God In The Equation Read online

  In his paper “Cosmological Considerations of the General Theory of Relativity,” presented to the Prussian Academy of Sciences early in 1917, Einstein set forth a shocking new picture of the universe. At the time, the general theory had proven itself capable of explaining only the precession of Mercury's orbit—an impressive achievement, but far from definitive proof of the theory's validity. Undaunted, Einstein declared that he could derive the size and structure of the entire cosmos from his puny equations. His answer contradicted every theory that had come before. “Even for Einstein,” writes biographer Ronald Clark, “this was playing for high stakes.”

  Once again, Einstein describes his intellectual work as a painful struggle, as if he were Jacob wrestling with an angel. The effort of writing this paper so taxed him that, he joked to his physicist friend Paul Ehrenfest, he felt he was facing “the danger of being confined to a madhouse.” He was already mentally exhausted from the mind-bogglingly complex mathematical details of four-dimensional space-time. Why embark on another grand venture?

  In part, Einstein had a specific scientific goal, based on his interpretation of Mach's ideas about the nature of inertia. Inertia is the tendency for an object to resist being moved; if you've ever tried rolling an automobile off the road, you have a good idea how much inertia there is in two tons of mass. But if there is no absolute space and absolute motion, how can there be absolute inertia? Mach's answer was that inertia “is nothing more or less than an abbreviated reference to the entire universe.” Einstein had echoed this view as early as 1912. This linkage connected every little instance of inertia, such as the leaning of a car around a tight curve or the difficulty of getting a bowling ball quickly down the lane, to the distribution of distant stars. If Einstein took Mach's ideas seriously, his theory of general relativity would be incomplete until he derived the overall structure of the cosmos.

  But another motivation behind Einstein's cosmological dabbling surely lay in his profound intellectual goals. He was interested in global solutions, physical theories so powerful and general that they could describe any and every location without producing any inconsistencies. Newton's unbounded universe struck him as irrational because it contained an infinite amount of mass scattered through an infinite space. In such an arrangement, every direction would lead to an object exerting a gravitational force, thereby producing an infinite gravitational field—a dynamical version of Olbers's paradox. All of these infinities would make the equations of relativity meaningless and the search for ultimate answers futile. When Newton recognized the problems with the infinite solution, he had called on God to bail him out. To Einstein, however, the physical laws and God were one and the same. Conflict between the two was not acceptable.

  Then Einstein revisited Newton's other picture of the cosmos, in which we reside within a finite clump of stars surrounded by a ceaseless stretch of nothingness. To the astronomers of the time, who roughly knew the structure of our Milky Way but were still unsure about the existence of other galaxies, this model seemed plausible. Einstein rejected it, however, recognizing that such an island universe is unstable. Slowly but surely the stars would interact with each other and disperse, so that the island would eventually evaporate and become like grains of sand scattered across the endless sea of empty space. He also considered the island universe arrangement wasteful, because “the radiation emitted by the heavenly system of the universe will, in part, leave the Newtonian system of the universe, passing radially outwards, to become ineffective and lost in the infinite.” He recoiled at the thought of such a “systematically impoverished” creation.

  By the process of elimination, Einstein returned to the finite universe of his ancient predecessors. The hard-edged, spherical construction envisioned by Eudoxus and Aristotle was no longer scientifically acceptable, of course. In its place, Einstein sought a seeming paradox, a universe that has a limited size but no physical boundaries. He wanted a new creation, a universe unlike anything ever imagined before.

  In this scientific article of faith, Einstein was indebted to Bernhard Riemann, a German mathematics wizard who had died young, his most innovative work largely unappreciated, thirteen years before Einstein was born. Riemann imagined warped geometries that follow very different rules from those of the familiar planar, Euclidian geometry. In his alternative worlds, parallel lines can meet and the angles of a triangle do not add up to 180 degrees, exactly as if somebody warped and twisted a page from a geometry textbook. Draw a big triangle on a basketball and you will see that the angles add up to more than what they would on a flat surface. Or draw the triangle on the inside of a bowl and all the corners look pinched in, with the totals of their angles equaling distinctly less than 180 degrees. Combining Riemann's geometry with concepts from his general theory of relativity, Einstein discovered he could construct exactly the kind of universe he sought.

