The Accelerating Universe: Infinite Expansion, the Cosmological Constant, and the Beauty of the Cosmos

Chapter One

Beauty and the Beast

What is beauty? What is it that makes certain works of art, pieces of music, landscapes, or the face of a person so appealing to us that they give us an enormous sense of excitement and pleasure? This question, with which many philosophers, writers, psychologists, artists, and biologists have struggled at least since the time of Plato, and which led (among other things) to the birth of the notion of aesthetics in the eighteenth century, is still largely unanswered. To some extent, all of the classical approaches to beauty can be summarized by the following (clearly oversimplified) statement: Beauty symbolizes a degree of perfection with respect to some ideal. It is strange, though, that something which has such an abstract definition can cause such intense reactions. For example, some accounts claim that the Russian writer Dostoyevsky sometimes fainted in the presence of a particularly beautiful woman.

    In spite of some changes in taste over the centuries (and some obvious differences among different cultures), the perception of what is beautiful is very deeply rooted in us. It suffices to look at a few paintings like Botticelli's Primavera or Leighton's Flaming June, or at a majestic mountain landscape, to realize this.

    The term beautiful has evolved from being identified with "good" and "real" or "truthful" in ancient Greece to a quality that is confined merely to its effects on our senses, in the second half of the eighteenth century. It is interesting to note, though, that theapproach of the philosophical book of Proverbs in the Bible has been rather dismissive and closer to the more modern definition in its assertion: "Charm is deceitful, and beauty is vain."

    But, even if only affecting our senses, the effects of beauty should not be underestimated. The ancient Greeks certainly did not underestimate them. Greek mythology contains the famous story of the goddess Eris, who, insulted because she was not invited to the wedding party of King Peleus and the sea nymph Thetis, decided to take revenge by throwing into the banquet hall a golden apple that carried this inscription: "For the Fairest." After a long debate among the goddesses, the choice was narrowed down to three contestants for the title: the goddesses Hera, Athena, and Aphrodite. The matter was brought for a decision to Zeus who (very wisely) passed the task on to Paris, the son of the King of Troy. As it turned out, Paris's job was reduced to an evaluation of the bribes offered to him by each of the goddesses.

    Hera whispered to him that she would make him the Lord of Europe and Asia; Athena promised him victory over the Greeks; and Aphrodite made him an offer he could not refuse—she promised him that the most beautiful woman in the entire world would be his. Paris gave the apple to Aphrodite, an action that can only be described as a mistake of historic proportions. The most beautiful woman on earth was Menelaus's wife, Helen, whose face "could launch a thousand ships." The end of the story is tragic. After Paris kidnapped Helen and brought her to Troy, a fierce and bitter war broke out, which led eventually to the total destruction of Troy.

    Helen's beauty is described as being so intense, and its effects so devastating, that when Menelaus decides to execute Helen, a Trojan hero's mother forces him to swear that he will perform the execution without looking into Helen's eyes, because "through the eyes of men she controls them and destroys them in the same way that she destroys cities."

    Some speculate that Helen's beauty was of the cold, unapproachable type, and that its overwhelming effects stemmed from the fact that Helen appeared as unattainable as the understanding of the concept of beauty itself.

    Sometimes one can understand a certain concept or quality better by examining something that epitomizes the opposite. This notion is partly responsible for such pairings as heaven and hell, Dr. Jekyll and Mr. Hyde, Stan Laurel and Oliver Hardy, and, indeed, as the title of this chapter implies, Beauty and the Beast. So far I have only discussed beauty, but where is the "beast"?

    The beast, in this case, is—physics! To many of my personal friends and to a large number of students of humanities whom I have met over the years, there is nothing more remote from the notion of beauty and more antithetical, from the point of view of the sensation that it induces in them, than physics. In fact, the disgust and fear that physics stimulates in some people is rivaled only by their feelings toward cockroaches. In an article in the Sunday Times (quoted in a BBC lecture by Richard Dawkins), columnist A. A. Gill compared observations in the sky to movie and theater stars by saying, "There are stars and there are stars, darling. Some are dull, repetitive squiggles on paper, and some are fabulous, witty, thought-provoking ..." Believe it or not, those "dull, repetitive squiggles" represented the discovery of pulsars, objects so dense that one cubic inch of their matter weighs a billion tons, and that take a fraction of a second to rotate, instead of the earth's twenty-four hours!

    I hope that this book will convince even skeptics that "beauty in physics and cosmology" is not an oxymoron. I remember a certain scene in the movie Good Morning, Vietnam in which a soldier is asked to which unit he belongs. His answer, "military intelligence," provokes an immediate reaction from the general: "There is no such thing!" In relation to science, the English poet Keats virtually accused Newton of ruining the beauty of the rainbow by his theoretical explanations of how it is formed, using the laws of optics. In Keats's words:

Philosophy will clip an Angel's wings Conquer all mysteries by rule and line, Empty the haunted air, and gnomed mine— Uneave a rainbow ...

Incidentally, some readers may find the latter story surprising, given that Keats is often quoted as having said: "Beauty is truth, truth beauty." In fact, Keats said no such thing. It is what he said the Grecian Urn depicts, in his criticism of works of art that deliberately eliminate existing difficulties of life.

