The Hunt for Vulcan

by Thomas Levenson

  Special relativity began to form as Einstein understood that the answer has to be “no.” The bystander sees the strikes as simultaneous, but the train’s rider does not. How can this be, given that both of them are describing the same events? Einstein’s answer is, in effect: think about how the constancy of the speed of light affects your ability to decide when something happens, when the event and the tick of your clock occur at the same time. The images of the lightning bolts from both ends of the train have to cross some distance from where they strike to where the two watchers happen to be. To reach the observer standing on the embankment, the signal—the light from each bolt—has to cover the same distance: half the length of the train. Each signal will take the same amount of time (the speed of light is a constant, fixed for both bolts) to cover an identical amount of ground—and the observer can clearly describe the event: he sees the two strikes as simultaneous.

  But for the watcher on the train, the situation is different. She’s still moving as the bolts hit. In the time it takes for the light to travel from the strikes to where she stands, she and the train will have traveled forward just a little. The light from the strike at the front will have a slightly shorter stretch to cross before it reaches the eye of the traveling observer than the light from the rear, chasing the advancing motion of the train. This observer will see first the flash from the forward bolt, and then after a moment, the flash from the trailing one. In other words, two strikes occur at different times for this observer, one slightly before her counterpart standing next to the train sees his “simultaneous” flashes, and one slightly after. These two people in two different states of motion cannot agree on the timing of the identical events.

  This version of Einstein’s simultanaeity thought experiment depicts what happens from the point of view of a trainspotter on the embankment. Lightning bolts strike both ends of the train as a passenger riding at the train’s midpoint passes the observer (top drawing). Einstein wanted to know if both the observer and the passenger would agree on the timing of the strikes. In the second drawing, the train moves forward, shortening the distance the light from the bolt at the front of the train has to travel to reach the passenger—which results in that light reaching the passenger before the trainspotter. In the third drawing, light from the forward bolt arrives at the trainspotter just as light from the back bolt reaches him too. To him, the lightning bolts hit simultaneously. Finally, in the bottom drawing, as the train moves yet farther, light from the rear bolt catches up to the passenger, who thus concludes that the two strikes hit at different times, disagreeing—in good faith—with the observer at trackside.

  What’s more, Einstein realized, this disagreement holds true for distance too. Following the same reasoning as he did for time, Einstein emphasized the importance of the measuring apparatus, in this case an ordinary ruler. Imagine the passenger measuring her legroom with a ruler as she passes the train-spotter beside the track. As the rider notes her result, her measuring stick is moving past the observer on the other side of the window. The observer on the embankment measures the time it takes for the front and the rear of the measuring stick to pass him by. But as we already know through the lightning bolt thought experiment, those times will be different from the measured flow of time on the train—and from that follows that each observer, with their differing measurements of time have to disagree about the length of the measuring rod. Space and time are relative.

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  One step more remained. In his version of relativity, Galileo had worked out a set of mathematical rules—now called the Galilean transformations—to enable two people in motion relative to each other to see that their different perceptions were really equivalent accounts of the same event. Einstein’s relativity established the physics within an updated set—called the Lorentz transformations—that use that single constant, the velocity of light, to reconcile any two observers’ differing observations.

  Newton’s God kept absolute time and absolute space throughout the universe, a divine clock striking the same hours at every point throughout all creation. That article of faith helped Newton to his genuinely revolutionary insight that the heavens above and the earth below are governed by a single set of laws, just one system of the world. As the flight of comets and the discoveries of planets seemed to prove, cosmic history seemed to possess a universal constancy, the same everywhere for all people at all times. Two centuries on, Einstein’s homely images of trains and timepieces and rulers laid waste to all that. His clocks tallied their seconds beautifully, but to a beat that varied in the eye of the beholder.

  The second half of Einstein’s relativity paper extended its concepts beyond the realm of the motion of material objects. Relativity, he showed, held for objects in space; for the constituents of atoms; for electromagnetic fields, seemingly, for everything. Three months later, Einstein wrote again to show how deeply the concept penetrated into the fabric of what had seemed to be settled physics. In just two pages, he investigated what happens when a chunk of matter emits energy—any form of radiation for example. From that starting point, he performed a brief calculation that revealed that within relativity energy and mass can be shown to be equivalent. He wrote his conclusion in a very different form from the way we now know it, as the most famous equation in science: E=mc2.

  On its own the demonstration that matter and energy were really two sides of a single coin would have been a fine piece of work. Though the calculation was easy—trivial, almost—it still seems almost unbelievable that an argument about kinematics, the properties of motion, could morph into such a deep claim. Common sense says that energy is something that happens to matter, the strike of a bat on a ball or the explosion that throws a shell out the cannon’s mouth. But common sense is wrong. Einstein’s equation forces us to conceive of matter and energy as intertwined, capable of transforming, one into the other.

