There is a moment in a dream when you realize that things don’t add up: You know that geese don’t speak English, and yet there you are chatting away with one about the price of gasoline. You’re sure that you never went to flight school, so why are you piloting this Cessna over Dubuque?
Then you wake up.
But sometimes, you don’t. Sometimes, as you confront two seemingly unassailable, clashing truths, you realize that you’re not dreaming at all—you’ve just encountered a paradox, and it’s a wake-up of an entirely different kind.
For centuries, paradoxes have helped physicists and philosophers challenge and deepen their understanding of how our world works. Paradoxes reveal assumptions and prejudices we never knew we had and open hidden hatches into new physics.
“There’s no getting around it: The universe is really strange, and paradoxes hit you with that,” says Anthony Aguirre, a physicist at the University of California, Santa Cruz.
For Aguirre, that’s a good thing: “That feeling of mystery is really what’s exciting in physics. You know there is something fun, interesting, and potentially important to be gained by going down that road.”
“Sometimes I consider that my knowledge is broken up into tectonic plates of understanding on the Earth of my total knowledge--a small part of the total universe of possible understanding,” says physicist Robert Nemiroff, who gives a special lecture on paradoxes to his students at Michigan Technological University. “Sometimes, I learn something that demands that two plates collide--both plates cannot be used to understand this new thought. This new thought can frequently be coined as a paradox. If resolved, these plates can lock into a larger plate of greater understanding, if I am lucky.”
“Paradoxes heighten what’s at stake conceptually,” says MIT science historian David Kaiser, who adds that physicists like Niels Bohr, John Wheeler, and Albert Einstein all deployed paradoxes strategically to underline mathematical contradictions that others deemed inconsequential. “Paradoxes are one way of grappling with what the equations really say.”
Here are the stories of three paradoxes from far-flung times and places in the history of physics and math. Though they have all been resolved, they remind us just how weird the universe really is. It is a dream from which we will never wake up. But who would want to?
How is it that anyone or anything ever moves from point A to point B? This simple question is the crux of a paradox first posed by the ancient Greek philosopher Zeno, and it has made generations of math students question the nature of reality every time they walk across a room.
Here’s one rendition of the paradox: Say you want to walk down the hallway from your bedroom to your bathroom. First, you have to cover half the distance between the rooms. Next, you’ll need to cross half the remaining distance. As you continue down the hall, you will always have half the previous distance left to cross. Though you will move ever closer to the threshold of the bathroom door, you will never actually reach it.
Obviously, we don’t spend our entire lives stranded in hallways. Why not? The answer is at the heart of calculus: It turns out that infinitely long sequences of numbers can actually have finite sums. This means that even though we must cross an infinite number of progressively smaller “chunks” of space on our way to the bathroom, the time it takes to do so is finite. That’s why we eventually get there.
Yet Zeno’s paradox also reflects one of the biggest questions in physics today: Is space—or spacetime—continuous, or is it broken up into discrete chunks? In Zeno’s world, space was continuous: It could be subdivided into smaller pieces on and on into infinity. Yet we know this isn’t how matter works. If you split a cookie in half over and over again (as many guilty sweet-tooths have no doubt tried at home) you will eventually be left with the indivisible components (electrons and quarks) of one atom. Eat them or don’t, but you cannot divide them in half.
This is also the moral of the story of quantum mechanics: The energy contained in all the particles that make up the universe is quantized. Why should spacetime be any different? In fact, some of the leading theories of quantum gravity predict that, on the tiniest scales, spacetime should break down into discrete chunks. Like a pointillist painting, spacetime may look perfectly smooth from afar, but up close it dissolves into pixels. There are currently experiments in the works to test this prediction.
Why is the sky dark at night? Before you say that it’s because the sun has set, consider that it took the greatest minds in science more than three hundred years to resolve this paradox. Everyone from Einstein to Edgar Allan Poe was swept up by this apparently simple puzzle.
The history of Olbers’ paradox goes back at least as far as Johannes Kepler, who posed it in 1610. If the universe is infinite, argued Kepler, containing an infinite number of stars distributed evenly across the sky, then every point on the night sky should be illuminated by starlight. The brightness of any individual star, as seen from Earth, fades in proportion to the square of its distance from Earth, but the number of stars at a given distance from Earth increases in proportion to the square of the distance from Earth, so it is a wash. The night sky, therefore, should be just as bright as the daylight sky. To Kepler, this meant that the universe must not be infinite after all.
In 1823, Heinrich Olbers drew up a different solution to the paradox that now bears his name. Olbers argued that as the light from each star makes its way toward Earth, it runs into interstellar dust and gas that absorb some its energy. Stars that are sufficiently far enough away from Earth would therefore be “cut off” from us.