    Keats's complaint merely reflects the general feeling that magicians' tricks often lose their charm once we know how they are performed. However, in physics, very often the explanation is even more beautiful than the question, and even more frequently, the solution to one puzzle helps uncover an even deeper and more intriguing mystery. I therefore hope to be able to demonstrate that reactions like Keats's merely represent a misunderstanding that is based on false myths.

What Is Beautiful?

Definitions are always difficult, especially when we are dealing with something that is (largely) subjective. In this sense, even the definition in the Oxford English Dictionary—"Impressing with charm the intellectual or moral sense, through inherent fitness or grace"—which surely does not involve all the intricacies originating from philosophical interpretations, is not particularly useful.

    I will attempt to make my task easier by answering at this point a much simpler question. Since my goal is to discuss beauty in physics and cosmology, I will address the question: When does a physicist feel that a physical theory is beautiful?

    Any endeavor aimed at answering this question is bound to result in the use of an entire vocabulary of concepts, mostly borrowed from the arts. The list of such concepts may include symmetry, coherence, unity, harmony, and so on.

    Probably not all physicists agree on which subset of concepts from this list should be used. However, I will argue later that at least three requirements are absolutely essential and must be fulfilled:

1. Symmetry 2. Simplicity 3. The Copernican principle

    A fourth element, elegance, is also considered by some to be an important ingredient of a beautiful theory, but, as I will explain later, I do not consider it essential.

    I realize that at this stage the meanings of all of these concepts are vague at best (if not totally obscure), but I will now explain in some detail what I mean by each one of them.

1. Symmetry: When Things That Might Have Changed Do Not

Everyone is familiar at least with symmetries of pictures, objects, or shapes. For example, our face and body have an almost exact bilateral symmetry. What this means is that if we reflect each half of our face, we obtain something that is nearly identical to the original (strangely enough, this is true even for the one-eyed giant Cyclops whom Ulysses encountered in his travels).

    Some shapes are symmetric with respect to rotation. For example, a circle drawn on a page remains the same if we rotate the page on the desk.

    Other arrangements are symmetric under certain displacements or translations. For example, if we stood in front of some of the row houses in Baltimore facing one unit, and someone were to displace the entire row by one unit, we would not notice any difference. Similarly, if we look at one Campbell soup can in a painting containing rows of identical cans by the pop artist Andy Warhol, and the painting is shifted slightly sideways or upward, we see an identical picture.

    Notice that in all of these examples the object or shape did not change when we performed the symmetry operation, reflection, rotation, or translation.

    The association of symmetry with beauty does not require elaborate explanations. Anyone who ever looked through a kaleidoscope has experienced the sensation of beauty that symmetry inspires. In fact, the word kaleidoscope itself comes from the Greek words kalos, which means "beautiful," and eidos, which means "form" (skopeein means "to look").

    It is important to emphasize, though, that when the concept of symmetry is introduced into physics, it is not the symmetry of shapes that we are interested in but rather the symmetry of the physical laws. As we shall soon see, in this case, too, the symmetry is associated with things that do not change. In order to explain this concept better, let me first describe briefly the nature of these entities that we call laws of physics.

    The laws of physics, sometimes referred to as the laws of nature, represent attempts to give a mathematical formulation to the behavior that we observe all natural phenomena to obey. For example, in classical physics, Newton's universal law of gravitation states that every particle of matter in the universe attracts every other particle through a force called gravity. It further gives a quantitative measure of how this attraction is larger the more massive the particles (doubling the mass of one particle doubles the force), and how it decreases when the distance between the particles is increased (doubling the distance weakens the force by a factor of four). To give another example of a law of physics, one of James Clerk Maxwell's equations, the laws that describe all the electric and magnetic phenomena, states that there are no magnetic monopoles (single magnetic poles). Namely, there cannot be a magnet that has only one, say, north pole. Indeed, we know that even if we take a bar magnet and chop it up into smaller and smaller pieces, each piece will have north and south poles.

    Now, what is the meaning of symmetries of the laws of physics? These symmetries are certain fundamental properties of the laws, which are somewhat similar to the symmetries described for shapes or objects. For example, all the laws of physics do not change from place to place. A simple but remarkable manifestation of this property is the fact that if we perform an experiment, or study any physical phenomenon, in Russia, in Alabama, or on the moon, we obtain the same results. This also applies to different parts of the universe; when we observe a star that is located trillions of miles away from us, it still appears to obey the same laws of physics that we find here on Earth. This means that we can apply the same laws that we have deduced from laboratory experiments to the understanding of the universe as a whole. This universal transportability of the physical laws is encapsulated in the statement that the laws of physics are symmetric under translations. This property is not to be confused, for example, with the fact that the strength of the force of gravity is not the same on the earth and on the moon. Gravity on the moon is different (weaker) because both the mass and the size of the moon are different from those of the earth. However, given that we know the mass and radius of the moon, we would use exactly the same formula to calculate the force of gravity there, as we do on Earth.

    The laws of physics also do not depend on the direction in space. For example, they would not change if the earth started to rotate in the opposite direction. Were this not the case, then experiments might yield different results in the Southern Hemisphere than they do in the Northern Hemisphere. Furthermore, we might obtain different results if we perform an experiment lying down, rather than standing up, or we might find that light travels faster to the north than it does to the east. Note again that I do not refer to the fact that, for example, different stars happen to be seen in the night's sky from Australia than from Alaska (nor to the fact that different rock music groups may be popular in the two places), but to the fact that the laws that describe all the natural phenomena do not have a preferred direction. Thus, the laws would not change if someone took our entire universe and rotated it somehow. This property is expressed by the statement that the laws of physics are symmetric under rotation.

