Ever-Changing Moods: Did Nature’s Constants Evolve?

*The Sefer ha-Temeneh (13th century mystical text) teaches that* *in the seventh millennium there sets in a gradual and progressive retardation in the movement of stars and the spheres, so that* *the measurements of time change and become longer in* *geometrical progression…. Hence [mystics] arrived at truly* *astronomical figures for the total duration of the world.*

Gershom Scholem (*Kabbalah*)

*Whether or not it is clear to you,*

*No doubt the universe is unfolding as it should.*

Max Ehrmann (*Desiderata*)

Physicists and astronomers like to frame natural laws in terms of simple mathematical equations. Labeling specific features of nature with symbols, they seek unambiguous relationships between such values, unalterable over time. Einstein’s famous equation relating energy to mass and the speed of light squared constitutes an example of this tendency, and there are countless others. The hallmark of such equations is that they can be tested again and again, never failing to yield the same result. Therefore, like a warm bowl of porridge

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5
Ever-Changing Moods: Did Nature’s Constants Evolve?
The Sefer ha-Temeneh (13th century mystical text) teaches that in the seventh millennium there sets in a gradual and progressive retardation in the movement of stars and the spheres, so that the measurements of time change and become longer in geometrical progression…. Hence [mystics] arrived at truly astronomical figures for the total duration of the world.
Gershom Scholem (Kabbalah)
Whether or not it is clear to you,
No doubt the universe is unfolding as it should.
Max Ehrmann (Desiderata)
THE LEXICON OF NATURE
Physicists and astronomers like to frame natural laws in terms of simple mathematical equations. Labeling specific features of nature with symbols, they seek unambiguous relationships between such values, unalterable over time. Einstein’s famous equation relating energy to mass and the speed of light squared constitutes an example of this tendency, and there are countless others. The hallmark of such equations is that they can be tested again and again, never failing to yield the same result. Therefore, like a warm bowl of porridge

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in the morning and a steaming cup of cocoa in the evening, they offer the comfort of regularity.
Within the framework of equations, scientists like to distinguish two types of elements: variables and constants. Traditionally, constants aren’t supposed to alter over time. For instance, when Einstein composed his energy-mass relationship, he fully expected that the speed of light, a constant, would be the one permanent factor. So for any type of material under any kind of circumstance, he posited that this value would never change.
Another well-known equation is Newton’s inverse-square law of gravity. It too contains a seemingly enduring fixture of nature, the gravitational constant. When Einstein proposed general relativity as a theory of greater scope than that of Newton, he kept the same constant. Both cases represent invariant relationships—first formulated theoretically and later confirmed by observation as being correct descriptions of our world.
However, there is another kind of law, logically distinct from the type to which Newton’s and Einstein’s theories belong. When we examine nature’s vast array of phenomena, we sometimes observe patterns of a wholly different sort. Rather than the products of predictable equations, they constitute much subtler relationships that sometimes only the remarkable organizational capacities of our minds can perceive.
Consider, for example, the intricate designs of seashells and the elaborate lacework of snowflakes. Neither of these is governed by immutable equations. Instead, these spectacles emerge through self-organization—wonderful instances of order stemming from chaos. In the first case, the Fibonacci sequence of numbers, formed by adding each pair to produce the next (1, 1, 2, 3, 5, etc.), serves to characterize the length of successive turns in a spiral. In the second, the molecular geometry of water delimits the six-pronged symmetries of icy shapes. In each case, mathematical features manifest themselves in surprising ways.