The problem with Olbers’ logic is that dust and gas must spit back out the energy that they have absorbed, leaving us with the same problem we started with.
This is where Poe enters the picture. In his prose poem “Eureka,” published in 1848, he inserted time into the equation: What if light from distant stars just hasn’t had enough time to reach us yet? Poe wrote:
...yet so far removed from us are some of the "nebulae" that even light, speeding with this velocity, could not and does not reach us, from those mysterious regions, in less than 3 millions of years.
He may have had the details wrong, but the idea was right: Because stars have not been shining forever, and because it takes time for light to travel from here to there, there is a certain horizon from beyond which light has not yet reached us. Today, we know that the universe had a beginning (the Big Bang) and that the universe is expanding, causing light from distant stars to get stretched out (“redshifted”) beyond the visible part of the spectrum and into the infrared and radio bands, compounding the dark-sky effect.
“Olbers’ paradox is based on such a mind-numbingly simple observation,” says Aguirre. “What impresses me is the sheer amount of time that went by as people came up with one complicated and wrong solution after the next.”
The Twin Paradox
When Einstein proposed that time and space were elastic, it was weird enough. But the twin paradox challenges our understanding of Einstein’s rules even further. Imagine two twins, one a space traveler and the other an avowed homebody. The traveler sets out on a mission to a distant planet in a newfangled rocket that zips along at close to the speed of light. It’s a round-trip journey, so when she gets back, she is eager for a reunion with her twin. She wants to share the amazing stories of her voyage, of course, but she’s also looking forward to gloating about one favorable side effect of life in the cosmic fast lane: Because time passes more slowly for objects traveling close to the speed of light, the traveling twin anticipates that though she has hardly aged a bit, her stay-at-home sister will be sporting many years’ worth of new wrinkles. (Both twins are a bit vain.)
The stay-at-home sister is just as excited to see her twin. She knows her special relativity, too, and reasons that from her point of view, she was traveling close to light speed aboard spaceship Earth, while her sister sat complacently aboard her stationary vessel. Therefore, she will be the young-looking one, and will have fun counting up all her sister’s gray hairs.
And here is the paradox: Einstein tells us that no observer is “more correct” than any other, but the sisters can’t both be younger than each other. Still, Einstein wasn’t wrong. The solution to the paradox is that the sisters’ journeys were not actually identical. The traveling twin did not keep up a constant velocity throughout her entire trip. She accelerated to get up to speed, and then she changed directions—another kind of acceleration—before decelerating to get back into orbit around Earth. So the traveler, not her stay-at-home sister, gets the anti-aging benefits of time dilation.
Paradox solved? In one sense, yes. But in another sense, says Aguirre, the twin paradox reveals a deeper conundrum within the laws of relativity. The crux of the resolution, he says, is that even while velocity is relative, acceleration is absolute. “Where does this absolute non-accelerated reference frame come from? Einstein tried to do away with it in general relativity, but even a century later the question is largely open.”
Will today’s paradoxes be tomorrow’s truisms? Paradoxes arise when equations clash with our intuition about reality, says Kaiser. But intuition can change. “Newtonian mechanics did not look or sound ‘intuitive’ in the 17th century,” Kaiser points out. Today, we take it for granted that Newton’s conceptions of speed, gravity, and mass are in harmony with human intuition, but perhaps intuition itself has been reshaped by Newton’s view of the world. Will the apparent paradoxes of quantum theory and relativity one day feel just as natural as Newton’s laws? Is it possible to “grow out” of a paradox?
“Even once you know the golden thread that unravels the seeming contradiction, paradoxes are still appealing,” says Kaiser, who thinks of them as a kind of mental bodybuilding. “Paradoxes are like a much more satisfying version of Sudoku or a crossword puzzle.” The best part: “In the process we might really learn something about how the universe works.”
Editor's picks for further reading
arXiv: Paradox in Physics, the Consistency of Inconsistency
In this article, Dragoljub A. Cucic classifies paradoxes in physics and reviews their utility.
Edge: The Paradox
In this essay, Anthony Aguirre argues that a better understanding of paradoxes would improve our cognitive toolkit.
FQXi: Black Holes: Paradox Regained
Stephen Hawking conceded his bet on the black hole information paradox, but the debate continues.
Nothing is not as simple as it seems.
The concept of nothing has fascinated philosophers and scientists throughout history. The search for an ever-deeper understanding of nothing has driven scientific discovery since the age of ancient Greece, and today the pursuit of nothing defines the frontier of modern particle physics. But before we talk about nothing, let’s talk about something: air.