  According to relativity, matter warps the structure of space-time so that it follows Riemann's rules. It is difficult to visualize what this means, but think about the bending of starlight as it passes by the sun. From the perspective of the light beam, it is not bending; it is following a linear path through the warp of space-time. At first, Einstein had thought that only a huge mass would produce a significant effect of this kind. While he was contemplating how the eclipsed sun would displace the light of the nearby stars, he realized that the widely scattered stars in the universe would collectively distort the overall geometry of space. It is helpful to think of a two-dimensional equivalent: a large, thin rubber sheet. A weight placed on the sheet causes it to sag. If you keep adding more weights in different places, the whole sheet begins to dip and assume a concave shape. Likewise, all the stars, planets, and rocks in the cosmos bend the geometry of space, so it resembles the inside of that bowl. If there is enough matter, Einstein concluded, the bowl would close in on itself. “The curvature is variable in time and place, according to the distribution of matter, but we may roughly approximate it by means of a spherical shape,” he wrote. In other words, the universe is a huge lumpy ball, finite but unbounded.

  To be precise, the real universe would be not a ball but a four-dimensional sphere of curved, three-dimensional space. Such higher dimensions are pretty much impossible to visualize, ironic for a model allegedly grounded in Mach's philosophy that science should be rooted in experience. Einstein's cosmology was more like a divine revelation, a description of something so far beyond human scales and human comprehension that we can talk about it but never truly know it.

  The simplest way to grasp Einstein's solution is to return to the rubber sheet analogy. He used a similar analogy in Relativity, a semi-popular account of his newly expanded theory that he published at the end of 1916. Our sheet has curved in on itself so much that it is now an enclosed sphere, basically a balloon. Now imagine a two-dimensional being living on this three-dimensional balloon, actually embedded in the balloon's surface, who is trying to understand his world. Our poor two-dimensional friend—let's call him Trevor—cannot imagine a direction called “up” because he lives entirely within the balloon's surface. As Trevor explores his world, he never comes to an edge, yet if he travels long enough in one direction, he will complete a circumference of the sphere and return to this starting point.

  Einstein, displaying a touch of false modesty, declared “with a moderate degree of certainty” that our universe resembles Trevor's, only in our case we inhabit three-dimensional space that follows a curved geometry in an imperceptible fourth dimension. (This type of language has become the standard rhetoric of sci/religion: make a whopper of a claim, then surround it with modest-sounding qualifiers.) If Einstein was correct, then an intrepid traveler in a speedy rocket could take off from the Earth, race away in an apparently straight line, and eventually return home without turning around, just like Trevor circumnavigating his balloon. Thus Einstein escaped the contradictions of Newton's infinite universe. And without returning to the absurdity of Aristotle's sharply drawn crystalline spheres, Einstei
n gave back a geometric sense of our place in the cosmos. It seemed inconceivable that a scientific theory could explain the entirety of an infinite universe. German physicist Max Born later gushed, “This suggestion of a finite, but unbounded space is one of the greatest ideas about the nature of the world which ever has been conceived.” By making the universe finite, Einstein opened up the possibility that all of existence could lie within the grasp of human conception—and then he confidently started down that path.