    I would like to further clarify the difference between a symmetry of a shape and of a law. The symmetry of the laws of physics under rotation does not mean, for example, as it was believed in ancient Greece, that the orbits of planets must be circular. A circle, as a shape, is indeed symmetric under rotation. But this has nothing to do with the symmetry of the law—in this case the law of gravity, which governs the motion of the planets around the sun. In fact, since the time of Johannes Kepler, a German astronomer who worked in Prague in the seventeenth century, astronomers have known that the orbits of the planets are not circular but elliptical. The symmetry of the law means that the orbit can have any orientation in space (Figure 3).

    Another symmetry that the basic laws of physics exhibit concerns the direction of time. Curiously, the laws would not change if time were to flow backward. This is true for both mechanical and electromagnetic phenomena at both the macroscopic and the subatomic levels. For example, there is nothing in the basic laws to indicate that the phenomenon of a plate falling from a shelf and shattering to pieces on the floor should be allowed, while that of the scattered pieces flying up from the floor and assembling to an intact plate on the shelf should be forbidden.

    Interestingly enough, as far as we know, the laws of physics also do not change with the passing of time. Astronomy proves very useful in demonstrating this property. We can observe, for example, galaxies at distances of millions, and even billions, of light-years. One light-year is the distance light can travel in one year, which is about six trillion miles. What this means is that the light that reaches us now from a galaxy that is over 100 million light-years away left that galaxy more than 100 million years ago. Therefore, what we see today is really the way the galaxy was 100 million years ago. Astronomy thus truly allows us to look back into the past. The main point to note is that by analyzing the observed light, we can establish the fact that the same physical laws that govern the emission of light by atoms today also applied in the distant past. In fact, we can now state with a high degree of confidence that the laws of physics have not changed at least since the time the universe was only about one second old (see chapter 3).

    As we have seen, when objects or shapes possess a certain symmetry, this is related to something that does not change, an invariant. For example, because of its left-right symmetry, a mirror image of the Notre Dame Cathedral in Paris looks identical to the cathedral itself. Symmetries are therefore related to the indiscernibility of differences. Since we are discussing symmetries of the laws that govern the behavior of all natural phenomena, the property of having things that do not change is in this case translated to universal entities we call conservation laws. A conservation law simply reflects the fact that there exist physical quantities in the universe that remain constant in time. Namely, if we were to measure the value of such a quantity today, one year from now, or a million years from now, we would obtain exactly the same value. This is to be contrasted, for example, with the stock market, where money is definitely not a conserved quantity—that is, on a given day everybody may lose, with no one gaining.

    The two symmetries of the laws of nature I have already mentioned, the symmetry under translations and the symmetry under rotations, indeed result in conservation laws. For example, the linear momentum of a body is equal to the product of its mass and its speed, and its direction is the direction of the motion. Thus, the linear momentum of a body defines in some sense the quantity of motion this body possesses; it is larger the larger the mass and the speed. A stampeding buffalo has a larger linear momentum than that of a man running at the same speed but a smaller one than that of a rocket moving much faster. The symmetry under translations is manifested in the fact that linear momentum is conserved. Namely, momentum can neither disappear nor be created; it can just be transferred from one body to another. In everyday life, we see directly the consequences of conservation of linear momentum—for example, in the trajectories of colliding cars, of colliding billiard balls, and in the motion of the puck in ice hockey. The speed and direction of all of these motions are determined in such a way that the total momentum of the system is conserved. The motion of rockets is also a consequence of the conservation of linear momentum. When the rocket is resting on the launching pad, its momentum is zero (because its speed is zero). This means that as long as external forces do not interfere, the momentum must remain zero. When the rocket starts to eject gases downward at a high speed, the rocket itself acquires an upward speed, to counterbalance the momentum of the gases.

    The angular momentum of a rotating body is a measure of the amount of rotation it possesses. For example, if two identical spheres are rotating around their axes, the angular momentum is larger for the one that rotates faster. If two spheres of the same mass are rotating at the same rate, the angular momentum is larger for the one with the larger radius. The symmetry of the laws of nature under rotations is manifested in the fact that angular momentum is also a conserved quality.

    Ice skaters make frequent use of the conservation of angular momentum. In one of the popular routines, skaters start spinning slowly with their arms stretched sideways, and then they bring their arms to the sides of their body, thus increasing dramatically the rate of their spin. I still have a picture in my mind of a young Scott Hamilton, with his red hair spread almost horizontally, as he spins incredibly fast. This behavior results from conservation of angular momentum—reducing the distance of the arms from the rotation axis results in an increase in the speed of rotation. Conservation of angular momentum is also responsible (among other things) for the fact that moving bicycles and spinning Hannukah dreidels do not fall, for the stability of the axis of gyroscopes (which are used to determine directions accurately), and for the stability of the orbits of the planets around the sun.

    Another symmetry that was mentioned above, the fact that the laws of nature do not change with the passing of time, is responsible for the existence and conservation of the quantity we call energy. We all have a certain intuitive understanding of what energy means; after all, we pay energy bills to gas and electric companies, and many of us still remember the energy crisis in 1979, when gasoline was expensive and hard to find. In some sense, energy reflects the ability to do work. Very broadly speaking, energy can be associated with motion (in which case it is called kinetic energy), can be stored in some form (e.g., chemical, electrical, gravitational, nuclear; in which case it is called potential energy) , or can be carried by light (radiative energy). Again, conservation means that energy is neither created nor destroyed. It can merely be transferred from place to place or be transformed from one form to another. For example, when we drop a spoon, gravitational potential energy is transformed into kinetic energy of motion, and the latter is transformed into heat and acoustic energy as the spoon hits the floor.