For millennia, philosophers thought that “empty” air was nothing. Aristotle and the ancient Greeks, though, recognized air as a “thing” in its own right. Wind, after all, is nothing but air, yet it can be felt powerfully. Indeed, the Greeks considered air to be one of the basic elements, along with earth, water, and fire. These elements, in turn, were believed made of some basic something which they called “ur-matter.” A familiar modern example, sucking on a drinking straw, seems to illustrate the impossibility of creating a vacuum: The straw doesn’t fill up with vacuum but instead “implodes,” apparently confirming the Greek belief that “Nature abhors a vacuum.”
About two millennia would pass before Galileo and others realized that the implosion is due to the external pressure of the air, and not a cosmic law against nothingness. This soon led to the invention of the barometer and a remarkable discovery: Air pressure decreases with altitude. The reason is that the atmosphere has a finite height and the nearer you get to the surface, the less air there is pressing down on you. This inspired the thought that above the atmosphere is nothing—or, at least, no air.
By the end of the 17th century, then, when people talked about “nothing,” they were no longer talking about air: They were talking about the void of space. Today, we know that though space is empty of air, it is filled with gravitational forces which guide the planets and order the galaxies. It is also full of electric and magnetic fields that give us sunlight and starlight in the form of electromagnetic waves.
This created great problems for 19th century scientists: Since the electromagnetic waves from the sun and stars were making it all the way to Earth, they must be traveling through something. After all, they knew that sound waves need a medium through which to travel. I speak and air molecules bump into one another until some hit your eardrums, making them vibrate, generating signals that your brain interprets as sound. The absence of air in space leaves the sun silent, yet we can see it.
To resolve this paradox, scientists argued that there must be some medium through which the electromagnetic waves traveled. “Waves in what?” was answered with: “The ether.” And so began one of the greatest wild goose chases in the history of science, as many of the leading lights in the field went in search of this weird ether that was capable of transmitting light at about 300,000 km every second while still allowing the planets to pass through as if there were nothing there at all. The search did not end until Einstein finally introduced his theory of relativity in 1905, which eliminated the need for the ether. (But that's a story for another day.) The tables had turned on nothing: Aristotle was wrong. Nothing could exist—or so we thought. And then came quantum mechanics.
In the quantum realm of tiny subatomic particles, the more closely you look at nothing, the more things you discover. What looks empty to our gross senses turns out to be effervescing with particles of matter and anti-matter. The apparent void is a medium filled with stuff, a froth of will-o’-the-wisp particles of matter and antimatter.
This new quantum mechanical view of nothing began to emerge in 1947, when Willis Lamb measured spectrum of hydrogen. The electron in a hydrogen atom cannot move wherever it pleases but instead is restricted to specific paths. This is analogous to climbing a ladder: You cannot end up at arbitrary heights above ground, only those where there are rungs to stand on. Quantum mechanics explains the spacing of the rungs on the atomic ladder and predicts the frequencies of radiation that are emitted or absorbed when an electron switches from one to another. According to the state of the art in 1947, which assumed the hydrogen atom to consist of just an electron, a proton, and an electric field, two of these rungs have identical energy. However, Lamb’s measurements showed that these two rungs differ in energy by about one part in a million. What could be causing this tiny but significant difference?
When physicists drew up their simple picture of the atom, they had forgotten something: Nothing. Lamb had become the first person to observe experimentally that the vacuum is not empty, but is instead seething with ephemeral electrons and their anti-matter analogues, positrons. These electrons and positrons disappear almost instantaneously, but in their brief mayfly moment of existence they alter the shape of the atom's electromagnetic field slightly. This momentary interaction with the electron inside the hydrogen atom kicks one of the rungs of the ladder just a bit higher than it would be otherwise.
This is all possible because, in quantum mechanics, energy is not conserved on very short timescales, or for very short distances. Stranger still, the more precisely you attempt to look at something—or at nothing—the more dramatic these energy fluctuations become. Combine that with Einstein’s E=mc2, which implies that energy can congeal in material form, and you have a recipe for particles that bubble in and out of existence even in the void. This effect allowed Lamb to literally measure something from nothing.
This suggests that the contents of the vacuum—the “stuff” of nothing—could be organized in different ways at different times in the history of the universe. Think of water molecules: They can roam freely in the liquid or lock tightly to one another in ice crystals. This analogy hints at an intriguing possibility: Could the contents of the quantum vacuum be in a different configuration in today’s cool universe than they were in the first moments after the hot Big Bang?
At creation, the thinking goes, particles had no mass and moved through the vacuum at the speed of light. Around a trillionth of a second after the Big Bang, the universe was cool enough that a mass-giving field called the “Higgs field” condensed in the vacuum, as water condenses from steam.
