He completed the theory in November 1915 and published it the following spring. Though its equations are complex, its central conception is startlingly simple. The force of gravitation disappears, and is replaced by the geometry of space itself: Matter curves space, and what we call gravitation is but the acceleration of objects as they slide down the toboggan runs described by their trajectories in time through the undulations of space. The planets skid along the inner walls of a depression in space created by the fat, massive sun; clusters of galaxies rest in spatial hollows like nuggets in a prospector’s bowl.
In marrying gravitational physics to the geometry of curved space, general relativity emancipated cosmology from the ancient dilemma of whether the universe is infinite and unbounded or finite and bounded. An infinite universe would be not just large but infinite, and this posed problems. The gravitational force generated by an infinite number of stars would itself be infinite, and would, therefore, overwhelm the local action of gravity; this prospect so troubled Newton that he resorted to invoking God’s infinite grace to resolve it. Moreover, the light from an infinite number of stars might be expected to turn the night sky into a blazing sheet of light; yet the night sky is dark.* The alternative, however—a finite euclidean universe with an edge to it—was equally unattractive: As Liu Chi posed the question, in China in the fourteenth century, “If heaven has a boundary, what things could be outside it?”39 The difficulty of imagining an end to space had been enunciated as early as the fifth century B.C., by Plato’s colleague Archytas the Pythagorean; Lucretius summed it up this way:
Let us assume for the moment that the universe is limited. If a man advances so that he is at the very edge of the extreme boundary and hurls a swift spear, do you prefer that this spear, hurled with great force, go where it was sent and fly far, or do you think that something can stop it and stand in its way?40
General relativity resolved the matter by establishing that the universe could be both finite—i.e., could contain a finite number of stars in a finite volume of space—and unbounded. The key to this realization lay in Einstein’s demonstration that, since matter warps space, the sum total of the mass in all the galaxies might be sufficient to wrap space around themselves. The result would be a closed, four-dimensionally spherical cosmos, in which any observer, anywhere in the universe, would see galaxies stretching deep into space in every direction, and would conclude, correctly, that there is no end to space. Yet the amount of space in a closed universe would nonetheless be finite: An adventurer with time to spare could eventually visit every galaxy, yet would never reach an edge of space. Just as the surface of the earth is finite but unbounded in two dimensions (we can wander wherever we like, and will not fall off the edge of the earth) so a closed four-dimensional universe is finite but unbounded to us who observe it in three dimensions.†
Two-dimensional inhabitants of a finite universe must confront the paradox of an “edge” to their cosmos. But if we add a dimension, curving the plane on which they live into a sphere, their world, though still finite, becomes unbounded. General relativity reveals a similar prospect for the four-dimensional geometry of the universe we three-dimensional creatures inhabit: hence Einstein’s “closed, unbounded” universe.
The question of whether the universe is hyperbolic and open or spherical and closed remains unanswered, as we shall see. But, thanks to Einstein, the problem is no longer clouded by paradox. By introducing the scientific prospect of a finite, unbounded cosmos, Einstein’s general theory initiated a meaningful dialogue between the human mind and the conundrums of cosmological space.
The theory was beautiful, but was it true? Einstein, having been to the mountaintop, felt supremely confident on this score. General relativity explained a precession in the orbit of the planet Mercury that had been left unaccounted for in Newtonian mechanics, and he did not doubt it would survive further tests as well. As he wrote his friend Besso, “I am fully satisfied, and I do not doubt any more the correctness of the whole system…. The sense of the thing is too evident.”41
The wider scientific community, however, awaited the verdict of experiment. There would be a total solar eclipse on May 29, 1919, at which time the sun would stand against the bright stars of the Hyades cluster. The English astronomer Arthur Stanley Eddington led an expedition to a cocoa plantation on Principe Island off west Equatorial Africa to observe the eclipse and see whether the predicted curvature of space in the region of the sun would distort the apparent positions of the stars in the briefly darkened sky. It was a scene of high drama—English scientists testing the theory of a German physicist immediately after the end of the Great War. As the time of the eclipse approached, rain clouds covered the sky. But then, moments after the moon’s shadow came speeding across the landscape and totality began, a hole opened up in the clouds around the sun, and the camera shutters were triggered. The results of Eddington’s expedition, and of a second eclipse observation conducted at Sobral, Brazil on the same day, were presented by the Astronomer Royal at a meeting of the Royal Society in London on November 6, 1919, with Newton’s portrait looking on. They were positive: The light rays coming from the stars of the Hyades were found to be offset to just the degree predicted in the theory.
