Coming of Age in the Milky Way

by Timothy Ferris

  With the special theory of relativity, Einstein had at last resolved the paradox that had occurred to him at age sixteen, that Maxwell’s equations failed if one could chase a beam of light at the velocity of light. His did so by concluding that one cannot accelerate to the velocity of light—that, indeed, the velocity of light is the same for all observers, regardless of their relative motion. If, for instance, a physicist were to board a spaceship and fly off toward the star Vega at 50 percent the velocity of light, and while on board measure the velocity of the light coming from Vega, he would find that velocity to be exactly the same as would his colleagues back on Earth.

  To quantify this strange state of affairs, Einstein was obliged to employ the Lorentz contractions. (At the time he knew little of Lorentz, whom he was later to esteem as “the greatest and noblest man of our times … a living work of art.”)30 In Einstein’s hands, the Lorentz equations specify that as an observer increases in velocity, his dimensions, as well as those of his spaceship and any measuring devices aboard, will shrink along the direction of their motion by just the amount required to make the measurement of light’s velocity always come out the same. This, then, was why Michelson and Morley had found no trace of an “aether drift.” In fact the aether is superfluous, as is Newton’s absolute space and time, for there is no need for any unmoving frame of reference: “To the concept of absolute rest there correspond no properties of the phenomena, neither in mechanics, nor in electrodynamics.”31 What matters are observable events, and no event can be observed until the light (or radio waves, or other form of electromagnetic radiation) that brings news of it reaches the observer. Einstein had replaced Newton’s space with a network of light beams; theirs was the absolute grid, within which space itself became supple.

  Observers in motion experience a slowing in the passage of time, as well: An astronaut traveling at 90 percent of the velocity of light would age only half as fast as her colleagues back home, so that at, say, a twentieth class reunion of interstellar astronauts, those who had served the most aboard relativistic spacecraft would be the youngest. Mass, too, is rendered plastic within the framework of the light beams; objects approaching the speed of light increase in mass. The effects of relativistic time dilation, mass increase, and change in dimension are minute at ordinary velocities like that of the earth in its orbit or the sun through space (which is why it had not been noticed sooner) but become pronounced as speeds increase, and go to infinity at the speed of light. If the earth could be accelerated to the velocity of light (a feat that would require infinite energy to achieve) it would contract into a two-dimensional wafer of infinite mass, on which time would come to a stop—which is one way of saying that acceleration to light speed is impossible.

  Nor are these effects illusory, or merely psychological: They are as real as the stone that Dr. Johnson kicked in his famous refutation of Bishop Berkeley, and have been confirmed in scores of experiments. The relativistic increase in the mass of particles moving at nearly the velocity of light is not only observed in particle accelerators, but is what gives the speeding particles most of their punch. Relativistic time dilation has been tested by flying atomic clocks around the world in commercial aircraft; the clocks were found to run slow by just the tiny amount the theory specifies. A NASA ground controller once threatened to dock astronauts in space a fraction of a penny of their flight pay, to compensate for the decrease in the passage of time they experienced as a result of their velocity in orbit.

  These and other implications of special relativity initially struck the lay public, and many scientists as well, as uncommonly strange.* But if Einstein’s approach was radical, his intention was essentially conservative. As is implied by the title of his original relativity paper, “On the Electrodynamics of Moving Bodies,” his aim was to redeem the laws of electrodynamics so that they could be shown to work in every imaginable situation, not just in a quiet laboratory in Zurich but in whirling dynamos and on moving worlds hurtling past one another at staggering speeds. The term relativity, coined not by Einstein but by Poincaré and applied to the theory by the physicist Max Planck, is somewhat misleading in this sense; Einstein, stressing, its conservative function, had preferred to call it Invariantentheorie—“invariance theory.”

  Relativity nonetheless cast its net wide, embracing the study not only of light and space and time, but of matter as well. The theory derives its catholic impact from the fact that electromagnetism is implicated not only in the propagation of light but also in the architecture of matter: Electromagnetism is the force that holds electrons in their orbits around nuclear particles to make atoms, binds atoms together to form molecules, and ties molecules together to form objects. Every tangible thing, from stars and planets to this page and the eye that reads it, carries electromagnetism in the fiber of its being. To alter one’s conception of electromagnetism is, therefore, to reconsider the very nature of matter. Einstein caught sight of this connection only three months after the first account of special relativity had appeared, and published a paper titled, “Does the Inertia Content of a Body Depend Upon Its Energy Content?” The answer was yes, and ours has been a sadder and wiser world ever since.

  In the first paper, as we have seen, Einstein demonstrated that the inertial mass of an object increases when it absorbs energy. It follows that its mass decreases when it radiates energy. This holds true, not only for a spaceship gliding toward the stars, but for an object at rest as well: A camera loses a (very) little mass when the flash goes off, and the people whose picture is being taken become a little more massive in the exchange. Mass and energy are interchangeable, with electromagnetic energy doing the bartering between them.