    Having briefly explained the concept of symmetry, I will now turn to the second requirement for beauty, that of simplicity.

2. Simplicity: Less Is More Beautiful

Simplicity is to be understood in the sense of reductionism. Namely, the goal of physics is to replace many questions by very few, basic questions; or a description of nature that involves many laws of physics by a complete theory that has only a few fundamental laws. Physicists have been driven for centuries by a feeling that underneath the enormous wealth of phenomena that we observe, there exists an underlying relatively simple picture. The great seventeenth-century French philosopher and scientist René Descartes once said: "Method is necessary for discovering the truths of nature. By method, I mean rules so clear and simple that anyone who uses them carefully will never mistake the false for the true, and will waste no mental effort." We have already seen an example of the application of this type of thinking in chapter 1, in the search for one mechanism to explain all the shapes of the nebulae.

    We can identify in this drive for reductionism some of the same elements that perhaps formed the basis for the notion (in the Judeo-Christian cultures at least) that monotheism represents a more advanced (more beautiful?) form of faith than polytheism. The order for one God is expressed very clearly in the first two commandments: "I am the Lord your God ..." and "You shall not make for yourself an idol ..." I remember that as a skeptical child, I used to be somewhat puzzled by the statement made by a teacher that the move to monotheism represented progress. After all, I thought, if it is all a matter of faith anyhow, then what difference does it make if you believe in one God or in many gods, each of whom is responsible for a different phenomenon in nature? Today, I can identify in that statement the same requirement for reductionism, for simplicity.

    Given two theories that explain a given phenomenon equally well, the physicist will always prefer the simpler one, for this aesthetic (and not just practical) reason. I want to emphasize that this drive toward reductionism does not mean that the physicist fails to recognize that there is beauty in the richness and complexity of phenomena. After all, physicists realize, too, as did the poet William Cowper in the eighteenth century, that "variety's the very spice of life." The emergence of complexity in our universe, with life being perhaps at the pinnacle of this complexity, is what makes it so beautiful. However, in evaluating the beauty of a physical theory, the physicist regards as an essential element of beauty the fact that all of this complexity stems from a limited number of physical laws.

    I would like to further clarify this idea with a simple example. Imagine that we draw a square, and then on each side of the square we draw another square with a side equal in length to one-third the side of the original square, and we repeat this process many times (Figure 4). Now, almost everyone will agree that the final pattern is quite beautiful to the eye (because of its symmetry). However, the physicist will recognize an additional element of beauty in the fact that underlying this relatively complex pattern, there is a very simple law (algorithm) for its generation.

    The great German philosopher Immanuel Kant had similar ideas (in the eighteenth century) concerning the ideal of human consciousness. He defined this ideal as the attempt to establish our understanding of the universe on a small number of principles, from which an infinity of phenomena emerge. He went on to identify a beautiful object as one that has a multitude of constituents, all of which at the same time obey a clear, transparent structure that provides the big picture.

    Quite frequently, the mutual influences between the arts and the sciences are exaggerated. As a physicist who also happens to be an art fanatic, I can testify that the direct, immediate, conscious influence is minimal. Nevertheless, it is true that in some epochs, people from different disciplines sometimes think along similar lines. For example, a part of the title of this section, "Less Is More," was a popular aphorism with the twentieth-century architect Ludwig Mies van der Rohe. The feeling that one has to search for the most fundamental characteristics of things, which has guided physicists during all ages and in particular in this century, found its way also into some of the art movements of this century. Specifically, the roots of minimal art and conceptual art are clearly in this type of feeling.

    An excellent example for this revolution in art is provided by the transition from the very realistic description of erotic attraction and the act of love, as in the sculptures Desire by the French artist Aristide Maillol or The Kiss of Auguste Rodin, to the very minimalistic description, as in the sculpture The Kiss by the French (Romanian-born) sculptor Constantin Brancusi. The entire train of thought (from realism to minimalism) is exposed in a series of tree paintings by the Dutch painter Piet Mondrian, in which one can literally follow the transformation of a very realistic rendition of a tree into a very abstract, minimalistic painting that involves a series of lines symbolizing the leaves.

    Another painting by Mondrian that encapsulates the idea of reductionism, of grasping the most basic characteristics, is Broadway Boogie Woogie (currently in the Museum of Modern Art in New York City), in which the essence of the title is captured in a collection of squares and rectangles in bright, almost luminous, neon-like red, yellow, and blue colors.

    Interestingly, while this reductionistic approach was adopted only by certain movements in Western art, by contrast, in Japanese art, as in physics, reductionism has been regarded as an element of beauty for centuries. It suffices to look at a lyric landscape painting by Toyo Sesshu from the fifteenth century, or to read a poem by Shikibu from the eleventh century

Come quickly—as soon as these blossoms open they fall. This world exists as a sheen of dew on flowers

to realize that the Japanese culture has brought simplicity and reductionism to aesthetic peaks. In fact, ever since the eighth century, the most popular poem structure in Japan has been the short poem (tanka), which has only five lines and thirty-one syllables (arranged in 5, 7, 5, 7, 7). In the seventeenth century, an even shorter structure appeared (haiku), with three lines and seventeen syllables (in a 5, 7, 5 pattern).