The Higgs field is believed to disturb the motion of fundamental particles like electrons as they move through it, producing the effect that we call mass. If this is correct, there should be particle manifestations of the Higgs field, known as Higgs bosons, just waiting to be discovered. The Large Hadron Collider (LHC) at CERN is hot on the trail of these particles, but decisive evidence of the Higgs boson—which is very massive and can only be produced in an enormous blast of energy—is still elusive. Scientists working on the LHC expect that they may see the first glimpse of the Higgs by the end of 2012. Whether this is the real deal or whether we are being fooled by some cruel, random throw of Nature’s dice, time will tell.
Aristotle was right: There is no thing that is nothing. Is the Higgs field part of the something? Within a few months we may know the answer.
Editor's picks for further reading
FQXi: Much Ado About Nothing
Ted Jacobson investigates the nature of the cosmic vacuum.
The New York Times: There’s More to Nothing Than We Knew
In this article, Dennis Overbye reviews why physicists believe that something—like our universe—can come from nothing.
World Science Festival: Nothing: The Subtle Science of Emptiness
Journalist John Hockenberry leads Nobel laureate Frank Wilczek, cosmologist John Barrow, and physicists Paul Davies and George Ellis in a discussion of the physics and philosophy of nothing.
Are beauty and truth two sides of the same coin? It is charming to believe so. As Nobel Prize laureate Paul Dirac, who helped lay the mathematical groundwork for quantum mechanics, put it:
It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress. If there is not complete agreement between the results of one's work and experiment, one should not allow oneself to be too discouraged, because the discrepancy may well be due to minor features that are not properly taken into account and that will get cleared up with further developments of the theory.
The poet John Keats expressed it more concisely:
Beauty is truth, truth beauty – that is all
Ye know on earth, and all ye need to know.
But, in science, does a beautiful hypothesis necessarily lead to deep truth about nature?
Several famous success stories suggest that it does, at least in physics:
James Clerk Maxwell arrived at his celebrated system of equations for electromagnetism by codifying what was thought to be known experimentally about electricity and magnetism, noting a mathematical inconsistency, and fixing it. In doing so, he moved from truth to beauty. The Maxwell equations of 1861, which survive intact as a foundation of today's physics, are renowned for their beauty. The normally sober Heinrich Hertz, whose experimental work to test Maxwell's theory gave birth to radio and kickstarted modern telecommunications, was moved to rhapsodize:
One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.
Albert Einstein, on the contrary, arrived at his equations for gravity—the general theory of relativity—with minimal guidance from experiment. Instead he looked for beautiful equations. After years of struggle, in 1915 he found them. At first, and for decades afterwards, few testable predictions distinguished Einstein's new theory of gravity from Newton's venerable one. Now there are many such tests, and it is amply clear that Einstein moved from beauty to truth.
Yet even in physics, the record is more mixed than is commonly known. Despite Keats and Dirac, beauty's seductions don't always give birth to truth. There have been fascinating theories that are both gorgeous and wrong: Beautiful Losers.
Like surgeons, physicists bury their failures. But the most beautiful of the Beautiful Losers deserve a better fate than oblivion, and here they'll receive it. I've written brief accounts of three Beautiful Losers: Plato's Geometry of Elements, Kepler's Harmonic Spheres, and Kelvin's Vortex Atoms.
Plato’s Geometry of Elements: Plato believed that he could describe the Universe using five simple shapes. These shapes, called the Platonic solids, did not originate with Plato. In fact, they go back thousands of years before Plato; you can find stone models (perhaps dice?) of each of the Platonic solids in the Ashmolean Museum at Oxford dating to around 2000 BC. But Plato made these solids central to a vision of the physical world that links ideal to real, and microcosm to macrocosm in an original, and truly remarkable, style. Read more
Kepler’s Harmonic Spheres: Like Plato, the German astronomer Johannes Kepler believed that five Platonic solids provided an essential blueprint for our universe. Six planets were known to Kepler, and he believed that they were carried around on nested globes that he called the celestial spheres. Kepler reasoned that five solids could correspond to six planets, if the solids—or more precisely, their bounding surfaces—marked the spaces between planetary spheres. He described this elegant construction in his Mysterium Cosmographium in 1596. Read more
Kelvin's Vortex Atoms: A tornado is just air in motion, but its ominous funnel gives an impression of autonomous existence. A tornado seems to be an object; its pattern of flux possesses an impressive degree of permanence. The Great Red Spot of Jupiter is a tornado writ large, and it has retained its size and shape for at least three hundred years. The powerful notion of vortices in fluids abstracts the mathematical essence of such objects, and led William Thomson, the 19th century physicist whose work earned him the title Lord Kelvin, to ask: Could atoms themselves be vortices in a ether that pervades space? Read more
It's wonderful, and comforting, that each of my Beautiful Losers, though wrong, was in its own way fruitful. Today more than ever physicists working at the frontiers of knowledge are inspired by beauty. In the alien realms of the very large, the very small, and the extremely complex, experiments can be difficult to perform and everyday experience offers little guidance. Beauty is almost all we've got!