  This new conception of the universe bore little resemblance to what contemporary astronomers thought they were seeing through their telescopes. Back then, many of those researchers estimated the Milky Way was about fifty thousand light-years across and believed it was the only galaxy in the universe. The Milky Way alone seemed huge; recall that the sun's neighbor 61 Cygnus is eleven light-years away, and even that distance is roughly seven hundred thousand times the span from the Earth to the sun. Einstein was envisioning a cosmos much, much larger still. His model also required that the matter be scattered more or less evenly through all of that copious space, not all clumped in one place. That smooth distribution, now known as the cosmological principle, meant that space had an overall uniform geometry and that the universe was dynamically stable. The cosmological principle was, in a way, an extension of Copernicus's idea that we do not live in a privileged position. He proposed that the Earth is just one of a group of planets. Einstein took that idea further and suggested that the properties of our part of the universe—including its density—largely resemble the properties of any other location. He thought this had to be true in order to build a universe that followed the rules of relativity and satisfied Mach's principle. How this beautiful model corresponded to the real universe was unclear at the time. “Whether, from the standpoint of present astronomical knowledge, it is tenable, will not here be discussed,” he wrote, seeming to brush aside such concerns. He had a vision, and he believed in it.

  But as with the prediction of curved starlight, Einstein wanted confirmation to back up his revelation. Because it was fully described by a set of equations relating the size of the universe to its density, Einstein's model of the universe suggested the possibility of plugging in numbers and restaging the ancient game of calculating the size of the outermost sphere of the heavens. “The exact number is a minor question,” Einstein insisted. In private, however, he told his reporter friend Alexander Moszkowski he could estimate the universe is a staggering one hundred million light-years across. This was a hugely daring claim at a time when the basic distance scale of the universe was entirely unknown. Einstein notably did not include his size calculation in the published paper. By the time he discussed his cosmological model before the Prussian Academy of Sciences in 1921, he had backed away from this line of argument, recognizing that “the distribution of the visible stars is extremely irregular. . . so it seems impossible to estimate the average density.” Still, this did not alter his essential point that the universe is finite and that, given the right information, he could derive its dimensions purely from his theory.

  In developing his “Cosmological Considerations,” Einstein had expressed no doubt about his conclusion that the universe had to be finite in size. The duration of the universe was a much more complicated matter, one that rapidly led Einstein to a paradox and to a fateful decision. His intuition told him that the universe must be eternal. But as Newton had learned centuries earlier, a universe that is finite in space tends not to be infinite in time. In the updated conception, the equations of general relativity implied that the curvature of space should change over time, producing either cosmic expansion or contraction. Einstein could not and would not renounce his beautiful theory, nor could he bring himself to renounce his classical belief in a static universe.

  Faced with this conundrum, Einstein conceived his clever, if arbitrary, way to explain the situation. What if, he supposed, space produces a mysterious repulsion—“a mass-density of negative sign, distributed evenly through space”—that makes itself known only over very large distances? Then everything could remain in balance, and the theory of general relativity could live in harmony with the happy reality that the sky is not falling. So Einstein modified his gravitational equation and added a “universal constant,” denoted by the Greek letter lambda. With this little mathematical trick, he created a universe that could stand still.

  Lambda gave empty space an outward pressure whose strength is proportional to distance. On small scales its effect would be too small to influence the well-studied motions of the planets, which would explain why nobody had observed it. Over large distances, however, its cumulative effect would negate the pull of gravity. Lambda was not based on any experiment or even on any physical theory. It existed only to keep the universe at rest without compromising general relativity. The invocation of Lambda demonstrated that Einstein was at last fully committed to his search for sci/religious transcendence. In his relentless drive for unity, he had tried to reconcile two incompatible ways of looking at the universe. In the short run his effort failed, but Lambda kept returning as spiritual aid for building a coherent model of the universe. When Einstein endowed space with structure, he gave authority to the intangible and thus helped to usher in the new sci/religious era. Gradually Lambda evolved a larger meaning, representing scientists' unshakable faith that a truly comprehensive cosmological model is possible, if only they can find the right X factor that will strip away the last veil of mystery and arrive at some kind of ultimate truth—what Einstein called “the secrets of the Old One.”