  If that weren’t strange enough, this little paper went deeper still, embodying an insight that Einstein grasped before anyone else: relativity wasn’t so much a specific law of nature as a condition to which all the more ordinary patterns in nature had to conform. E=mc2 translated the concept of inertia into relativistic terms. Through Einstein’s further work and that of others, the laws of motion fell into line. The electromagnetic equations of Maxwell had to be reinterpreted to account for the relativity of space and time…and so on. In the metaphor of the day, relativity was an imperialist, colonizing ever greater tracts of physics. The logic of empire is to grow, and the special theory’s next target was, if not obvious, then inevitable.

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  Einstein would remain at the patent office into 1909—among other virtues, it paid better than entry-level academic jobs—but long before he finally moved into the professoriat, his miracle year made it clear that he was a rising star, just coming into the full play of his powers. Thus, in the autumn of 1907 it made perfect sense that he would be asked to write a retrospective, charting the progress of relativity over the previous two years. That kind of invitation is an honor—but this one came with a barb. The request reached him late, leaving him only two months before the December 1 deadline. At first, that didn’t seem to be a problem. The larger half of the review went quickly. In four sections Einstein described the application of the relativistic worldview to measurements of time, the study of motion, the behavior of electromagnetic fields, and the implications of Einstein’s discovery of the equivalence of energy and matter. That was, in fact, more or less all that he’d been asked to do. His editor had sought a survey of current developments in relativity theory, and he’d written one.

  He hung up on one last question, though. The “special” in the special theory of relativity refers to the fact that it is a limited concept. It described the behavior of space and time perfectly for almost all physical situations, with one major caveat: as Einstein understood it at the time, the special theory applied only to motion at a constant rate. That left those circumstances in which speeds change, accelerating o
r slowing down.*3 For Einstein, this left an intolerable gap. Acceleration is ubiquitous throughout the cosmos. Most important: anything subject to gravity accelerates.

  With all the weight of the Einstein legend between us and that very young researcher—he was still just twenty-eight—stealing time for science in the ebb and flow of patent applications, it’s hard to recapture now just how much raw intellectual confidence was required to take this step. Thinking beyond special relativity would inevitably force him to confront the most famous idea—universal gravitation—produced by the most famous physicist in history: Newton. But if his theory really did form a part of the logic of nature, then no idea, not even the most iconic, should be able to escape it.

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  November 1907. Einstein reported each working day to the Patent Office. He wrote; he thought; he stared into space. Through his window he looked over at the rooftops of Bern. One day—we don’t know exactly when—he saw that anonymous roofer; he imagined the accident; and the missing piece, his happy thought, burst into his mind. The realization that a falling man won’t feel his own weight provided the crucial hint that led Einstein to think about gravity along similar lines to those he used to analyze the relativity of time and space. Einstein formalized this insight as the “equivalence principle”—an axiom that would become as important to his thinking as the relativity principle had proved to be in 1905. In its simplest form, equivalence simply holds that a person in free fall—like the imaginary roofer—cannot distinguish between two possible descriptions of his circumstances. He can’t say whether he is falling under the influence of gravity, or just floating in a gravity-free region of space.

  In other words, never mind that from where Einstein sits, the roofer is accelerating, speeding up as he plummets. The roofer himself feels no change (until he hits the ground)—only weightlessness, no push or tug of any sort—which is the signature of uniform, inertial motion, the sort the special theory of relativity describes. The two states—free-fall and un-accelerated motion—thus had to be seen as equivalent, both accurate descriptions of the same phenomenon. The inverse is also true: it’s impossible for someone (in a closed room) to decide whether the tug she feels as she stands on the floor is that of the earth’s gravitational field, or something—say a rocket motor—accelerating beneath her, pushing upward on the soles of her shoes.

  Epiphany! This new principle pointed Einstein directly to the essential connection special relativity on its own couldn’t make: the link between inertia, which is the measure of mass, and weight, which is mass multiplied by whatever force affects an object. Someone visiting the moon retains the same mass she has anywhere else. But because the moon exerts about 16 percent of the earth’s gravitational tug, her weight will be just one sixth what it would measure back home. More generally, weight can be understood as the perception of a change in the motion of any object, no matter whether that shift comes from acceleration or gravity. Free fall produces the same experience as the weightlessness of empty space, far from any source of gravity; acceleration produces the identical perception—that of weight—as does standing still in the grip of the earth’s gravitational field. That tumbling roofer and the equivalence principle he inspired told Einstein the bare minimum of what any relativistic theory of gravity would have to include: a mathematical account that could express the still undiscovered physics that established the connection between the inertial path of anyone or thing and the pull of gravity that exerts its hold on all of us as we travel across the universe.

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  November came to its end. Einstein finished his paper, including its final section on the equivalence principle and what it implied for a relativistic theory of gravity. It was barely a sketch, a hint toward something richer to come. All he really knew at this point was that it was possible to think about gravity in relativistic terms, in the language of inertia and acceleration.

