God In The Equation

by Corey S. Powell

  When he worked out this global application of general relativity, Friedmann discovered that the static picture of the universe made no sense; the nature of the cosmos is to expand or contract. He kept Lambda in the equations and, like Einstein—but unlike de Sitter—assumed a uniform distribution of matter. Yet still space erupted into motion. Friedmann set out his new solutions in two stunning papers, 1922's “On the Curvature of Space” and 1924's “On the Possibility of a World with Constant Negative Curvature,” both published in the prominent German physics journal Zeitschrift fur Physik. Where de Sitter polished the equations of general relativity and found a second, ambiguous way to interpret their cosmological meaning, Friedmann gave the equations a more determined rub and resoundingly unleashed the genie that Einstein had tried to keep corked in a bottle.

  In Friedmann's hands, the corrected equations of relativity allowed a dazzling array of possible cosmologies, every single one of them alive with motion: expanding universes, contracting universes, even oscillating universes that grow and shrink as if following the exhalations and inhalations of a cosmic Creator. Each of his solutions corresponded to a different geometry of space, a particular warping of the four-dimensional analogue of our distorted rubber sheet. An oscillating universe would be concave, or bowl shaped, as in Einstein's initial cosmological mode. A contracting universe would also have this shape, but a depressingly limited life span. Expanding universes might be convex, shaped somewhat like a riding saddle, in which case they would expand forever, or flat, in which case the force of gravity would exactly balance the expansion.

  Friedmann did more than undermine Einstein's belief in stability. He also showed that curved space did not necessarily imply a finite universe. A concave universe would fold in on itself; this shape corresponds to Einstein's unbounded cosmos of limited extent. But flat and convex universes could be infinite in dimension. As de Sitter's work hinted, Einstein had failed to appreciate how much lay untapped within his equations. All of a sudden the relativity equations were birthing not just one or two but myriad possible universes, from which the theorist could pluck the most attractive ones like a shopper selecting a natty new tie. And unlike de Sitter's vague interpretations of the uncurling, empty cosmos, Fried-mann's solutions were not just academically interesting descriptions of space. They were genuinely plausible models of space plus matter that could be connected to the real world. Friedmann lacked the skills or knowledge to follow through on his desire to link his theoretical models to astronomical observations, but that advance came soon enough. Scientists still refer to his three basic geometries of space and argue over which one best fits the latest data like Catholics and Protestants debating Scripture in sixteenth-century Switzerland.

  Of all these cosmic possibilities, Friedmann considered the oscillating universe the most appealing, for it suggests a potentially endless cycle of rebirth. It thus avoids the notion of a beginning of time, a seeming impossibility that vexed Einstein, not to mention Saint Augustine, Newton, and, in recent years, Stephen Hawking. One could imagine that each expansion allows time for stars, planets, and life to appear; then everything collapses down to a point, resetting the clock so that the universe can rebound, expand again, and begin another existence. The modest Friedmann lacked Einstein's propensity for broad theological pronouncements, but now he could hardly ignore the religious implications of his work. “This brings to mind what Hindu mythology has to say about cycles of existence, and it also becomes possible to speak about 'the creation of the world from nothing,'” Friedmann wrote in The World as Space and Time, his 1923 book summarizing his work.

  How long such a cycle would last depends on the mass of the universe, a number that was completely unknown at the time. “All this should at present be considered as curious facts which cannot be reliably supported by the inadequate astronomical experimental material,” he wrote. But Friedmann was sufficiently intrigued that he took a crack at estimating the age of the universe, just as Einstein couldn't keep himself from speculating about its size.

  Friedmann even cited Einstein's unpublished values for the radius and density of the universe. If we live in oscillating universe, Friedmann guessed that the expanding cycle would last “of the order of 10 billion years.” Other scientists had attempted to measure the age of the Earth or the ages of the stars, but here was something utterly new. Friedmann, in his matter-of-fact manner, presumed that science could derive a time for the beginning of existence, treading onto the territory once held securely by the biblical chronologists.

  Like de Sitter, Friedmann seems to have viewed his cosmological calculations more as mathematical idealizations than as descriptions of the real universe. As he put it, “The data available to us are completely inadequate for any kind of numerical estimates and for solving what kind of world our universe is.” But the genie was out of the bottle. From the 1930s on, astronomers gained more knowledge about the overall density and dynamics of the universe and started to speak more literally about the size, age, and rate of expansion of the universe. Soon cosmic speculation knew no bounds.

  The other two Friedmann solutions are less dramatic. The concave, or saddle-shaped, universe does not contain enough matter to pull itself back together, so it keeps expanding and never turns back. The third possibility is a flat universe, one in which the geometry is just like the Euclidian planes taught in primary school. Returning to the two-dimensional analog, a flat universe is like a rubber sheet in which the local bumps and ripples average out so that the sheet as a whole resembles an even tabletop. Owing to the exact balance between space and matter, the expansion grows slower and slower, eternally approaching but never quite reaching a complete stop.