When Einstein received a telegram from Lorentz announcing the outcome of the Eddington expedition, he showed it to a student, Ilse Rosenthal-Schneider, who asked, “What would you have said if there had been no confirmation?”
“I would have had to pity our dear Lord,” Einstein replied. “The theory is correct.”42*
Subsequent experiments have further vindicated Einstein’s confidence. The curvature of space in the vicinity of the sun was established with much greater accuracy, by bouncing radar waves off Mercury and Venus when they lie near the sun in the sky, and the extent of curvature matched that predicted by the general theory of relativity. A light beam directed up a tower in the Jefferson Physical Laboratory at Harvard University was found to be shifted toward the red by the earth’s gravitation to just the anticipated degree. Maelstroms of energy detected at the centers of violent galaxies indicate that they harbor black holes, collapsed objects wrapped in infinitely curved space that shuts them off from the rest of the universe; the existence of black holes was another prediction of the general theory. And the theory has been tested in many other ways as well—in examinations of entombed dead stars, the whirling of active stars around one another, the wanderings of interplanetary spacecraft well past Jupiter, and the slowing of light as it climbs up out of the sun’s space well—and all these trials it has survived.
Too modest to be immodest, Einstein had written when publishing his completed account of general relativity that “hardly anyone who has truly understood this theory will be able to resist being captivated by its magic.”44 But, even if only those mathematicians and physicists who have mastered general relativity are in a position properly to understand it, still we can all appreciate it to some degree, if, while keeping in mind its basic concepts, we contemplate the universe of effortlessly wheeling galaxies deployed across the blossom petals of gently curving space. Einstein’s epitaph could be Christopher Wren’s: If you seek his monument, look around.
*If, instead, the earth dragged the aether along with it, like a ship gathering up seaweed as it plows through the Sargasso Sea, then the aberration of starlight that Bradley had first observed (the effect of moving through starlight like a woman running in the rain) would not occur.
*I have adopted this metaphor from one employed by Einstein’s colleague Banesh Hoffmann.2
*The American mathematician Ernst Straus was treated to an example of Einstein’s tenacity one afternoon while working as his assistant at the Institute for Advanced Study in Princeton in the 1940s. “We had finished the preparation of a paper and we were looking for a paper clip,” Straus writes. “After opening a lot of drawers we finally found one which turned out to be too badly bent for use. So we were looking for a tool to straighten it. Opening a lot more dr
awers we came on a whole box of unused paper clips. Einstein immediately started to shape one of them into a tool to straighten the bent one. When I asked him what he was doing, he said, ‘Once I am set on a goal, it becomes difficult to deflect me.’”14
*Maxwell found that the speed with which electromagnetic fields are propagated is equal to the ratio between the electrical force exerted by two electrical charges when at rest and the magnetic force they exert when in motion. As this turned out to be nothing other than the velocity of light, Maxwell concluded that light itself is an electromagnetic field. Since popular accounts of the special theory of relativity sometimes convey the mistaken impression that the velocity of light is an arbitrary speed limit, like that set by legislatures for public highways, it is helpful to keep in mind Maxwell’s finding—that the velocity of light results from a fundamental constant in the equations that describe the behavior of electromagnetic fields.