  Einstein knew that Lambda looked like an arbitrary embellishment of general relativity. In his 1917 paper he confessed, “In order to arrive at this consistent view, we admittedly had to introduce an extension of the field equations of gravitation which is not justified by our actual knowledge of gravitation. . . necessary only for the purpose of making a quasi-static distribution of matter.” As for why the universe had to be quasi-static, he pointed to “the fact of the small velocities of the stars.” Historians often take this explanation at face value. After all, Einstein was no astronomer. Leading scientists of the day believed that the universe was at rest, that the Milky Way was the only galaxy, and that the stars within it were not racing away from us, or toward us for that matter, at high speeds. If Einstein blundered, the common argument goes, it was not because he placed too little trust in observational data but because he placed too much. In other words, his faith had faltered.

  But there was much more to Einstein's imaginative leap. In 1917, astronomers were in the midst of a heated debate about the nature of spiral nebulae, those wispy swirls of light that William Herschel had studied a century earlier. Modern analyses showed that some of the nebulae were moving away from us at hundreds of miles per second, far faster than any known stellar motions. Many, though not most, scientists believed these nebulae were in fact other galaxies like our Milky Way. In that case, the measured motions of nearby stars within the Milky Way would reveal nothing about the behavior of the universe as a whole. Einstein should have been receptive to these arguments. As a student he had been an avid reader of Kant, who pictured the Milky Way as just one of myriad island universes. And the existence of a multitude of galaxies scattered through space would in fact have bolstered his belief that matter must be evenly distributed in the universe over very large scales. So why did he ignore the provocative new astronomical findings while formulating his cosmology?

  Perhaps bits of Einstein's conservative schooling had come back to haunt him. His writings drew freely from the history of philosophy, as when he expressed the general theory of relativity as a confirmation of Descartes's notion that there is no such thing as empty space. His deep-buried classical instincts told him that the universe could not have a beginning or an end. In this he may also have been guided not by Aristotle but by Baruch Spinoza, the seventeenth-century Jewish philosopher who described God as an impersonal, eternal force defined by natural law. Many philosophers, theologians, and sc
ientists had imagined that the cosmos might be finite in scale, but even the most devoted biblical chronologists did not believe that God sprang into existence just six thousand years ago. Even if God existed outside of conventional space and time, as Saint Augustine proposed, God's eternal essence had to exist somewhere from which He could create the universe. As Stephen Hawking notes, the basic tools for describing an expanding (or contracting) universe have existed for hundreds of years. “This behavior of the universe could have been predicted from Newton's theory of gravity at any time in the nineteenth, the eighteenth, or even the late seventeenth centuries,” he writes. But the cult of stasis held sway and may have drawn Einstein under its seductive spell.

  Perhaps Einstein fell victim to one of his bouts of arrogant detachment, in which his dedication to the pursuit of pure thought made it seem unnecessary to consult with experts in related fields. He expressed this attitude with shocking frankness in a 1906 paper: “In view of the fact that the questions under consideration are treated here from a new point of view, I believed I could dispense with a literature review which would be very troublesome for me.”

  Or perhaps Einstein felt that providing a scientific framework for the construction of the universe was all that really mattered. Maybe the overreaching that inspired him to stick Lambda in the equations was, at the time, at least, a reasoned strategy that let him show the world such a cosmic model was possible. Einstein was growing increasingly confident in the beauty and simplicity of his theories, regardless of whether the supporting data yet existed. The 1917 cosmology paper marked an important turning point in that regard. Up to then he had focused on theories to explain specific phenomena—the photoelectric effect, the propagation of light, the nature of gravity. Afterward he shifted his attention to broader, unifying themes—first his cosmology, then his search for a theory that would show the underlying similarity of gravity, electromagnetism, and the two forces governing the behavior of atomic nuclei. And around this time he started speaking much more expressively about the romantic and religious underpinnings of his scientific work. “The supreme task of the physicist is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. . . The state of mind which enables a man to do work of this kind is akin to that of the religious worshiper or the lover,” he said at a 1918 scientific gathering in celebration of Max Planck's sixtieth birthday.