  Friedmann never explicitly described a flat universe, though it is an implicit possibility in the range of cosmological models he portrayed in his two papers. This scenario struck Einstein as the one closest to the static universe he had initially preferred. When he finally had to abandon Lambda and pick one of the dynamic cosmologies, he flirted with Friedmann's oscillating universe but finally settled on a flat geometry, which he formalized in a 1932 paper co-written with de Sitter. Over time, a majority of cosmologists followed suit, and this “Einstein-de Sitter universe” became the leading contender. Modern theorists still consider a flat universe the most appealing version and sought support for it in their speculative equations long before astronomical observations showed it to be a plausible answer. Those new observations, ironically, are the ones that have resurrected the long-derided Lambda.

  “Friedmann's papers laid the foundation for cosmology based on general relativity,” reflects MIT's Alan Guth, who helped establish the modern incarnation of the flat universe. But the revelatory nature of Friedmann's work, so clear in retrospect, was not much appreciated at the time. After the publication of the 1922 paper, Einstein wrote a dismissive reply to Zeitschrift fur Physik. “The results concerning the nonstationary world. . . appear to me suspicious. In reality it turns out that the solution given in it does not satisfy the field equations,” he wrote. A curious use of the word reality, as the two men debated their mathematical idealizations of the universe! Einstein believed that Friedmann's debunking of the static universe arose out of a conceptual error and that, when corrected, “the significance of the work is precisely that it proves this constancy.” This response struck a sour tone, suggesting Einstein's deeper objection was that Friedmann's dynamic universe did not sound the gong of divine truth.

  In December of that year, Friedmann wrote to him to explain how he arrived at his conclusions but received no response. By the time the letter arrived, Einstein was off traveling in Japan. In the immediate aftermath of his international celebrity, he was suddenly deluged with more mail and callers than ever before, and he probably never saw Friedmann's note even after he returned. Einstein habitually retreated from distractions when immersed in his work. Earlier, while he was completing general relativity, he felt so overwhelmed with mail that he snared his letters on a meat hook h
anging from the ceiling of his apartment. According to Einstein's friend Erwin Finlay-Freundlich, he burned the lot of them when the hook got full.

  Friedmann persisted, attempting to visit Einstein in Berlin, but to no avail. His ideas might have vanished into oblivion but for the intervention of Yuri Krutkov, a colleague of Friedmann's from Petrograd University. Krutkov met Einstein in Leiden in the spring of 1923 and called his attention to Friedmann's work. Forced to reconsider, Einstein revisited the arguments of his 1917 paper and finally had to admit that the unknown Russian had defeated the famous German at his own game. “My criticism. . . was based on an error in calculations. I consider that Mr. Friedmann's results are correct and shed new light,” he wrote in a follow-up letter to Zeitschrift. “It has turned out that the field equations allow not only static but dynamic. . . solutions for the space structure.” Einstein still doubted that Friedmann's equations described that elusive thing known as the “real universe.” But he was having serious second thoughts about Lambda, which he now considered “a complication of the theory, which seriously reduces its logical simplicity.” Like Newton, Einstein tried to craft a balanced and divinely beautiful universe, only to find his own equations fighting him. It was becoming increasingly evident that Friedmann was correct: general relativity required the cosmos to move.

  Friedmann was elated. “Everybody was very impressed by my struggle with Einstein and my eventual victory,” he wrote. He was especially excited that he would now find it easier to get his papers published. Nobody will ever know how far Friedmann might have continued with his incisive cosmological analysis. In the summer of 1925 he set out on a scientific ballooning experiment designed to collect meteorological and medical data; along the way, he set an altitude record of twenty-four thousand feet. Perhaps the stress of the expedition was too much for him, because he died in the fall, just thirty-seven years old. The official diagnosis was typhoid fever, although his student George Gamow believes Friedmann actually was done in by pneumonia contracted during his balloon flight. Despite his earlier response, Einstein did not follow up on Friedmann's ideas, perhaps because of his profound dislike of the concept that the universe could have emerged from a single point, the seeming implication of an expanding Friedmann universe. Fried-mann's work remained little noticed in theoretical circles for nearly a decade, and utterly unknown to the astronomers who could have helped confirm his ideas. But the expanding universe soon resurfaced from an unexpected corner.

  In the mid-1920s, a Belgian cleric named Georges Lemaitre began looking into the global implications of relativity in much the same way Friedmann had several years earlier. Like his predecessor, Lemaitre had fought in World War I—in this case, as an artillery officer in the Belgian army—before returning to the world of science and engineering. During the war he participated in fierce urban fighting and witnessed one of the first military attacks using poison gas. In other ways, Lemaitre led a very different kind of life than did Friedmann. A gregarious, kind-faced man, he enjoyed a comparatively comfortable lifestyle and close contact with some of the major players in the nascent field of cosmology. He also sampled both the religious and scientific sides of life in a way that would have been unthinkable in Friedmann's postrevolutionary Russia. After the war Lemaitre attended the University of Louvain but then entered the seminary and was ordained in 1923. Refusing to abandon his interest in mathematics and physics, he went on to study at the University of Cambridge and at the Massachusetts Institute of Technology. His studies persuaded him that the universe operates according to simple, knowable rules that may be obscured by complicated appearances. He wrote a comment in his notes that plainly expresses a sentiment both scientific and religious: “Behind objects that can be touched or looked upon, should be something hidden.”