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Certain musical refrains seem more pleasing to our ears, certain artistic conceptions more beautiful to our eyes. We would recognize the first four notes of Beethoven’s fifth as played on everything from an accordion to a tin whistle—instruments with radically different material compositions. Michelangelo’s most famous statue could be carved out of ivory or soldered from sheets of steel; we’d still know the lad as David. No physical law allows us to recognize and enjoy such creations; rather our magnificent gray matter somehow does the job.
Sometimes the predilections of mathematics and the preferences of our senses even coincide. For example, the Greeks favored the Golden Ratio in art and architecture because they knew that this unique number produced pleasing relationships. Thus, the grandeur of the Parthenon attests to both the masterful structures wrought by mathematics and our unique abilities to perceive and make use of such constructs. We can’t always rely, however, on our pension for detecting patterns. In some cases it can prove misleading instead of enlightening. A gambler might notice that the first five spins of a roulette wheel land on the corresponding numbers of the Fibonacci sequence. He bets his fortune on the sixth, then glumly empties his pockets after his hypothesis proves wrong.
Einstein’s equations of general relativity harbor many distinct solutions. Some of these represent real aspects of our universe. Others apparently constitute false leads. It is our pattern-discerning ability that helps us choose among these. But only the test of time— continued observation using increasingly powerful experimental tools—will corroborate our theories.
In the film Pi, the mentally prodigious protagonist begins to see patterns in everything—from biblical writings to the stock market. Convinced he has found a grand scheme that underlies all creation, he pushes ever further. All material comforts succumb to his frantic quest. Finally suffering a nervous breakdown, he decides to trade his numerical search for inner peace.

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Whenever we perceive natural patterns, we must reasonably ask ourselves if they represent deep truths or are simply the products of our overactive imaginations. Sometimes our hunches guide us to extraordinary discoveries. Other times they lead us to dead ends. Still our curiosity drives us to their very limit, with the hope of arriving at ultimate knowledge.
DIRAC’S BOLD IDEA
Oxford mathematician Ioan James has detected a curious pattern among members of his own profession. Surveying their personalities and achievements, he has arrived at the conclusion that particular kinds of social deficits, possibly neurological in origin, often correlate with the focus needed for monumental discoveries. Perhaps certain personal, physical, or psychological limitations help concentrate the mind. Or, alternatively, perhaps development in some areas of the brain comes at the sacrifice of other areas. Thus, for instance, mathematical geniuses may not always make the best conversationalists.
The physicist Paul Adrien Maurice Dirac, one of the subjects of James’s study, was a notoriously inscrutable 20th century thinker. He was a mystery even to many of his closest friends. “Nobody knew him very well,” recalled physicist Engelbert Schucking, who encountered Dirac at various conferences. His colleagues joked that he rarely said anything more than “Yes,” “No,” or “I don’t know.” Legends swirled around him like tales spun about uncharted islands. They typically focused on his economy of words and his solemn dedication to pure science. In one such story, Dirac had just finished reading Dostoevsky’s Crime and Punishment. Asked about his impressions of the classic Russian novel, he had only one comment: “It is nice, but in one of the chapters the author made a mistake. He describes the Sun rising twice on the same day.”
Another time Dirac was delivering a lecture in his usual crisp and precise style. He never minced words and always planned each

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sentence meticulously. After the talk, there was a question-and-answer session. Someone raised his hand and said, “Professor Dirac, I do not understand how you derived the formula on the top left side of the blackboard.”
“This is not a question,” Dirac curtly responded. “It is a statement. Next question, please.”
Dirac’s rigidity in conversation contrasted with his extraordinary brilliance and creativity in discerning the properties of the universe. In the early days of quantum mechanics, his agile mathematical mind rapidly encompassed radical new ways of interpreting physics. He developed theories and ideas so fantastic, such as a negative energy sea that fills all of space, that they knocked the breath out of his colleagues. This concept relates to arguably his most important contribution to physics, the Dirac equation, proposed in 1928. The Dirac equation offered a quantum, relativistic description of an electron, encompassing properties such as its mass, charge, and spin. It predicted the existence of positively charged counterparts to electrons. Known as positrons, they were first experimentally detected in 1932. For his pivotal scientific contributions, Dirac was awarded a Nobel Prize the following year.
In 1937, Dirac applied his prodigious talents in an attempt to explain an astonishing physical coincidence in cosmology. Comparing the strength of the electrical and gravitational forces acting between the proton and electron in a hydrogen atom, he noticed that the ratio is an immense number, approximately 1040 (one followed by 40 zeros). The fact that this value is so large is related to what is now known as the “hierarchy problem.” Curiously, Dirac found that the present age of the universe as expressed in atomic units (the time for a light particle to trek across a hydrogen atom) is roughly the same size. In what is known as the Large Numbers Hypothesis (LNH), Dirac suggested that the two numbers are in fact equal.
Sometimes apparent coincidences mask fundamental truths. For example, when physicist Murray Gell-Mann discovered that he could