    I will now explain the third element that I regard as essential for a physical theory of the universe to be beautiful—the Copernican principle.

3. The Copernican Principle: We Are Nothing Special

Many recognize Nicolaus Copernicus as the Polish astronomer who lived in the sixteenth century and reasserted the theory (based on a suggestion by Aristarchus some eighteen hundred years earlier) that the earth revolves around the sun. However, Copernicus was in fact (although not intentionally) responsible for a much more profound revolution in human thinking. The early models of the universe followed religiously the ideas of Aristotle, and were all geocentric. Namely, they all assumed that the earth was at the center of the universe. The most detailed, and most successful, model along these lines (in terms of explaining the observed paths of the sun, the moon, and the known planets) was due to the Greek astronomer Ptolemy, who lived in the second century A.D. This model survived, amazingly enough, for nearly thirteen centuries. One can only assume that it was the withering of intellectual curiosity during the Dark Ages, combined with the dominance of the Catholic Church, which regarded Aristotle's teachings as entirely consistent with its own doctrines, that granted the Ptolemaic model its longevity. In fact, following the integration of Aristotle's teachings into Christian theology, which is credited to Saint Thomas Aquinas (in the thirteenth century), Aristotle achieved an almost reverential status. Copernicus was the first to point out clearly that we do not occupy a privileged place in the universe. He discovered that we are nothing special. This has evolved to become known as the Copernican principle. In retrospect it is very easy to understand why there should be a Copernican principle in relation to the existence of "intelligent" creatures. After all, of all the places in the universe where intelligent creatures could emerge, assuming that there are many such places, very few, by definition, are "special." Therefore it is infinitely more likely for us to find ourselves in a nonspecial rather than in a special place. Put differently, the Copernican principle is a principle of mediocrity.

    Since the time of Copernicus, the Copernican principle has been substantiated even further. Not only has the earth been dethroned from its central position in the universe, in fact, at the beginning of this century, the astronomer Harlow Shapley demonstrated that our entire solar system is not even at the center of our own Milky Way galaxy. Indeed, it is about two-thirds of the way out from the center, completing a revolution around the center in about 200 million years. As we shall see in the next chapters, this vulgarization of the earth's location continued even much further.

    The Copernican principle can be expanded and generalized to include theories of the universe in general. In other words, every time that a certain theory would require humans to occupy a very special place or time for it to work, we could say that it does not obey the generalized Copernican principle. To give a specific example, if a theory were suggested in which the origin and evolution of humans was entirely different from that of all the other species, such a theory would not have obeyed the generalized Copernican principle. Darwin's theory of the origin of species by means of natural selection is thus a perfect example of a theory that does obey the Copernican principle (I will always mean the generalized principle from here on) and is therefore, from this point of view, beautiful.

    I would also like to note that some theories are considered "ugly" because they violate something that can be regarded as intermediate between simplicity and an even more general interpretation of the Copernican principle. I include here all the theories that are extremely contrived, or that necessitate some very special circumstances or fine-tuning for their validity (even if they do not involve an explicit role for humans). The reluctance to associate beauty with such theories is a bit like the disbelief that we would surely feel if someone told us that he can flip a coin and make it land on its side. We will encounter examples for such fine-tuning in chapters 5 and 6.

    I will now explain the concept of elegance, which, as I said, I personally do not regard as a necessary ingredient for beauty in a physical theory, but which can certainly enhance the beauty of certain theories.

4. Elegance: Expect the Unexpected

In mathematics and physics, and indeed in almost any discipline, it sometimes happens that a very simple, unexpected new idea resolves an otherwise relatively difficult problem. Such brilliant shortcuts lead to what are considered to be very elegant solutions. Interestingly, in chess, prizes for beauty are given for precisely this type of exceptional quality. It is amusing to note that one of the most elegant games of the brilliant American chess player Paul Morphy was played in Paris in 1858, in a box at the Paris Opéra, while Rossini's Barber of Seville was being performed on stage!

    It is important to understand that elegance has nothing to do with reductionism (what I called simplicity). For example, the Ptolemaic model for the motion of the planets offered in fact an elegant solution to a difficult observational problem, in that it found a clever way to explain remarkably well the observed motions of the planets. The model, however, was not simple at all. It required each planet to move around a small circle, called epicycle, the center of which moved around the earth on a large circle (called deferent). To explain all the observations, the Ptolemaic model required no fewer than eighty circles! It was only following Kepler's discovery that the planetary orbits around the sun are elliptical that a simple model for the solar system emerged.

    An example of elegance can be found in the following, well-known mathematical puzzle. Suppose we are asked: can one cover the board shown in Figure 5 by dominoes (each one having the area of exactly two squares), so that only one corner is left uncovered? The answer is very simple: no. Since the board has an even number (sixty-four exactly) of squares, and since each domino covers two squares, we can only cover an even number of squares and therefore we cannot leave one square open. Suppose, however, that we are now asked: can we cover the board in such a way that we leave two diagonally opposite corners uncovered? Clearly, in this case we will be covering an even number of squares, and so it is less trivial to determine immediately if this can be done or not (try thinking about this a little). It turns out, however, that with the help of an extremely simple trick we can answer the question right away. The idea is to think of the board as if half of the squares are painted black, as in a chessboard. Now, since each domino piece covers one black square and one white square, it is clear that it is impossible to leave uncovered two diagonally opposite squares, since they have the same color! So, the extremely elegant idea of turning the board mentally into a chessboard helped us to solve immediately a problem that appeared otherwise to be much more complex. Isn't this elegant? However, I want to emphasize again that elegance, which I consider superfluous in a beautiful theory, should not be confused with reductionism, which I regard as absolutely essential.