  In England Lemaitre studied under Arthur Eddington, the redoubtable champion of Einstein's relativity, who informed him about the apparent high velocities of certain spiral nebulae, which Eddington thought might be signs of the curious reddening of light predicted by de Sitter's cosmology. The following year, Lemaitre continued his education in the United States, with a specific goal of learning more about “the astronomical consequences of the Principle of Relativity.” While at the Massachusetts Institute of Technology, he took a trip down to Washington, D.C., and attended a meeting of the National Academy of Sciences. There, he heard the American astronomer Edwin Hubble lecture on his new discoveries revealing the scale of the universe. Lemaitre also spent time at Harvard College Observatory studying under Harlow Shapley, another brilliant cosmic cartographer, and visited the Lowell Observatory and Mount Wilson Observatory, the leading centers for the study of the enigmatic spiral nebulae. These experiences bolstered Lemaitre's faith that Einstein's equations could embody the hidden rules that would explain the whole of the universe.

  In 1925, Lemaitre returned to Brussels and took a position at the University of Louvain, where he wrote up a cosmological paper, “Note on de Sitter's Universe,” that incorporated the lessons from his travels abroad. Most significant, he explored the “non-statical” nature of de Sitter's cosmology and described it as “a possible interpretation of the main receding motions of spiral nebulae.” Although he apparently had not read Friedmann's work, he then retraced it almost step by step and arrived at very similar results. Once again, he rejected Einstein's insistence on a static universe. Even with Lambda in place, Lemaitre discovered (or rather, rediscovered) that there are whole families of solutions to the equations of relativity that do not produce a static universe. In fact, the static solution hangs in precarious balance; even a slight change in physical conditions could upset the equilibrium and cause the whole system to start expanding. He pictured a universe that expanded indefinitely outward from an initial state, which prompted him to ponder what happens if you run the clock backward and consider where this expansion began. He started to think that there must have been a time when all of existence was packed together in a compact, formless blob, which at some point came undone and begat the modern universe. Slowly but surely, Lemaitre was edging past Friedmann and developing a full-fledged, physical description of a cosmos in motion.

  These fresh ideas came just as astronomers were launching into their own Great War over the size and state of the universe. Such information was vital for evaluating the mind-bending cosmologies Einstein had formulated. When he developed his cosmic model in 1917, most scientists still believed that the Milky Way, our home galaxy, was unique and all-encompassing. The stars within the Milky Way follow leisurely orbits, and our galaxy is indeed quite stable, which allowed Einstein to envision an eternal, immobile universe—but was such a picture correct? On the other hand, Einstein's universe worked only if the shape of space-time folded in on itself. That required a high density of matter spread evenly throughout an enormous cosmos—but could the universe actually be like that? And there was that century-old puzzle, Olbers's paradox, hanging in the background. Could Einstein's finite universe explain why the night sky is dark?

  By 1932, the consensus view had changed and many of the key questions were being answered. The fifteen-year period of scientific development after Einstein's cosmology paper was reminiscent of the upheaval of the early seventeenth century, when Galileo produced potent evidence in favor of the Copernican system and Kepler finally ended the cult of spherical motion. Theory fed on observation, observation fed on theory, and in the end science ended up much grander and more powerful than before. This time around, astronomers recognized the Milky Way as one of countless galaxies strewn through a vast and dynamic universe, each one composed of many billions of stars. And the cosmologists were ready to make sense of it all, to demonstrate they could explain our place among the fleeing galaxies as readily as their ancestors had put the Earth in motion among the planets.

  The path to enlightenment was achieved not just through better instruments but also through passionate philosophical debate. There is no need for a powerful telescope to see another galaxy. In fact, all it
takes to spot a galaxy is good eyesight. Go outside on a crisp fall night and run your gaze along the bent narrow “V” of stars that marks the constellation Andromeda. Just above the arc you will perceive a wispy glow, a little smaller than the full moon. That glow is the Andromeda galaxy, a near twin of the Milky Way now estimated to lie about two million light-years away; it is the most distant object visible to the unaided human eye. A decent pair of binoculars will easily reveal a dozen other galaxies, if you know what you are looking for.

  The difficulty in spotting other galaxies was one of conception more than perception. Astronomers simply could not accept the vastness of the universe, and the diminution of our place within it, that necessarily followed from accepting that the Milky Way is just one galaxy among a multitude. If anything, they had made backward progress since the eighteenth century, when William Herschel had begun cataloging the different types of luminous nebulae strewn across the sky. Herschel's argument that some of these nebulae were actually giant systems of stars, dimmed by their tremendous distance from us, carried considerable authority for a time. In the mid-nineteenth century, William Parsons, the third earl of Rosse, noticed that some of the nebulae had a lovely spiral form. A few decades later, the new technique of spectroscopy showed that the light from these spirals resembled that from stars, not from luminous gas.