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arrange the properties of elementary particles into curious arrays, he speculated that these patterns stemmed from groupings of yet more fundamental objects called quarks. If he had turned out to be wrong, his methods would have been deemed numerological hokum. But he was right, and his insight led to the modern field of quantum chromodynamics—the theory of the strong nuclear interaction that binds protons and neutrons together.
Following a similar hunch, Dirac bet that the coincidence he discovered between various large numbers in the universe stemmed from a fundamental principle of nature. He proposed that the ratio of the strengths of the gravitational and electromagnetic forces was equal in the cosmic beginning but diminished proportionally with each atomic interaction. That is, each time the “clock” of a hydrogen atom ticked, gravity would become slightly weaker. Consequently, by 1040 ticks, gravity would be that much scrawnier than still-brawny electromagnetism—the unequal match we witness today. In general form the LNH states that large dimensionless numbers should vary with the epoch of the Universe.
Is Dirac’s result profound or simply prestidigitation? In purest numerical form it is almost certainly not correct, since it does not match up with any known gravitational theory. However, there are compelling ways of altering general relativity to produce a changing gravitational strength that have attracted their share of supporters over the decades.
CHANGING GRAVITY
Spurred by Dirac’s curious notion, other physicists have attempted to develop explanations in cosmology and particle physics for why the Newtonian gravitational constant would vary. This parameter, G, an important component of general relativity as well as Newton’s law of gravitation, sets the scale of gravity. If G drops in value, the gravitational attraction between any set of masses grows correspondingly weaker.

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One of the first such G-variability theories was proposed by German physicist Pascual Jordan in the early 1940s. He recognized that he could modify Einstein’s theory by removing explicit references to G and replacing it with a scalar field (representing a spin zero particle). The result is that gravitational strength diminishes over time. Jordan’s ideas received little notice at first, likely because of the political situation of the time. During World War II, Jordan was an active member of the Nazi Party in Germany while at the same time generalizing theories developed by an Englishman (Dirac) and a Jewish expatriate (Einstein). This bizarre combination of political and scientific pursuits endeared him to neither side. After the war his theories were finally published, but his record was tainted by his former nationalistic allegiances.
It was not until the early 1960s that Carl Brans and Robert Dicke rediscovered Jordan’s notion and granted it more of a hearing. They came upon the idea in a different manner, by developing a novel interpretation of Mach’s principle. Brans and Dicke wondered what would happen if G was related to the mass distribution of an expanding universe. Like Jordan, they constructed such an arrangement by replacing G with a variable scalar field. Unlike their predecessor’s approach, however, they took extra steps to ensure that energy would be conserved for all times. In Jordan’s theory, energy is not conserved. In honor of the developers, the combined proposal is now called the Jordan-Brans-Dicke scalar-tensor theory. The “scalar-tensor” appellation refers to the combination of a scalar field with the geometric tensor of general relativity.
Another model of changing gravity was developed several years later by Hoyle and Narlikar. Similarly based on Mach’s principle, it combines the original steady state cosmology with the scalar-tensor theory. It defines the mass of each particle as a field, enabling that value to change from point to point. Thus, electrons could have different masses in different parts of space. This field is a function of the masses of all the other particles in the universe. Because the universe changes, the strength of the gravitational interaction similarly

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alters from time to time and is also allowed to differ from place to place.
Yet another variable-G cosmology, called scale-covariant theory, was proposed in the 1970s by Vittorio Canuto of the City College of New York. Canuto called for a reframing of Dirac’s LNH by means of modifying a number of physical laws, including general relativity and the principle of the conservation of energy. Then, the LNH enters the model as a special condition.
In 1989, Steinhardt and Daile La incorporated aspects of the Jordan-Brans-Dicke theory into extended inflationary cosmology, a variable-G version of the standard inflationary scenario. More recently, a changing gravitational constant has been suggested as a possible solution to the dark-energy conundrum. A diminution in gravity’s strength would offer a natural way of explaining the acceleration of the cosmos—a weaker hold allowing for faster expansion.
Researchers have developed numerous tests to distinguish the various contenders for a possible new theory of gravitation and to determine if standard general relativity requires modification. Each of the variable-G models offers specific predictions in the fields of astrophysics and geophysics, consequences that experimenters can readily assess.
ASTROPHYSICAL CLUES
Astrophysics provides us with myriad examples of gravitationally bound systems. Each would be profoundly affected if G happened to vary. Researchers would notice discrepancies on every scale—from individual stars (such as the Sun) to star clusters (such as Messier 67) to galaxies (such as Andromeda) to clusters of galaxies (such as the one in the constellation Hercules) and finally to superclusters (such as the Local Supercluster).
A slow decrease in the gravitational constant would engender a multitude of long-term consequences. Objects in orbit would