  There was one more mundane matter that caught his attention shortly after he sent off his review. Nowhere in that paper did he mention any real-world challenge to Newton’s gravitation, no anomalies or suspicious phenomena. Instead, then and for the next eight years, the central issue remained one of theoretical consistency, of reconciling the formal disagreement between special relativity and Newton’s ideas. Privately, though, Einstein had a perfectly fine grasp of the tactics as well as the strategy of intellectual combat. He knew that he could win if and only if he could clearly demonstrate that his theory modeled reality better than Newton’s. On Christmas Eve he wrote to Conrad Habicht, an old friend, not a physicist, that he was working on a new, relativistic law of gravitation. His aim? “To explain the still unexplained secular changes in the perihelion of Mercury.”

  Vulcan had long since drifted to the far penumbra of possibility. But now, Albert Einstein, constructing a cosmos on a foundation of relativity, was taking dead aim at the undiscovered planet. From the beginning of his investigation of gravity, Einstein grasped the crucial either/or of Vulcan’s existence or absence. He wouldn’t mention Mercury again for several years, even in his private correspondence. But he didn’t forget it either.

  * * *

  *1 It would take another book—and at least thousands have already been written on this theme—to trace the tendrils of quantum mechanics in contemporary life. From the electrical phenomena that allow my computer to turn the motion of my fingers into letters on a screen and thence a page, to the specific properties of the materials within the seat upon which I sit, to the high poetry of the theories of the cosmos with which this book is partly concerned…quantum ideas are implicated everywhere. Most of the way we move around our world is classical, Newtonian, accessible. The secret life that underpins that overt one is inconceivable except as expressed in the language of the quantum revolution. Here the sermon endeth.

  *2 In more detail: Maxwell’s equations yield a constant velocity for an electromagnetic wave (of which visible light is simply a particular set of wavelengths). By Galilean invariance, the assertion that the laws of physics remain the same for every observer in uniform motion, that constant velocity remains a constant in every frame of reference. But as Newton developed his mechanics based in part on Galileo’s work, the speed of light was no more a constant than any other velocity, and the claim of Galilean invariance did not apply; hence the conflict.

  *3 Special relativity does apply to accelerated systems; the famous twin “paradox,” in which a twin accelerating away from and then returning to the earth ages more slowly than her stay-at-home brother, is an example of a special relativistic analysis of nonuniform motion. But in his initial framing of special relativity, Einstein considered uniform motion, and as he first began to think about generalizing the theory, he continued to think about this distinction between different states of motion.

  “HELP ME, OR ELSE I’LL GO CRAZY”

  There is an idea—utterly strange at the time—shot through the fabric of special relativity. In the century since it was first revealed, it has woven itself through the warp of popular culture as much as it flows through formal cosmology. When he first encountered it, though, Albert Einstein was unimpressed. “Now that the mathematicians have seized on relativity theory,” he declared, “I know longer understand it myself.” The offending mathematician? Einstein’s former teacher, Hermann Minkowski. The offending idea? In Minkowski’s own words:

  “Gentleman, the concepts of space and time which I wish to present to you have sprung from an experimental physical soil and therein lies their strength. They are radical. Henceforth space by itself and time by itself are doomed to fall away into mere shadows, and only a kind of union between the two will preserve an independent reality.”

  We now call that union “space-time.” The old notion that space occupies three dimensions—our familiar height, width, and depth—and time ticks on regardless, Minkowski argued, could no longer hold, not if you take seriously the discovery that one’s state of motion affects measurements of both. His response: to propose a world
that exists in four dimensions, three of space, one of time, all intertwined with each other.

  Most important, Minkowski provided the mathematical apparatus with which to explore space-time. He showed how any two points, any two events—me, sitting as I write this; me rising to grab another mug of coffee—could be combined into a single, absolute picture, one that both observers in motion and at rest could accept as the true description of their differing measurements. The detailed geometrical argument is somewhat complex, but Minkowski’s work defined the single path that marks out the shortest track between any two points in four-dimensional space-time. That path is called the absolute interval, one measurement that combines the distances traveled in both space and time between two events. It is a map that does not change. (To make manipulating space-time a little easier, physicists have developed a trick to express measurements of both space and time using the same units. The speed of light provides the ultimate yardstick. How long does a meter last in time? Just so long as light takes to cross that distance—3.3 billionths of a second. How far is a second? It is the distance light travels in one tick of the clock, 186,000 miles, or 300 million meters.)

  Einstein had always affirmed that while the two observers’ measurements would differ, there was only one sequence of underlying phenomena, and all observers would find the same laws of physics behaving in the same ways as they conducted their experiments. Minkowski’s accomplishment was to make that sequence explicit—plain for anyone to see. For Minkowski, this was revolutionary. For Einstein, not so much. Four-dimensional geometry he said, was “superfluous erudition.”