  Einstein, contemplating this fact, concluded that energy and inertial mass are the same, and he expressed their identity in the equation

  in which m is the mass of an object, E is its energy content, and c is the velocity of light. In composing this singularly economical little equation, which unifies the concepts of energy and matter and relates both to the velocity of light, Einstein initially was concerned with mass. If instead we solve for energy, it takes on a more familiar and more ominous form, as

  E = mc2

  Viewed from this perspective, the theory says that matter is frozen energy. This of course is the key to nuclear power and nuclear weapons, though Einstein did not consider these applications at the time and rejected them as impractical once they were proposed by others. In the hands of the astrophysicists, the equation would be used to discern the thermonuclear processes that power the sun and stars.

  But for all its protean achievements, special relativity was silent with regard to gravitation, the other known large-scale force in the universe. The special theory has to do with inertial mass, the resistance objects offer to change in their state of motion—their “clout,” or “heft,” so to speak. Gravitation acts upon objects according to their gravitational mass—i.e., their “weight.” Inertial mass is what you feel when you slide a suitcase along a polished floor; gravitational mass is what you feel when you lift the suitcase. There would appear to be distinct differences between the two: Gravitational mass manifests itself only in the presence of gravitational force, while inertial mass is a permanent property of matter. Take the suitcase on a spaceship and, once in orbit, it will weigh nothing (i.e., its gravitational mass will measure zero), but its inertial mass will remain the same: You’ll have to work just as hard to wrest it around the cabin, and once in motion it will have the same momentum as if it were sliding across a floor on Earth.*

  Yet for some reason, the inertial and gravitational mass of any given object are equivalent. Put the suitcase on the airport scale and find that it weighs thirty pounds: That is a result of its gravitational mass. Now set it on a sheet of smooth, glazed ice or another relatively friction-free surface, attach a spring scale to the handle, and pull it until you get it accelerating at the same rate at which it would fall (i.e., 16 feet per second, on Earth), and the scale will register, again,
thirty pounds: That is a result of its inertial mass. Experiments have been performed to a high degree of precision on all sorts of materials, in many different weights, and the gravitational mass of each object repeatedly turns out to be exactly equal to its inertial mass.†

  The equality of inertial and gravitational mass had been an integral if inconspicuous part of classical physics for centuries. It can be seen, for instance, to explain Galileo’s discovery that cannonballs and boccie balls fall at the same velocity despite their differing weight: They do so because the cannonball, though it has greater gravitational mass and ought (naively) to fall faster, also has a greater inertial mass, which makes it accelerate more slowly; since these two quantities are equivalent they cancel out, and the cannonball consequently falls no faster than the boccie ball. But in Newtonian mechanics the equivalence principle was treated as a mere coincidence. Einstein was intrigued. Here, he thought, “must lie the key to a deeper understanding of inertia and gravitation.”32 His inquiry set him on his way up the craggy road toward the general theory of relativity.

  Einstein’s first insight into the question came one day in 1907, in what he later called “the happiest thought of my life.” The memory of the moment remained vivid decades later:

  I was sitting in a chair in the patent office at Bern, when all of a sudden a thought occurred to me: “If a person falls freely he will not feel his own weight.” I was startled. This simple thought made a deep impression on me. It impelled me toward a theory of gravitation.33

  To appreciate why this seemingly straightforward picture should have so excited Einstein, imagine that you awaken to find yourself floating, weightless, in a sealed, windowless elevator car. A diabolical set of instructions, printed on the wall, informs you that there are two identical such elevator cars—one adrift in deep space, where it is subject to no significant gravitational influence, and the other trapped in the sun’s gravitational field, plunging rapidly toward its doom. You will be rescued only if you can prove (not guess) in which car you are riding—the one floating in zero gravity, or the one falling in a strong gravitational field. What Einstein realized that day in the patent office was that you cannot tell the difference, neither through your senses nor by conducting experiments. The fact that you are weightless does not mean that you are free from gravitation; you might be in free fall. (The “weightlessness” experienced by astronauts in orbit is precisely of this sort: Though trapped by the earth’s gravitational field they feel no weight—i.e., no effect of gravitation—because they and their spaceship are constantly falling.) The gravitational field, therefore, has only a relative existence. One is reminded of the joke about the man who falls from the roof of a tall building and, seeing a friend looking aghast out a window on the way down, calls out encouragingly, “I’m okay so far!” His point was Einstein’s—that the gravitational field does not exist for him, so long as he remains in his inertial framework. (The sidewalk, alas, is in an inertial framework of its own.)

  The same ambiguity applies in the opposite situation: Suppose that when you awaken you find yourself standing in the elevator car, at your normal weight. This time the instructions say that you are either (1) aboard a elevator stopped on the ground floor of an office building on Earth, or (2) adrift in zero-gravity space, in an elevator attached by a cable to a spaceship that is pulling it away at a steady acceleration, pressing you to the floor with a force equal to that of Earth’s gravitation—at one “G,” as the jet pilots say. Here again, you cannot prove which is the case.