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gradually move farther and farther away from the bodies around which they are revolving. Locally, the average distance between Earth and the Sun would slowly get bigger, cooling our planet over time. Naturally this would be true not just for our own world but for any other planet in space. Hence, if many years in the future we decided to escape our frigid planet, we should expect that any world we settled on would eventually suffer the same fate.
To make matters worse, the Sun and other stars would become less luminous (intrinsically bright) over time. The rate at which a star like the Sun burns hydrogen and other elements depends on its internal temperature, which in turn is governed by the delicate balance of gravity and pressure inside the star. If G decreases, this balance becomes altered and the star becomes less efficient in its fusion. Ergo, it dims like a lightbulb during a power shortage.
With billions of slowly fading stars, galaxies too would gradually lose their luster. New stars would be born, but they’d similarly become less efficient—as the “dimmer switch” of G steadily lowered. Moreover, stars would orbit galactic centers at orbits farther and farther away, making galaxies appear more diffuse. Clusters and superclusters would spread out, in an effect that would add to their existing cosmological recession. In short, the cosmos would become, over the eons, fainter and more dispersed—like the scattered embers of a once-raging fire. (Such a grand cooling process would happen anyway, due to the irreversible law of entropy and the expansion of the universe, but any reduction in G would hasten it.)
We need not wait for eons. There are already a number of astrophysical ways of determining if G has decreased. If the Sun’s luminosity has gone down, for instance, meteorites falling on Earth would have been warmer, on average, in the past, affecting the results of their impact and nature of their debris. Thus, meteorite remnants could be investigated with the goal of determining their original temperatures—thereby attempting to find out if the Sun has dimmed over time.

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Another astrophysical test of G-variability involves the rhythms of pulsars. These dense, rapidly spinning remnants of massive stars emit energy in a periodic fashion. Like lighthouse beacons, they send off pulses as they turn, which astronomers detect as oscillating radio waves (or sometimes X-rays). Gradually, this rate of emission slows, in a process called “spin-down.”
In 1968, shortly after the discovery of pulsars, MIT radio astronomers Charles Counselman and Irwin Shapiro demonstrated how their spin-down rate could be used to calculate changes in the gravitational constant. Their paper, published in Science, was very much ahead of its time, given that pulsar spin-down had not yet been observed. More than two decades later, at a conference held in Rome, Israeli astrophysicist Itzhak Goldman announced results for the radio pulsar PSR 0655+64. By measuring its spin-down, he placed strong limits on the variability of G. He found that G changed less than one-billionth of a percent per year.
Although compact objects such as pulsars have extremely high densities, they are able to resist complete collapse into black holes. The bounds of such resistance, set by what is called the Chandrasekhar limiting mass, depend on gravity’s strength. Subsequently, if G varies, the number of dense stars that end up as black holes would gradually alter. For instance, if gravity grew weaker over time, this rate would plummet. By counting relative numbers of black holes compared to pulsars at various distances (by looking farther out, we see further back in time), astronomers could conceivably detect this effect.
On the grandest scales, a changing G would manifest itself in altered patterns of star clusters, individual galaxies, and large-scale galactic distributions. If G varied, their gravitational tourniquets would slowly ease up, offering them freer circulation. The result would be a gradual—and potentially discernable—spreading out of these systems.