She Walks in Beauty

The beauty (or absence thereof) of physical theories is the main theme of this book. Beauty, like love, or hatred, is also almost impossible to define properly. In fact, probably any definition is likely to raise some objections. I therefore feel that it is worthwhile, even at the risk of some repetitiveness, to attempt to reconstruct briefly the thought process that led me to my requirements. I should first note that because of the fundamental role I think beauty does play in physics, one cannot be satisfied merely with the attitude "I'll know it when I see it," which some physicists have adopted toward beauty in physical theories. I claimed to have identified three elements that are absolutely necessary for a physical theory to be beautiful. These are symmetry, simplicity, and a generalized Copernican principle. It is perfectly legitimate to question this identification and ask: (1) what is it that makes these elements essential to beauty? and (2) are there other elements that may be equally important?

    The answers to these questions are not trivial, partly because of the fact that the recognition of beauty in a physical theory is to a large extent dependent on the scientist's intuition and sometimes even on taste (not unlike the dependence on the artist's aesthetic sensibility and taste in relation to a work of art). Nevertheless, I will attempt to outline now a certain logical process that can at least give us some guidance toward an answer. Instead of starting with the elements and trying to justify them, let us try to work our way backward, starting from a beautiful theory and retracing our steps in the direction of its basic ingredients.

    First, it is important to realize that since the ultimate goal of physical theories is to describe the universe and all phenomena within it in as perfect a way as possible, a theory cannot produce a real sensation of beauty unless it can be regarded as a major step toward perfection. We therefore need to identify which combination of properties can be regarded as constituting perfection. Since the universe involves an immense number of phenomena, and may generally appear quite chaotic, it is clear that what is required is the introduction of some regularity, organization, balance, and correspondence into the description of nature. These properties can allow more encompassing perceptions, which eventually enable scientists to identify common characteristics of different phenomena. Next, since we are interested in beauty, we want to identify classical aesthetic constituents or concepts that can contribute. Adopting ideas from the arts, the list of such concepts may include symmetry, simplicity, order, coherence, unity, elegance, and harmony. The question now is which of these truly plays a central role in science. In order to answer this last question I will attempt to rank-order these concepts in terms of their contribution to scientific thinking.

    Symmetry definitely occupies the top position in this hierarchy, since it literally forms the foundation on which physical laws are built, as I explained in the last section (and as we shall see further in chapter 3).

    Simplicity comes next, since it allows scientists to choose from among all the different possible hypotheses and ideas the most economical ones. Albert Einstein himself wrote once: "Our experience until now justifies our belief that nature is the realization of the simplest mathematical ideas that are reasonable."

    I did not list order, coherence, harmony, and unity as separate elements that are essential for beauty since in physics these are not independent concepts. For example, what is meant by order is that similar physical circumstances should produce similar consequences. However, we have already seen that symmetry and simplicity achieve precisely this goal. Furthermore, in the next section we will encounter an excellent example of the unifying power of symmetry and simplicity. Werner Heisenberg, one of the founding figures of quantum mechanics, the theory of the subatomic world, stated once as a criterion: "Beauty is the proper conformity of the parts to one another and to the whole." As we shall soon see, this is what symmetry and simplicity (reductionism) are all about.

    Finally, as I discussed in the last section, elegance can certainly contribute to the feeling of attractiveness that is associated with a certain solution to a scientific problem. However, I do not regard it as an essential element of beauty in a physical theory. In this assertion I humbly disagree somewhat with the sixteenth-century philosopher and statesman Francis Bacon, who was described by the poet Alexander Pope as "the brightest, wisest, meanest of mankind." Bacon wrote that "there is no excellent beauty that has not some strangeness in the proportion." Since "strangeness in the proportion" is to be understood at least partly as an element of surprise, Bacon's "definition" is more closely associated with what I called elegance than with beauty. However, Bacon's criterion refers also to the unification of otherwise seemingly independent concepts, and as such to symmetry and reductionism. My point of view on elegance is best expressed by a famous quote from the nuclear physicist Leo Szilard: "Elegance is for tailors."

    I hope that the above discussion clarifies the first two elements in my definition of beauty in physics. The third element, however, requires some further explanation. I have included in my essential ingredients the generalized Copernican principle, which is not traditionally linked with aesthetics. This is really a property that is peculiar to the sciences, and to theories of the universe in particular. In general, scientists absolutely detest theories that require special circumstances, contrived modeling, or fine-tuning. In this sense, as particle physicist Steven Weinberg puts it in his book Dreams of a Final Theory, a beautiful theory must be seen as essentially inevitable. A violation of the generalized Copernican principle, in the form of statements like "the universe must be so-and-so because we humans are so-and-so," or in the form of fine-tuning, is certainly a slap in the face of all encompassing inevitability, and is therefore ugly. I will return to this question in the discussion of intelligent life and the anthropic principle in chapter 9. At this point I will conclude by noting that placing humans center stage may be regarded as a desirable property from theological, psychological, or even theatrical perspectives, but it is not a property most scientists would like to associate with a beautiful theory.