This essay is part of the series Beautiful Losers.
Plato believed that he could describe the Universe using five simple shapes. These shapes, called the Platonic solids, did not originate with Plato. In fact, they go back thousands of years before Plato; you can find stone models (perhaps dice?) of each of the Platonic solids in the Ashmolean Museum at Oxford dating to around 2000 BC, as pictured below. But Plato made these solids central to a vision of the physical world that links ideal to real, and microcosm to macrocosm in an original, and truly remarkable, style.
AN1927.2727-31 Neolithic Carved Sandstone Balls, Copyright Ashmolean Museum, University of Oxford.
Let me explain, first, what the Platonic solids are. To begin, consider something simpler: regular polygons. Regular polygons, by definition, are two-dimensional shapes bounded by sides of equal length, each making the same angles with its neighbors. Equilateral triangles, squares, regular pentagons, and so on are all regular polygons. Platonic solids are the three-dimensional analog of regular polygons, and prove to be far more interesting. Platonic solids are bounded by regular polygons, all of the same size and shape. One can prove mathematically that there are exactly five Platonic solids. Here they are:
The Platonic solids and their proposed identification with fundamental world-elements
The tetrahedron has four triangular faces, the cube six square faces, the octahedron eight triangular faces, the dodecahedron twelve pentagonal faces, and the icosahedron twenty triangular faces. Plato proposed that four of these solids built the Four Elements: sharp-pointed tetrahedra give the sting of Fire, smooth-sliding octahedra give easily-parted Air, droplety icosahedra give Water, and lumpish, packable cubes give Earth. The dodecahedron, at last, is the shape of the Universe as a whole. Later Aristotle emended Plato's system, suggesting that dodecahedra provide a fifth essence—the space-filling Ether.
Plato's ideas lent dignity and grandeur to the study of geometry, and greatly stimulated its development. The thirteen and final book of Euclid's Elements, the grand synthesis of Greek geometry that is the founding text of axiomatic mathematics, culminates with the construction of the five Platonic solids, and a proof that they exhaust the possibilities. Scholars speculate that Euclid planned the Elements with that climax in mind from the start.
From a modern scientific perspective, of course, Plato's mapping from mathematical ideals to physical reality looks hopelessly wrong. The four (or five) ancient “elements” are not simple substances, nor are they usable building blocks for constructing the material world. Today's rich and successful analysis of matter involves entirely different concepts. And yet...
In its general approach, and its ambition, Plato's utterly mistaken theory anticipated the spirit of modern theoretical physics. His program of describing the material world by analyzing (“reducing”) it to a few atomic substances, each with simple properties, existing in great numbers of identical copies, coincides with modern understanding.
Deeper still penetrates his insight that symmetry defines structure. Plato sensed enormous potential in the fact that asking for perfect symmetry leads one to discover a small number of possible structures. Based on that foundation, and a few clues from experience, the outlandish synthesis that his philosophy suggested should be possible, to realize the World as Ideas, might be achievable. And clues were there to be found: Near-coincidence between the number of perfect solids (five) and the number of suspected elements (four); suggestions of how observed qualities might reflect underlying shapes (e.g., the sting of fire from the sharp points of tetrahedra). One must also admire the boldness of genius in seeing an apparent defect in the theory—five solids for four elements—as an opportunity for crowning creation, either with the Universe as a whole (Plato) or with space itself (Aristotle).
Modern physicists, when seeking equations to describe the unfamiliar laws of the microcosm, must make guesses based on fragmentary information. Optimistically—and lacking constructive alternatives—they have turned, as Plato did, to symmetry as their guide. Symmetry of equations is perhaps a less familiar idea than symmetry of shapes, but there is nothing obscure or mystical about it. We say an equation, like a shape, displays symmetry when it allows changes that make no change. So for instance the equation
X = Y
has a nice symmetry to it, because exchanging X for Y changes it into this:
Y = X
and this transformed equation expresses exactly the same content as the original. On the other hand X = Y + 2, say, turns into Y = X + 2, which expresses something else entirely. As this baby example demonstrates, symmetric equations can be rare and special, even when the symmetry involves quite simple transformations.