*A sense of the allure of the dynamo was preserved by the American historian Henry Adams in his The Education of Henry Adams. Describing his visit to the “great hall of dynamos” at the Paris Exposition in 1900, he writes, “To Adams the dynamo became a symbol of infinity. As he grew accustomed to the great gallery of machines, he began to feel the forty-foot dynamos as a moral force, much as the early Christians felt the Cross. The planet itself seemed less impressive, in its old-fashioned, deliberate, annual or daily revolution, than this huge wheel.”22
†Research like Herr Weber’s was being applied with dispatch to the execution of convicts and the punishment of malingering conscripts. The first electrocution of a criminal in the United States occurred in 1890, less than ten years after the first public power station in America started operating; the method was purportedly humane, but it took the victim fifteen long minutes to die. Shell-shocked German soldiers in the trenches of the First World War were administered jolts of electricity and then sent back to the front; if they returned, they were given still more severe shocks, in a closed circuit of fear and pain that drove some to suicide.24
*Einstein shared this fate with Newton, whose ideas were routinely characterized as incomprehensible. A student who saw Isaac Newton passing in his carriage is said to have remarked, “There goes the man that writ a book that neither he nor anybody else understands.”
*I was treated to an inadvertent demonstration of this effect one day, aboard a DC-3 in a violent storm over the Bahamas, when a doctor’s iron scale, standing about four feet tall, tore loose from its moorings in the aft end of the cabin. The plane then plunged into a downdraft, rendering everything momentarily weightless, and the scale rose into the air and drifted toward me. I fended it off with my feet, thus briefly experiencing its inertial mass absent its gravitational mass. The fact that the menacing object happened to be a weightless device for measuring weight invested the lesson with a certain ironic intensity.
†The definitive experiments were conducted by Baron Roland von Eötvös in Budapest in 1889 and 1922. Eötvös suspended objects of various compositions from threads and looked for deviations in these plumb lines caused by differences between their gravitational mass (which was being pulled straight down) and their inertial mass (which was being pulled sideways, by the rotation of the earth). “In no case,” he wrote, “could we discover any detectable deviation from the law of proportionality of gravitation and inertia.” This remains the case today, although one recent reenactment of the experiment did produce subtle anomalies that could not immediately be accounted for.
*The very term “fourth dimension” called to mind the enthusiasms of eccentrics and ecstatics like Charles Hinton, who sought to enhance his appreciation of its subtleties by manipulating 81 cubes that represented the units of a 3-by-3-by-3-by-3-unit euclidean hypercube. Hinton’s career was interrupted—and his subject cast into further ignominy—when he was convicted of bigamy for living out the free-love philosophy of his father, who liked to say that “Christ was the Savior of men, but I am the savior of women, and I don’t envy Him a bit!” Hinton fits dropped dead at a banquet of the Society of Philanthropic Inquiry in Washington, D.C., moments after delivering a toast in honor of femininity.35
*This disturbing puzzle, known today as Olbers’s paradox after the nineteenth-century German astronomer Wilhelm Olbers, was discovered independently by other astronomers, among them Halley, who lectured on it at a Royal Society meeting in 1721. Newton chaired that meeting, but for some reason never wrote about the paradox. The historian of science Michael Hoskin suggests that the old man was napping while Halley spoke.38
†Alternately, general relativity allows that the universe might be structured like a four-dimensional hyperbola, in which case it would be both infinite and unbounded. This possibility resurrects some of the difficulties that afflict all infinite-universe models, but they could perhaps be resolved, should the observational data indicate that space is indeed hyperbolically rather than spherically curved.
*Einstein once astonished Ernst Straus by saying of Max Planck, the father of quantum physics, “He was one of the finest people I have ever known and one of my best friends; but, you know, he didn’t really understand physics.” When Straus asked what he meant, Einstein replied, “During the eclipse of 1919, Planck stayed up all night to see if it would confirm the bending of light by the gravitational field of the sun. If he had really understood the way the general theory of relativity explains the equivalence of inertial and gravitational mass, he would have gone to bed the way I did.”43
11
THE EXPANSION OF THE UNIVERSE
Nature lives in motion.