    I do not wish to leave the reader with the impression that in their quest for beauty in the theory physicists lose sight of the beauty of the universe itself. This is certainly not the case. When Einstein once said that the only incomprehensible thing about the universe is that it is comprehensible, he referred precisely to these two "beauties." The reality of Marc Chagall's Lovers with Half-Moon (currently at the Stedelijk Museum, in Amsterdam) is much more than the chemical composition of its paint. Humans are on one hand able to access the beauty of the universe and on the other able to comprehend the beauty of its workings.

I have always admired the theater. I regard the art of presentation of ideas succinctly through dialogues and monologues as being a close cousin to the "art" of scientific presentations. Consequently, I have decided that in a few places in this book, I will abandon the more expository style in favor of a more theatrical style. The aim of these short "scenes" is to provide an introduction to new concepts, on the basis of the ideas that have already been developed.

Galileo

In a dark room, lit only by a few candles, five men dressed in red sit behind a long table. The grave expressions on their faces make them look almost identical.

Man at the center of the table (Grand Inquisitor): Bring him in! [The steps of a guard echo from the stone walls and the ceiling. The guard enters, pushing in front of him a bearded old man, who clearly has trouble walking.]

Grand Inquisitor: We would like to ask you again a few questions about your crazy and, may I add, dangerous ideas.

Old Man: All of my ideas merely represent the progress of scientific knowledge.

Grand Inquisitor: We will be the judges of that. The proposition that the sun is in the center and immovable from its place is absurd, philosophically false, and heretical. Do you agree that the earth stands still at the center of the universe?

Old man [in a weak voice]: Certainly not. The earth rotates around its axis and at the same time revolves around the sun. The entire solar system is not even at the center of our own galaxy.

Inquisitor at far left end of the table [with surprise]: Galaxy? What galaxy? What does the Milky Way have to do with it?

Old Man [straightening himself up a bit]: While the faintly luminous band seen across the sky at night is referred to as the Milky Way, a galaxy is really a collection of about one hundred billion stars like our sun.

Grand Inquisitor: Are you out of your mind? Where are all these suns?

Old Man: As I said, our own solar system belongs to such a galaxy. Most
of these stars are too faint to be observed with the naked eye.

Grand Inquisitor: Why do I not see that the earth revolves around the sun as you say? I see everything here standing still, while the sun is moving around the earth.

Old man [somewhat scornfully]: That is because you can only see relative motions. Everything on the earth is moving with the earth, so you do not see it moving with respect to you.

Second Inquisitor from right: This is the greatest nonsense I have ever heard, even if I were to ignore the desecration in your words. Soon you will be telling us that the sun is not revolving around the earth but around something else.

Old man: Indeed, the sun is rotating around its own axis, and it is revolving around the center of our galaxy.

Grand Inquisitor [raising his voice in great anger]: Do your ears hear what your mouth utters? Everything is rotating! [With great scorn.] What else do you think is rotating?

Old man [now clearly hesitant]: Well, the electrons in the atom, for example, revolve around the nucleus.

[The inquisitors are visibly baffled by this statement and start whispering among themselves. Finally, the Grand Inquisitor resumes the interrogation.]

Grand Inquisitor: It is becoming more and more obvious to us that you have lost your faculties. While we do not have the faintest idea of what you are talking about, I point out to you that the word atom in Greek implies that it is indivisible and therefore [raising his voice] there cannot be anything in it!

[The old man remains silent.]

Grand Inquisitor: Well?

Old man [feebly]: Matter as we know it is made of atoms, this is true. But the atoms themselves have a very dense and compact nucleus at their centers. In this nucleus there are particles called protons and neutrons. Other tiny particles, called electrons, revolve around this nucleus.

Grand Inquisitor [clasping his hands and looking upward in despair, then, after looking at his colleagues, turning to the old man with a resentful face]: At least I hope that your protons and electrons are not rotating around their axes?

[All the inquisitors laugh loudly.]

Old man [with some determination]: Well, the electrons and the protons have a quantum mechanical property called "spin," which in some respects can be thought of as if they are rotating around their own axes.

Grand Inquisitor [shouts in rage]: Shut up! I've had enough of this. The earth does not revolve, it is at the center of the universe, and all of these so called electrons and protons do not even exist. [Turning now to the guard.] Put him in a damp cell in solitary confinement, where he will have time to think until his own head will start spinning!

[The guard starts to drag the old man out of the room. The old man, his eyes open wide with fear, can hardly keep up with the guard's huge steps. As he is being dragged past the large wooden door he murmurs to himself]: Eppur si muove. (And yet it moves.)

Needless to say that in Galileo Galilei's time (1564-1642) galaxies, atoms, nuclei, quantum mechanics, and electrons had not yet been discovered. But had they been, it would not surprise me if Galileo would have used about them similar phrases to the ones I have put in his mouth.

Spinning

Electrons are the smallest (in mass) known particles that are electrically charged. Galaxies are huge collections of tens of billions stars. Yet precisely the same physical laws govern the behaviors of both. This is the true meaning of simplicity and symmetry—of beauty in physics. How do we know this? The following example describes these principles in action.

