, anything that can move does move, spontaneously. As a result, the distance between two points fluctuates. Upon combining general relativity with quantum mechanics, we calculate that space is a kind of quivering Jell-O, in constant motion.

  When the distance between two points is not too small, those quantum fluctuations in distance are predicted to be a negligible fraction of the distance itself. Then we can ignore them, as a practical matter, and return to the comfort of Euclidean geometry. But when we refine our focus below about 10−33 centimeters—a tiny distance known as the Planck length—then typical fluctuations in the distance between two points can be as large or larger than the distance itself. Two lines from William Butler Yeats’s apocalyptic vision spring to mind:

  . . . the center cannot hold;

  Mere anarchy is loosed upon the world. . . .

  Writhing rulers and dancing compasses undermine the foundations of Euclid’s approach to geometry, and ultimately Einstein’s, too. The ideas of GPS can’t be scaled down, because the orbits of satellites in Planck-length detail are noisy and unpredictable. What will replace them? Nobody knows for sure. There’s little prospect of guidance from experiment, because the Planck length is thousands of trillions times smaller than distances we know how to resolve. For me, though, it is difficult to resist the idea that space-time is not essentially different from matter, which we understand much more deeply. If so, it will consist of vast numbers of identical units—“particles of space”—each in contact with a few neighbors, exchanging messages, joining and breaking apart, giving birth and passing away.

  2

  THERE’S PLENTY OF TIME

  PRELUDE: MEASURE AND MEANING

  Frank Ramsey (1903–1930) blazed bright, though briefly. Before dying of liver problems at the age of twenty-six, Ramsey made seminal contributions to mathematics, economics, and philosophy. Despite his youth he was a central figure in intellectual life at Cambridge in the 1920s. He collaborated and argued with both John Maynard Keynes and Ludwig Wittgenstein, who are widely regarded as the greatest economist and the greatest philosopher of the twentieth century, respectively, on their home turfs. “Ramsey theory” is a thriving, entertaining corner of mathematics that grew out of his work.

  (Here’s a classic little gem that will give you a taste of Ramsey theory: Among any group of six people, each pair of whom are either friends or enemies, there will either be a set of three people who are all mutual friends, or a set of three people who are all mutual enemies.)

  Frank Ramsey is a thinker to be reckoned with. His objection to the significance of the physical world’s superhuman proportions deserves serious attention:

  My picture of the world is drawn in perspective and not like a model to scale. The foreground is occupied by human beings and the stars are all as small as three-penny bits. I don’t really believe in astronomy, except as a complicated description of part of the course of human and possibly animal sensation. I apply my perspectiv
e not merely to space but also to time. In time the world will cool and everything will die; but that is a long time off still and its present value at compound discount is almost nothing.

  A famous New Yorker cover expresses a similar thought. It shows a “map of the world” where most of the drawing is devoted to Manhattan while the rest of our planet gets squeezed into a cramped, sketchy background.

  Ramsey’s attitude is a healthy corrective to cosmic sizeism. Equal volumes of space are equal in their potential for accommodating matter and motion, but that does not mean they are of equal importance. The undifferentiated, empty regions are less interesting. Similarly, equal intervals of time are equal in their ability to accommodate the ticking of clocks, but that does not mean they are of equal importance. To most of us, most of the time, nearby events matter more. It is an attitude that comes naturally to us, as children, as a strategy to cope in the world.

  But Ramsey, in retaining that attitude, takes it too far. When he says he does not believe in astronomy, I do not believe him. His statement hints to me instead that the outrageous hugeness of cosmic space and time bothered him deeply, as it had bothered Pascal. Sadly, by denying their significance, he cut himself off from a potential source of inspiration. He missed the opportunity to become a great cosmologist, as well as mathematician, economist, and philosopher.

  We can recognize both that there’s plenty “out there” and that there’s plenty “in here.” Neither fact contradicts the other, and we do not have to choose between them. From different perspectives, we are both small and large. Both perspectives capture important truths about our place in the scheme of things. To get a full and realistic understanding of reality, we must embrace them both.