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TAKING THE EARTH’S PULSE
In looking for aberrations, sometimes it’s wise to start with the familiar. To determine whether or not the gravitational constant is changing, we need look no farther than Earth itself. Aspects of Earth’s dynamics, such as its rate of spin, are sensitive measures of gravity’s strength. Hence, geophysics provides us with another credible means of testing Dirac’s hypothesis and comparing its various incarnations.
By virtue of its rotation, Earth represents a kind of cosmic clock that has been ticking for about 4.5 billion years. During that time it has been gradually slowing down by a rate of about two milliseconds per century. By considering the various processes that could possibly contribute to this lag, we could theoretically deduce information about long-term cosmological effects—such as changes in G. In practice, this is a complicated problem because there are numerous mundane effects that contribute to the slowdown. Scientists believe that much of the deceleration stems from ocean tides, caused by the Moon’s gravitational attraction. Over time these tugs dissipate energy and gradually decrease Earth’s rotational speed.
Newton’s laws, applicable to the Earth-Moon system, mandate that angular momentum (the mass of each body times its rotational velocity times its distance from the center of rotation) must be conserved. Hence, as Earth has slightly slowed down, the Moon has compensated by speeding up a bit. This has resulted in the Moon receding, ever so slightly, over the eons. Consequently, by measuring the Moon’s orbital motion, we can obtain a precise record of Earth’s rotational slowdown, which we can then use to measure any change in the strength of gravity.
There are a number of ways to track the Moon’s behavior. The most direct method goes back to the APOLLO project, mentioned earlier, and its predecessors. By beaming a laser pulse to a mirror on the Moon (placed there by astronauts in 1969) and measuring the return time, scientists have developed precise records of the Moon’s

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supernova results of Schmidt’s and Perlmutter’s groups, among other measurements, would require reinterpretation. Consequently, the universe might not be accelerating after all (or accelerating at a different rate than previously thought). This would again change our understanding of the cosmological constant.
Magueijo vividly describes how he and Albrecht developed this radical approach. Before they began the collaboration, Albrecht had a “lifelong obsession, the need to find an alternative to inflation.” The paper with Steinhardt had been Albrecht’s first (it was his doctoral dissertation work), so he felt it was time to examine other possibilities. Together, they explored the changes to the equations of physics needed to realize their idea. These turned out to be quite significant, considering that their theory contradicted not just general relativity but also special relativity. Even long-accepted formulations, such as Maxwell’s equations of electromagnetism, required modification. Nevertheless, they ardently pressed on—not knowing where the fruit of their efforts would lead.
Because of the controversial nature of the VSL hypothesis, the researchers had difficulty getting their findings published at first. Journal editors were reluctant to touch material that seemed to challenge the maxims of modern science. It took a year of revisions before their initial article was accepted. Even once their work was in print, many mainstream physicists shied away from it. Soon, this controversy was fueled even further by clashing experimental results pertaining to yet another natural “constant.”
ALTERING ALPHA
The gravitational constant and the speed of light are not the only fundamental parameters that researchers have asserted could change with cosmic time. Another popular candidate for variability is the fine-structure constant, known by the Greek letter (alpha), which is basically the square of the electron’s charge, combined with other parameters, including the speed of light. Thus, either a changing

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electric charge or a changing light speed would cause alterations in alpha as well. Although quantum theory asserts that at ordinary energies alpha remains approximately 1/137 for all times, physicists have reason to believe it could vary under extreme conditions, such as those in the very early universe.
The fine-structure constant lies at the heart of quantum electrodynamics; it gauges the strength of interactions between charged particles. Because at higher energies, virtual particles arise that shield the charges of real particles, some theoretical models suggest that alpha could have a different value in such regimes. If researchers established that it was not only energy dependent but also time dependent, this would imply a slow change in the properties of the vacuum. Earth’s evolution could consequently be affected over long periods. Hence, like changing G, variations in alpha could possibly be detected through geophysics.
In 1999 a team of astrophysicists led by John K. Webb of the University of New South Wales found evidence for evolution of the fine-structure constant in the absorption spectra of very distant quasars, extremely remote, superpowerful sources of energy, believed to serve as the dynamos of young galaxies. Webb and his colleagues found that alpha could have varied as much as 2 percent since the time of the Big Bang.
For those scientists who are of the opinion that at least some of nature’s firm footholds are really slippery sands, Webb’s results offered the tantalizing prospect of vindication. They seemed to reveal a past landscape significantly different from that of today. The study of “variable constants” kicked into high gear, with an increasing number of researchers eager to explore its exotic terrain. Among these innovative scientists was Cambridge cosmologist John Barrow, who, along with Magueijo and Håvard Sandvik, developed a model of the universe based on changing alpha.
Recent findings, however, have cast doubt on Webb’s results. In 2004 a group headed by Nobel Prize–winning physicist Theodor Hänsch reported that its four-year study of atomic emissions