  Einstein reasoned that if the effects of gravitation are mimicked by acceleration, gravitation itself might be regarded as a kind of acceleration. But acceleration through what reference frame? It could not be ordinary three-dimensional space; the passengers in the elevator in the New York skyscraper, after all, are not flying through space relative to the earth.

  The search for an answer required brought Einstein to consider the concept of a four-dimensional spacetime continuum. Within its framework, gravitation ts acceleration, the acceleration of objects as they glide along “world lines”—paths of least action traced over the slopes of a three-dimensional space that is curved in the fourth dimension.

  A forerunner here was Hermann Minkowski, who had been Einstein’s mathematics professor at the Polytechnic Institute. Minkowski remembered Einstein as a “lazy dog” who seldom came to class, but he was quick to appreciate the importance of Einstein’s work, though initially he viewed it as but an improvement on Lorentz. In 1908 Minkowski published a paper on Lorentz’s theory that cleared away much of the mathematical deadwood that had cluttered Einstein’s original formulation of special relativity. It demonstrated that time could be treated as a dimension in a four-dimensional universe. “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality,” Minkowski predicted.34 His words proved prophetic, and the special theory of relativity has been viewed in terms of a “spacetime continuum” ever since. Einstein initially dismissed Minkowski’s formulation as excessively pedantic, joking that he scarcely recognized his own theory once the mathematicians got hold of it. But he came to realize that if he were to explore the connection between weight and inertia, he would do well to travel farther up the trail Minkowski had blazed.

  Minkowski’s spacetime continuum, though suitable for special relativity, would not support what was to become general relativity. Its space was “flat”—i.e., euclidean. If gravitation were to be interpreted as a form of acceleration, that acceleration would have to occur along the undulations of curved space. So it was that Einstein was led, however reluctantly, into the forbidding territory of the noneuclidean geometries.

  Euclidean geometry, as every high school math student is taught, has different characteristics depending upon whether it is worked in two dimensions (“plane” geometry) or three (“solid” geometry). On a plane, the sum of the angles of a triangle is 180 degrees, but if we add a third dimension we can envision surfaces such as that of a sphere, on which the angles of a triangle add up to more than 180 degrees, or a saddle-shaped hyperbola, on which the angles add up to less than 180 degrees. The shortest distance between two points on a plane is a straight line, but on a sphere or a hyperbola the shortest distances are curved lines. In the noneuclidean geometries a fourth dimension is added, and the rules are changed in a similarly consistent manner to allow for the possible curvature of three-dimensional space within a four-dimensional theater. Two categories of curved space then can be imagined (or at least calculated): spherical, or “closed” space, in which three-space obeys geometrical rules analogous to those of the two-space on the surface of a sphere, and hyperbolic, or “open” space, analogous to the surface of a three-dimensional hyperbola. (One can also work out a flat, euclidean four-dimensional geometry, but in that case the rules do not change, just as plane two-dimensional geometry obeys the same rules if the planes happen to be the sides of a three-dimensional cube.)

  Triangles in flat two-dimensional space have interior angles that always add up to 180 degrees. But when two-space is curved into a third dimension, the angles always total either less than 180 degrees (if the curvature is hyperbolic, or “open”) or more than 180 degrees (if the curvature is spherical, or “closed”). Similarly, the geometry of the three-dimensional universe may be either flat (euclidean), or open or closed (noneuclidean), when viewed in the context of Einstein’s four-dimensional spacetime continuum.

  By the time Einstein came on the scene, the rules of four-dimensional geometry had been worked out—those of spherical four-space by Georg Friedrich Riemann and those of the four-dimensional hyperbolas by Nikolai Ivanovich Lobachevski and János Bolyai. The whole subject, however, was still regarded as at best difficult and arcane, and at worst almost disreputable.* The legendary mathematician Karl Friedrich Gauss had withheld his papers on noneuclidean geometry from publication, fearing ridicule by his colleagues, and Bolyai conducted
his research in the field against the advice of his father, who warned him, “For God’s sake, please give it up. Fear it no less than the sensual passions because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life.”36

  Einstein rushed in where Bolyai’s father feared to tread. With the aid of his old classmate Marcel Grossmann—“Help me, Marcel, or I’ll go crazy!” he wrote—Einstein struggled through the complexities of curved space, seeking to assign the fourth dimension to time and make the whole, infernally complicated affair come out right. He had by now begun to win professional recognition, had quit the patent office to accept a series of teaching positions that culminated in a full professorship in pure research at the University of Berlin, and was doing important work in quantum mechanics and a half-dozen other fields. But he kept returning to the riddle of gravitation, trying to find patterns of beauty and simplicity among thick stacks of papers black with equations. Like a lost explorer discarding his belongings on a trek across the desert, he found it necessary to part company with some of the most cherished of his possessions—among them one of the central precepts of the special theory itself, which to his joy was ultimately to return as a local case within the broader scheme of the general theory. “In all my life I have never before labored so hard,” he wrote to a friend. “… Compared with this problem, the original theory of relativity is child’s play.”37 Nowhere in human history is there to be found a more sustained and heroic labor of the intellect than in Einstein’s trek toward general relativity, nor one that has produced a greater reward.