The equations of interest for physics are considerably richer, of course, and the "changes that make no change" we hope they allow are much more extensive and elaborate. But the central idea inspiration remains, as it was for Plato, the hope that symmetry defines a few interesting structures, and that Nature chooses one (or all!) of those most beautiful possibilities.
Plato's Beautiful Loser was, in hindsight, a product of premature, and immature, ambition. He tried to leap directly from beautiful mathematics, some imaginative numerology, and primitive, cherry-picked observations to a Theory of Everything. In this his ambition was premature. Also, Plato failed to draw out specific consequences from his ideas, or to test them critically. He sketched an inspiring world-model, but was content to "declare victory" without engaging any serious battles. The mature and challenging form of scientific ambition, which aspires to understand specific features of the world in detail and with precision, emerged only centuries later.
This essay is part of the series Beautiful Losers.
Like Plato, the German astronomer Johannes Kepler believed that the five Platonic solids provided an essential blueprint for our universe. Six planets were known to Kepler, and he believed that they were carried around on nested globes that he called the celestial spheres. Kepler reasoned that five solids could correspond to six planets, if the solids—or more precisely, their bounding surfaces—marked the spaces between planetary spheres. He described this elegant construction in his Mysterium Cosmographium in 1596.
Kepler proposed that Mercury's sphere supports a circumscribed octahedron, which is inscribed within Venus's sphere. Then we have icosahedron, dodecahedron, tetrahedron, cube interpolating respectively Venus-Earth, Earth-Mars, Mars-Jupiter and at last Jupiter-Saturn. This revelation of cosmic order was, for Kepler, rapturous:
I wanted to become a theologian; for a long time I was unhappy. Now, behold, God is praised by my work even in astronomy.
It immediately suggests a plan of construction that human artists can mimic—as Kepler did himself—in worthy, gorgeous models:
Photo by Sam_Wise. Source.
A model of Kepler's solar system, on display at the Technical Museum, Vienna.
Though equally (that is, completely) wrong, Kepler's conception reaches a higher level, scientifically, than Plato's speculations, for it makes concrete numerical predictions about the relative sizes of planetary orbits, which can be compared to their observed values. The agreement, while not precise, was close enough to convince Kepler he might be on the right track. Encouraged, he set out, courageously, to prove it.
Thus Kepler's discovery set him on his storied career in astronomy. As his work developed, however, problems with his original model emerged. Late in the 16th century, the astronomer Tycho Brahe was making exquisitely accurate observations of the motions of the stars and planets. As Kepler strove to do justice to Tycho's work, he discovered that the orbit of Mars is not circular, but follows an ellipse. This and other discoveries fatally undermined Kepler's beautiful system of celestial spheres. (Eventually even the numerology collapsed, with the discovery of Uranus in 1781, though Kepler was spared that ignominy.) Through the arduous, devoted labor his vision inspired he found other regularities among the orbits of the planets—his famous three laws of planetary motion—whose accuracy could not be doubted. In the end he'd arrived at a different universe than the one he first envisioned, and hoped for. He reported back:
I write the book. Whether it is to be read by the people of the present or of the future makes no difference: let it await its reader for a hundred years, if God himself has stood ready for six thousand years for one to study him.
Kepler's hoped-for reader emerged not quite a hundred years later: Isaac Newton. To me, the first illustration of Newton's Principia (1687) worthily transmits the most consequential thought-experiment ever. With a few cartoonish strokes, it both presages a new, universal theory of gravity and embodies a new concept of scientific beauty: dynamic beauty.
Newton's dynamical thought-experiment for the universality of gravitation. Via Wikimedia Commons
Imagine standing atop a tall mountain on a spherical earth, throwing stones horizontally, harder and harder. To keep things clear, remove from thought the damping influence of the atmosphere. At first it is clear what will happen, based on everyday experience: The stones will travel further and further before landing. Eventually, when the initial velocity becomes large enough, the stones will pass over the horizon; then its landing point will circuit the planet. Visualizing the developing situation, as in Newton's diagram, it is easy to imagine the progress of trajectories leading to a circle (duck!). In this way we begin to see how the same force that pulls bodies to Earth might also support orbital motion. We see that orbiting is a process of constantly falling, but toward a (relatively) moving target.
I like to think that the images in this diagram reveal the deep inspiration, pre-mathematical and even pre-verbal, behind young Newton's program of research (parallel to how young Kepler's harmonic spheres inspired his). It contains the germ of universal gravitation: With (imaginary) taller mountains as launching pads, we fill the sky with possible orbits. Might the Moon occupy one of them? And if Earth's Moon, why not Jupiter's moons, or the Sun's planets? And this question too begs for an answer: Throw harder still—what are the shapes of the resulting trajectories? The answer, which Newton's mathematics allowed him to answer, is that they make more and more eccentric ellipses—the very shape that Kepler had used to fit planetary orbits!