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uncovered no sign of alpha changing over time. Their experiment was precise enough that it would have revealed a variation in alpha as little as one part per quadrillion per year; nevertheless, alpha didn’t blink. Another team, headed by Raghunathan Srianand of the Inter University Centre for Astronomy and Astrophysics in Pune, India, conducted a survey of distant quasars using the Very Large Telescope. Analyzing these quasars’ absorption spectra, the team established constraints on alpha variation at least four times stricter than those of Hänsch’s group.
Physicists and astronomers continue to examine the stony face of alpha, looking for any signs of a twitch. Probing the widest possible range of objects, from atoms to quasars, they are assessing its sturdiness (or flexibility) with ever-sharper tools. A host of contemporary physical theories await their results.
A MATTER OF SCALE
Each cosmological model rests on the bedrock of particular fundamental principles. Even if certain “constants” actually turned out to vary, other aspects of the cosmos could well transcend time’s capriciousness and remain true forever. They need not involve actual physical parameters, such as charge or mass, but might represent simple mathematical rules.
Some modern thinkers, inspired in part by mathematician Benoit Mandelbrot’s concept of fractals, have suggested that the universal guiding principle is “self-similarity.” Self-similarity, the hallmark of fractal structures, means that a portion of something, sufficiently enlarged, resembles the whole thing. Mandelbrot discovered numerous examples of self-similar geometries in nature—from the delicate patterns of snowflakes to the jagged profiles of coastlines.
Consider, for instance, the shapes of trees. Trees generally have a few main limbs extending from their trunks. From these major branches grow smaller branches; from those, tiny twigs; and so forth.

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If you clip off a cluster of branches from the end of a limb and hold it up by its own “trunk,” it resembles a miniature tree in its own right. Thus, trees are often self-similar—the parts resemble the whole.
Not that such match-ups would necessarily be exact. Nature yields only approximate fractals. True fractals, with perfect self-similarity, are found only in mathematics. Famous examples are the intricate Koch curve (formed by repeatedly removing the middle thirds of triangles’ sides and replacing them with smaller triangles) and the exuberant Mandelbrot set (a lacy design etched out through a special algorithm). Still, the existence of almost scale-free natural structures could reveal critical clues about the hidden architecture of reality’s cathedral.
A number of researchers have suggested that the universe itself is a fractal. One of the pioneers of such a hierarchical approach is Robert Oldershaw of Amherst College, who has published numerous papers on the subject. Through comparing the properties of systems on many scales, he “found that there was a considerable potential for physically meaningful analogies among atomic, stellar, and galactic scale systems.”
Oldershaw has speculated that nature’s hierarchy continues indefinitely—like an unlimited succession of Russian dolls, nested one inside the other. Why assume that galactic superclusters are the highest form of organization in the kingdom of all possibilities? Perhaps, he has suggested, the observable universe comprises but a metagalaxy in a greater realm—a meta-metagalaxy, so to speak. The meta-metagalaxy, in turn, would constitute part of an even larger entity, and so on.
In this spirit, let’s construct our own cosmic hierarchy. We divide astronomical objects into seven major classes, covering an enormous range of sizes. The first class includes asteroids, comets, and other types of “minor planetary objects,” ranging from several feet to hundreds of miles across. Second come planets, with radii spanning

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thousands to tens of thousands of miles. Stars come next, with radii ranging from hundreds of thousands to tens of millions of miles. (Note that such gargantuan figures represent living stars; extinguished ones, such as white dwarfs, neutron stars, and black holes are much smaller.) Systems in the fourth class, stellar clusters, have sizes best expressed in terms of light-years. Typically, they are on the order of 10 light-years across. (Recall that a light-year is approximately 6 trillion miles.)
Now let’s take a colossal leap and turn to the fifth category— galaxies. These objects are ordinarily hundreds of thousands of light-years across. Clusters of galaxies, the sixth rung on this astronomical ladder, generally contain between 50 and 1,000 galaxies within a region roughly 10 million light-years across. Finally, the seventh level includes superclusters and even larger structures, such as filaments, bubbles, and walls. Superclusters typically have total populations of as many as 10,000 galaxies, housed in a sector about 100 million light-years across. Astronomers used to think they were the largest structures in the universe, until in the 1980s a team led by Margaret Geller and John Huchra of the Harvard-Smithsonian Center for Astrophysics mapped out a three-dimensional slice of space, revealing vast, spongy arrangements of galaxies. In their cosmic map, stringy, bubbly and sheetlike arrays of galaxies—called filaments, bubbles and walls, respectively—bounded relatively empty regions, called voids. The largest structure they found was the “Great Wall,” a sheet of galaxies stretching out more than half a billion light-years across.
If nature’s operating principle is self-similarity, it behooves us to search for commonalities on all scales. One natural place to look is in the density distribution of various astronomical structures, which indicates how much of their material lies at their centers and how much is peripheral. Clearly, because these systems have mass, the inverse-square law of gravitation constitutes one part of the picture. Additionally, because many of these systems are rotating, their total