—James Hutton
Eyesight should learn from reason.
—Kepler
When Einstein began to investigate the cosmological implications of the general theory of relativity, he found something strange and disturbing: The theory implied that the universe as a whole could not be static, but must be either expanding or contracting. This was a completely novel idea, and one for which there was, at the time, no observational evidence whatever: The astronomers he consulted informed Einstein that stars wander more or less randomly through space, but display no concerted motion of the sort that would suggest cosmic expansion or contraction. Faced with this disjunction between his theory and the empirical data, Einstein reluctantly concluded that there must be something wrong with the theory, and he modified its equations by adding a term that he called the cosmological constant. Symbolized by the Greek letter lambda, the new term was intended to make the radius of the universe hold steady with the passing of time.
Einstein never liked the cosmological constant. He called it “gravely detrimental to the formal beauty of the theory,” pointing out that it was nothing more than a mathematical fiction, without any real physical basis, one that had been introduced solely to being the theory into accord with the observational facts. As he wrote in 1917:
[W]e admittedly had to introduce an extension of the field equations of gravitation which is not justified by our actual knowledge of gravitation…. That term is necessary only for the purpose of making possible a quasi-static distribution of matter, as required by the fact of the small velocities of the stars.1
Moreover, as soon became apparent, the term did not even accomplish its avowed function of making the relativistic universe stand still. The Russian mathematician Aleksandr Friedmann found that Einstein in introducing the term had made an algebraic error, dividing by a quantity that could be zero. When Friedmann corrected the error, general relativity broke free of its fetters and the relativistic universe, to Einstein’s frustration, once again took on wings.
Connoisseurs of irony’s serrated edge will appreciate that it was in 1917, the very year that Einstein besmirched his general theory of relativity by introducing the cosmological term, that the American astronomer Vesto Slipher published a paper containing the first observational evidence that the universe is in fact expanding.
Slipher knew nothing of general relativity. He was a nose-to-the-grindsto
ne staffer at Lowell Observatory, in Flagstaff, Arizona, an isolated and idiosyncratic private institution so remote from the theoretical physics community that it might as well have been on the far side of the moon. His employer was Percival Lowell, of the Boston Lowells, a loftily unconventional thinker remembered chiefly for having charted the (illusory) canals of Mars, which he took to be global waterways dug by a parched alien civilization desperately importing water from the polar ice caps. Like many astronomers of his day, Lowell thought that the spiral nebulae were Laplacian solar systems aborning. To test this thesis, he assigned Slipher to take spectra of a number of spirals, using a new and more efficient spectrograph, in order to search for the rotation velocities characteristic of Laplacian nebulae eddying their way into stars and planets. Slipher did, indeed, find evidence of rotation in the spirals—as Edwin Hubble would find, this was actually the motion of billions of stars orbiting in spiral galaxies—but he also found, superimposed on the rotation velocities, an enormous displacement in the spectral lines of most spirals toward the red end of the spectrum.
The only reasonable explanation for this astonishing finding was that Slipher was observing Doppler shifts. The name comes from the Austrian physicist Christian Johann Doppler, who noted in 1842 that light, sound, or other radiation coming from a moving source is received at a higher frequency if the source emanating it is approaching, and at a lower frequency if the source is receding. (It is owing to this “Doppler shift” that an automobile horn sounds higher in pitch if the car is approaching and lower if it is speeding away.) Astronomers had long been making use of Doppler shifts in spectra to measure the velocities of stars: The spectral lines of stars that were moving toward the sun would be displaced toward the blue, while those of stars moving away from the sun would be displaced toward the red, or lower frequency, end of the spectrum. Indeed, it was by virtue of just such measurements that astronomers had been able to inform Einstein that stellar motions in the Milky Way were generally random.
Coming of Age in the Milky Way Page 21