    Electrons and protons, the basic building blocks of atoms, have a property called spin. Strictly speaking, this property can be described only by quantum mechanics—the theory that governs the subatomic world. However, for some purposes spin can be thought of as a rotation of the electron or the proton around its own axis. Therefore, as for all rotating bodies, the properties of this spin are described by the conservation of angular momentum. Because of the fact that electrons and protons also have an electric charge, this spin makes them behave like small bar magnets, since a magnetic field is generated when electric charges are moving. Now, a simple experiment can be performed with two small magnets, such as two compass needles. Imagine that you suspend the needles on thin threads attached to their middle points. If you place the two needles along a straight line, with two equal (say, north) poles facing one another, you will observe that one of the needles will spontaneously flip, so that two opposite (one north and one south) poles will face one another. Energetically speaking, when a system undergoes such a spontaneous transition, this means that the second configuration is at a lower energy state than the first, since physical systems tend to be in their lowest possible energy state (as I am sure everybody recognizes from their own tendencies).

    In an analogous manner, when the hydrogen atom, which is composed of one proton around which one electron is orbiting, is in its lowest orbital energy level, the spins of the electron and proton can be parallel, which is equivalent in some sense to the electron and proton rotating in the same sense, or antiparallel (namely, the electron and proton rotating in opposite senses; quantum mechanics only allows two possible states for the spins of the electron and the proton). Consequently, the hydrogen atom can spontaneously undergo a spin-flip transition, from the parallel to the antiparallel state. Since the latter is at a lower energy state, the difference in the energy is emitted in the form of radio waves. Waves in general are characterized by a property called wavelength. For example, when we throw a pebble into a pool, we can observe a series of concentric waves, and the distance between two crests is the wavelength.

    The wavelength of the radio waves emitted by the spin-flip transition is 21 centimeters. So, how is all of this connected to galaxies? As it turns out, this 21-centimeter radio wave plays a crucial role in our exploration of galaxies. For example, in order to determine the global structure of our own galaxy, the Milky Way, and the motions within it, observations must be made to distances of the order of 30,000 light-years. The problem is, however, that the interstellar medium, diffuse gas and dust in which clouds are dispersed, prohibits optical observations to such large distances because it is opaque to visible light. The idea is therefore to identify a wavelength of electromagnetic radiation to which the interstellar matter is relatively transparent, so that the source of the radiation can be seen to large distances. In addition, the wavelength has to be such that either stars or cold gas clouds, which are the main and most abundant participants in the motions in the galaxy, would be strong emitters in that wavelength, to allow its detection.

    The 21-centimeter radiation proves perfect for this purpose. Not only does it allow observations to penetrate to the farthest corners of the Galaxy, it is also emitted by neutral hydrogen atoms, which constitute the main component of interstellar matter, in the form of cold gas clouds.

    There is one more effect that needs explanation in relation to observations aimed at determining the speeds of gas clouds, and this is known as the Doppler effect, after the Austrian physicist Christian Doppler, who identified it in 1842. We are familiar with this effect in everyday life, in relation to sound waves. When a source of sound (e.g., a car or a train) is approaching us, the sound waves bunch up and we receive waves at a higher frequency (higher pitch). The opposite happens when the source of sound is receding from us. The waves are spread out and we hear a lower pitch. The effect is particularly noticeable when the source of the sound passes us by rapidly, since the sound changes from a high to a low pitch (recall, for example, the "eeeeeeoooooo" sounds of cars on the Indy 500 race track). A similar phenomenon occurs with light or any electromagnetic radiation. Therefore, when a source that emits the 21-centimeter radiation is receding from us, the radiation will be detected at a longer wavelength (lower frequency), while when it is approaching us, it will be observed at a shorter wavelength (higher frequency). From the detected shift in the wavelength, the speed of the source can be determined, since the shift is larger the higher the speed.

    The stars and the gas clouds in the Galaxy participate in a global rotation around the galactic center. Radio observations of the 21-centimeter radiation allowed astronomers to determine the structure and the rotation pattern of the entire galaxy interior to the sun's orbit—namely, between the solar system and the center of the Galaxy.

    A detailed mapping of our galaxy reveals that it is a disklike, spiral galaxy. Namely, it has the shape of a flattened pancake, with a spiral pattern observed on the face of the disk. The disk structure is probably a consequence of the formation process of the Galaxy, and it is related at least in part to the amount of angular momentum (amount of rotation) possessed by the gas cloud from which the Galaxy formed. The exact process by which galaxies form is still not fully understood; however, the following general remarks can be made. When a rapidly rotating cloud of gas collapses to form a galaxy (due to the force of gravity), it will tend to form a flattened disk. This is a consequence of the centrifugal force, which tends to push material away from the rotation axis. The centrifugal force is very familiar to anyone who has tried to negotiate a sharp turn with a car moving at a high speed—a strong force is felt, pushing the car off the road. The centrifugal force is stronger when the "amount of rotation" of the cloud is higher. Consequently, for high angular momentum clouds, the collapse forms a flat, disklike structure.

    So, what have we have discovered here? This same entity, called angular momentum, the conservation of which resulted from the symmetry of the laws of physics under rotation, explained to us something about the electron, which has dimensions of about [10.sup.-13] (0.0000000000001) centimeter, and about the Galaxy, which has dimensions of about [10.sup.23] (100,000,000,000,000,000,000,000) centimeters. What's more, we used one (the spin-flip of the electron) to discover the other (the structure of the Galaxy). Now, I am asking you, isn't this absolutely BEAUTIFUL?