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angular momentum must play a strong role. Moreover, because of the elusiveness of dark matter, we know that the first two factors, applied just to visible material, cannot tell the whole story.
Consider, for instance, the density distribution of a cluster of galaxies. Like a sprawling metropolis, it has a greater concentration of galaxies packed within its central district than it does strewn way out in its suburbs. Hence, at least for visible material, clusters exhibit a density pattern that peaks at its center and drops off with radial distance. Superclusters display similar arrangements among the distribution of their member galaxies. They contain, however, several individual clusters and smaller groups of galaxies. Also, they may not be in equilibrium, meaning their forms are not settled.
Links between density distributions at various scales suggest that relative mass has more meaning than absolute mass in describing the state of the cosmos. After all, absolute mass is but a human invention. When we stand on a scale, we are comparing our bulk to a particular fixed amount. In metric units that standard is one kilogram—originally defined as the mass of a special platinum-iridium cylinder protected in an underground vault in Paris. Surely, galaxies don’t stop off in Paris when deciding how to arrange themselves.
Strange as it may seem, by temporarily abolishing the kilogram (and all other mass units) the theory frees up to have no particular scale. Instead, relative mass can be defined as a function of two fundamental quantities—the gravitational constant and the speed of light—as well as of the size of a particular region. This combination of distinct parameters could be an important clue to solving the mystery of why the naturally occurring laws of galaxy distribution comprise but a small subset among all possible arrangements. With these special assumptions in mind, we can construct self-similar cosmological solutions of Einstein’s equations that could represent the scale-free organization of material in the universe. By matching them up with the distributions of galaxies in clusters and superclusters, we could explain commonalities between those two scales.

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What about smaller scales, from asteroids, meteors, and comets up to planets, stars, stellar clusters, and galaxies themselves? Could there be a simple scale-free rule that would unite these disparate shapes and sizes? At first glance these objects appear as different as can be. Yet each has at least two things in common: gravitation and rotation. Each has mass, and each spins about an axis. Not to say that these masses and spins are at all identical—in fact, they are very different. However, what if there were a simple combination of mass and spin that is itself scale-free? Such a construction would represent a neat way of categorizing the properties of a vast range of astronomical objects.
OUT FOR A SPIN
Many amusement parks have rotor rides, best avoided after a hearty lunch. If you haven’t eaten recently, let’s take a spin on one of these contraptions. After handing over your ticket and waiting in a half-hour queue, you walk into a large metal cylinder. Following the lead of others, you stand with your back against the wall. Soon the contraption begins to rotate—slowly at first, then faster and faster. As the floor beneath you starts to drop, you find your back pinned against the metal surface. Inertia, you realize, as you remember the story of Newton’s bucket. If it weren’t for the wall, you’d be flying off toward the roller coaster. The surface against your back acts to keep you moving in a circle, exerting what physicists call a “centripetal force,” which creates an inward directed “centripetal acceleration,” enabling your circular motion. Then, before you can work out the equations, you see the ride operator pull the lever to end the ride. The floor rises, the whirling stops, and you get off. Had fun?
As you disembark, you realize to your dismay that everything still seems to be spinning—from the churning of your stomach to the agitation in your head. The delicate fluid in your inner ear presses uncomfortably upon special nerves, producing the loathsome sensa-