In Newton's dynamical approach, the beauty of planetary motion is not embodied in any one orbit, or in the particular set of orbits realized in the Solar System. Rather it is in the totality of possible orbits, which also contains trajectories of falling bodies. Putting it another way: The deepest beauty lies in the equations themselves, not in their particular solutions. Classical physics, initiated by Newton's brilliantly successful celestial mechanics, suggests that it is misguided to expect, as Kepler and Plato did, to find ideal symmetry embodied in any individual physical object, be it the Solar System or an elemental atom. Astronomers in recent years have identified dozens of extrasolar planetary systems, and found that they come in a wide variety of shapes and sizes. And yet...
Physical requirements can privilege, among the infinity of possible solutions to beautiful dynamical equations, special ones—often especially beautiful ones. Consider crystals: They are of course quite real and tangible natural objects; they can be grown in controlled, reproducible conditions; and their form is often highly symmetric. Kepler himself wrote a monograph featuring the six-fold symmetry of snow crystals.
We discover the same thing, spectacularly, in the quantum theory of atoms. An electron interacting with a proton obeys the same species of force law as a planet orbiting the Sun. Schrödinger's equation, no less than Newton's, allows an enormous infinity of complicated solutions. (In fact, much more so!) But if we focus on the solutions with the lowest energies—the solutions that coldish hydrogen atoms will settle into, after radiating—we pick out a special few. And those special solutions exhibit rich and intricate symmetry. They fulfill, as they transcend, the visions of Plato, Kepler, and Newton.
The quantum mechanical wave function for a typical stationary state of the hydrogen atom. Image from http://vqm.uni-graz.at, used with permission of Bernd Thaller.
This essay is part of the series Beautiful Losers.
A tornado is just air in motion, but its ominous funnel gives an impression of autonomous existence. A tornado seems to be an object; its pattern of flux possesses an impressive degree of permanence. The Great Red Spot of Jupiter is a tornado writ large, and it has retained its size and shape for at least three hundred years. The powerful notion of vortices in fluids abstracts the mathematical essence of such objects, and led William Thomson, the 19th century physicist whose work earned him the title Lord Kelvin, to ask: Could atoms themselves be vortices in an ether that pervades space?
Kelvin's idea was inspired by the work of Hermann Helmholtz, who first realized that the core of a vortex—analogous to the eye of a hurricane—is a line-like filament that can become tangled up with other filaments in a knotted loop that cannot come undone. Helmoltz also demonstrated that vortices exert forces on one another, and those forces take a form reminiscent of the magnetic forces between wires carrying electric currents.
To Thomson, these results seemed wonderfully suggestive. At the time, evidence from chemistry and the theory of gases had persuaded most physicists that matter was indeed composed of atoms. But there was no physical model indicating how a few types of atoms, each existing in very large numbers of identical copies—as required by chemistry—could possibly arise.
In seemingly unrelated work, physicists were discovering that space-filling entities are an essential tool in Nature's workshop. Today we accept those entities—known as electric and magnetic fields—on their own terms, as fundamental; but Thomson and his contemporaries believed them to be manifestations of an underlying fluid: an updated version of Aristotle's Aether.
Thomson's bold ambition, and instinct for unity, led him to a propose a synthesis: The theory of vortex atoms. The Ethereal fluid, being so fundamental, should be capable of supporting stable vortices, he reasoned. Those vortices, according to Helmholtz' theorems, would fall into distinct species corresponding to different types of knots. Multiple knots might aggregate into a variety of quasi-stable "molecules." All this remarkably fits the heart's desire, in a theory of atoms: Naturally stable building-blocks, whose possibilities for combination seem sufficiently rich to do justice to chemistry.
Thomson himself, a restless intellect, moved on to gush other ideas, but his friend and colleague Peter Guthrie Tait, enthralled by the vortex atom theory, set to work. Thus inspired, he did pioneering work on the theory of knots, producing a systematic classification of knots with up to 10 crossings.
A table of knots. The 'Unknot' was thought to represent hydrogen; to its right, the knot thought to represent carbon. By Jkasd (Own work, Public domain), via Wikimedia Commons
Alas this beautiful and mathematically fruitful synthesis is, as a physical theory of atoms, a Beautiful Loser. Its failure was not so much due to internal contradictions—it was too vague and flexible for that!—but by a certain sterility. Above all, it was put out of business by more successful competitors. Eventually the mechanical Ether was discredited by Einstein's relativity, and the triumphant Maxwell equations for electric and magnetic fields do not support vortices. The modern, successful quantum theory of atoms is based on entirely different ideas. And yet...