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tion of dizziness. You feel absolutely queasy, and need to sit down on a nearby bench.
Just as linear motion, in the absence of force, tends to continue indefinitely, rotational motion has its own way of lingering. This kind of persistence is called “conservation of angular momentum.” Recall that angular momentum is the product of an object’s mass, its rotational velocity, and its distance from the axis of rotation. One outcome of this conservation law is that if a rotating body pulls in its bulk, reducing its radius, it tends to spin faster—the total angular momentum remains constant—so if one quantity, say the distance, decreases, one of the others, say velocity, must increase. Due to conservation of angular momentum, therefore, a pirouetting skater is able to whirl at impressive speeds after drawing her arms closer to her torso. Similarly, pulsars spin considerably faster than the much larger stars from which they evolved.
The conservation of angular momentum is the reason that Earth and the other planets rotate today. The solar system, in its youth, was a whirlpool of gas and dust. As it coalesced into the Sun and the nine planets, they each retained a portion of the original whirlpool’s angular momentum—hence, each must also rotate. Similarly, in the amusement park example, if you had a sensitive frictionless gyroscope in your hand while the rotor was moving, it could continue to spin after the ride stopped.
In our search for a scale-free law uniting the cornucopia of astronomical systems, angular momentum provides an important clue. Along with mass, it is a fundamental way of classifying objects in the cosmos. Indeed some objects, such as black holes, are distinguished only by these two parameters (supplemented in certain cases by electric charge).
With the goal of a scale-free principle in mind, we can construct a simple relationship between angular momentum and mass that appears to describe many rotating astronomical systems on a variety of scales. The rule says that angular momentum is proportional to

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mass squared, at least approximately. Supported by well-known data, the accuracy of this relationship is continually being refined by ongoing observations of galaxies. Peter Brosche, of the University of Bonn, has suggested that a closer approximation involves a power of 1.7 instead of 2. Jagellonian University researchers Wlodzimierz Godlowski and Marek Szydlowski, along with two colleagues from Poland’s Pedagogical University, have proposed a slightly more complex rule that includes an additional term. Regardless of the actual equation, all these researchers agree there is a connection between rotation and mass that could well have been set in the inaugural stages of the universe.
Extrapolating this relationship to larger and larger systems leads to the startling prediction that the universe as a whole might be rotating, though data from WMAP and other background radiation surveys place strict limits on this possibility. Nevertheless, as Godlowski and Szydlowski have recently suggested, perhaps a small (hitherto undetected) spinning of the universe may manifest itself as a component of its acceleration. As the rotor example shows, circular motion implies a centripetal force. Maybe rotation offers the universe at least part of the extra push detected in the supernova observations. Although Godlowski and Szydlowski assert that they’ve found data to support this hypothesis, it remains a highly speculative idea— disputed by many researchers.
The notion that the cosmos is one vast merry-go-round dates back to a 1946 proposal by Gamow, who wondered if “all matter in the visible universe is in a state of general rotation around some centre located far beyond the reach of our telescopes.” Three years later the renowned mathematician Kurt Gödel followed up with the first rotating cosmological solution of Einstein’s equations. Gödel was quite proud of his result and discussed it with Einstein during their walks together in Princeton.
One curious aspect of Gödel’s spinning universe model is that by circumnavigating its rotational axis an astronaut could travel back

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in time, because time’s axis tilts during space-time rotation. With enough tilting, the law of cause and effect would break down, allowing an event from the future to “touch” an occurrence from the past. These causation-violating loops are called closed timelike curves (CTCs).
A good analogy to this idea is a standing circle of dominos, representing time’s directional arrows, placed carefully around the edge of a turntable. Suppose that by rotating the turntable slightly one of the dominos would become unstable, topple over, and hit the next, which would strike and tip over the next one, and so on. In short order, all the other dominos would fall, with one pushing over the next in sequence. Typically, the last domino to topple would lie on top of the first. Because the arrows of time would now connect up in a loop, a causal connection would establish itself between the future (the final domino to fall) and the past (the first domino). This would represent a potentially navigable CTC.
So if the universe is rotating and you would like to visit the past, simply hop on a spaceship and take a trip around the axis of spin— the ultimate rotor ride, suitable for those with no sense of vertigo or inconvenient present-day attachments. And unlike journeys into black holes, you wouldn’t have to dodge hideous singularities. But if you’ve already completed your voyage, you knew all this ages ago— or ages from now, as the case may be.

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