It's easy to understand the appeal of vortex atoms, not only as fascinating mathematics, but as potential elements for world-building. When we turn from understanding the natural world to designing micro-worlds on our own, we might come to treasure their virtues. Vortices can have an impressive degree of stability; they can be knotted into topologically distinct forms, which are also quasi-stable; and their interactions are complex and intricate, yet reproducible.
Those attractive features can be embodied in artificial “atoms” specifically designed to be building blocks for quantum engineering. For quantum theory, though it made the vortex theory of natural atoms obsolete, provides us with a variety of far more reliable, and far more perfect, aethers than the old Aetherial fantasies. Classical fluids, whether they are real liquids or speculative substrates, are inherently imperfect. Any motion in them will stir up little waves that carry away energy, and eventually dissipate the flow. Quantum fluids, such as superfluid helium and a variety of superconductors, on the contrary, support flows that, in theory, will persist unchanged forever. And in practice, too—that’s why we call them “super”! The deep point is that in quantum mechanics energy comes packaged in discrete lumps (quanta). If you operate at low temperatures, where there’s very little energy available, it can become impossible to stir up those little waves that bedevil classical fluids at all. In quantum fluids, vortices really are forever.
There is lots of room for creativity in designing and constructing artificial aethers. Many materials become perfect (quantum) fluids at low temperature.
By choosing the right media, we can tailor our fluids to have useful properties. Physicists and engineers have become quite adept at designing useful fluids, such as the liquid crystals that enable computer monitors and LCD displays. In those examples the fluids have internal structure, which can be manipulated electrically to change their appearance. So far most of the effort has gone into classical fluids, but physicists are beginning to awaken to some promising new possibilities offered by quantum fluids. Though the details can be quite different—as I said, there’s lots of room for creativity here—the basic inspiration, to make fluids that we can manipulate externally to make them do something useful, is the same.
Designer quantum fluids can offer us a variety of vortex atoms, and the opportunity to design new chemistries that accomplish something we want done. Perhaps the most intriguing possibility is to embody, in real materials, the so-far theoretical concept of anyons. Anyons are particles that interact in a special, peculiarly quantum-mechanical way. Anyons don’t exert any forces upon one another, but when you wind one anyon around another, you make interesting, predictable changes in the wave function that describes your system. Quantum computers are, in principle, nothing but machines that process wave functions. (Since wave functions can simulate a tape, or more generally a collection of tapes, that encode data, operations on wave functions can be massively parallel operations on data.) On paper, at least, theorists have proposed ways whereby one might orchestrate the motion of anyons to construct a general-purpose quantum computer. The future will tell whether this beautiful idea blossoms into reality, or proves another seductive Beautiful Loser.
In topological quantum computing, information is processed by braiding anyons. Image courtesy of the University of Glasgow.
When I was seven, I understood relativity perfectly. I got it. At least I thought I did. And it was glorious.
It all started with The Flight of the Navigator, that movie about the kid who accidentally stumbles upon a space ship and has a grand old afternoon traversing the galaxy in the company of its cargo of friendly creatures. The problem? When he returns to Earth he finds that eight years have elapsed. He has been declared missing; when he rings the bell at his old house, a stranger answers the door.
The Flight of the Navigator is a story about time dilation. The ship was traveling close to the speed of light, so time passed significantly more slowly for its passengers than it did for those back on Earth. That’s special relativity for you. It seemed easy, elegant, amazing.
For the next few years, that was my relationship with relativity. Then came the equations, and suddenly relatively didn’t seem so easy any more. Elegant, maybe. But that first spark of intuition got lost in symbols, equations, calculations and miscalculations. So I wondered: Why can’t we have it both ways? Is it possible to be both wide-eyed and rigorous, methodical and amazed?
That's a balance even physicists struggle with. As MIT theorist Max Tegmark told NOVA, "When I was in high school, physics was my most boring subject....It was only later that I realized that physics isn’t at all about just solving some equations. It’s about figuring out what reality is all about."
That’s what we're doing on this blog: taking on the biggest questions of all, the ones that get at the very nature of reality. What happens to information inside a black hole? What are we really talking about when we talk about teleportation? What if the laws of physics can't be unified? Every week, we’ll have a new post from a scientist or science writer--look for James Stein, Sean Carroll, and Frank Wilczek in the coming weeks and months--with plenty of room for you to join the discussion with your own questions and ideas.
We hope you'll let us know what subjects in fundamental physics you want to hear about—the stranger, the better. Can neutrinos really go faster than the speed of light? What is this spacetime thing anyway? Is our universe just one of countless others in a multiverse? Join us as we dive deep and discover how physics is changing what we thought we knew about the nature of reality.