Brilliant Blunders: From Darwin to Einstein - Colossal Mistakes by Great Scientists That Changed Our Understanding of Life and the Universe

by Livio, Mario

  Convinced in 1917 that the cosmos was unchanging and static on its largest scales, Einstein had to find a way to keep the universe described by his equations from collapsing under its own weight. To achieve a static configuration with a uniform distribution of matter, Einstein guessed that there had to be some repulsive force that could balance gravity precisely. Consequently, just a little over a year after he had published his theory of general relativity, Einstein came up with what appeared, at least at first glance, to be a brilliant solution. In a seminal paper entitled “Cosmological Considerations on the General Theory of Relativity,” he introduced a new term into his equations. This term gave rise to a surprising effect: a repulsive gravitational force! The cosmic repulsion was supposed to act throughout the universe, causing every part of space to be pushing on every other part—just the opposite of what matter and energy do. As we shall soon discover, mass and energy warp space-time in such a way that matter falls together. The fresh cosmological term effectively warped space-time in the opposite sense, causing matter to move apart. The value of a new constant that Einstein introduced (on top of the familiar strength of gravity) determined the strength of the repulsion. The Greek letter lambda, Λ, denoted the new constant, now known as the cosmological constant. Einstein demonstrated that he could choose the value of the cosmological constant to precisely balance gravity’s attractive and repulsive forces, resulting in a static, eternal, homogeneous, and unchanging universe of a fixed size. This model later became known as “Einstein’s universe.” Einstein concluded his paper with what turned out to be a pregnant comment: “That term is necessary only [my emphasis] for the purpose of making possible a quasi-static distribution of matter, as required by the fact of the small velocities of the stars.” You’ll notice that Einstein talks here about “velocities of stars” and not of galaxies, since the existence and motions of the latter were still beyond the astronomical horizons at the time.

  With few exceptions, hindsight is usually 20/20. Cosmologists tend to emphasize the fact that by introducing the cosmological constant, Einstein missed a golden opportunity for a spectacular prediction. Had he stuck with his original equations, he could have predicted more than a decade before Hubble’s observations that the universe should be either contracting or expanding. This is certainly true. However, as I shall argue in the next chapter, the introduction of the cosmological constant could have constituted an equally significant prediction.

  You may wonder how Einstein could add this new repulsive term into his equations without spoiling general relativity’s other successes in explaining several perplexing phenomena. For instance, general relativity elucidated the slight shift in the orbit of the planet Mercury in each successive passage around the Sun. Einstein was, of course, aware that his cosmological constant could undermine agreement with observations, so to avoid undesired consequences, he modified his equations in such a way that the cosmic repulsion increased proportionally to the spatial separation. That is, the repulsion was imperceptible over the distance scales of the solar system, but it became increasingly appreciable over vast cosmological distances. As a result, all the experimental verifications of general relativity (which relied on measurements spanning relatively short distances) could be preserved.

  Inexplicably, Einstein did make one surprising mistake in thinking that the cosmological constant would produce a static universe. While the modification did formally allow for a static solution of the equations, that solution described a state of an unstable equilibrium—a bit like a pencil standing on its tip or a ball on the top of a hill—the slightest departure from rest resulting in forces moving the system even further away from equilibrium. One can understand this point even without the aid of sophisticated mathematics. The repulsive force increases with distance, while the ordinary attractive force of gravity decreases with distance. Consequently, while one can find a mass density at which the two forces balance each other precisely, any slight perturbation in the form of, say, a small expansion would increase the repulsive force and decrease the attractive one, resulting in accelerating expansion. Similarly, the slightest contraction would result in total collapse. Eddington was the first to point out this mistake in 1930, and he credited Lemaître with the original perspicacity. However, by then, the fact that the universe was expanding had become widely known, so this particular shortcoming of Einstein’s static universe was no longer of any interest. I should also add that in his original paper, Einstein specified neither the physical origin of the cosmological constant nor its precise characteristics. We shall return to these intriguing questions—and, indeed, to the subject of how gravity can exert a repulsive push at all—in the next chapter.

  Despite these unresolved issues, Einstein was generally pleased with having succeeded (or so he thought) in constructing a model for a static universe—a cosmos that he regarded as compatible with the prevailing astronomical thinking. Initially, he was also satisfied with the cosmological constant for another reason. The new modification to the original gravitational field equations seemed to attune the theory with some philosophical principles that Einstein had used previously in conceiving general relativity. In particular, the original equations (without the cosmological constant) appeared to require what physicists call “boundary conditions,” or specifying a set of values of physical quantities at infinite distances. This was at odds with “the spirit of relativity,” in Einstein’s words. Unlike Newton’s concepts of absolute space and time, one of general relativity’s basic premises had been that there is no absolute system of reference. In addition, Einstein insisted that the distribution of matter and energy should determine the structure of space-time. For instance, a universe in which the distribution of matter is trailing off into nothingness would not have been satisfactory, since space-time could not be defined properly without the presence of mass or energy. Yet to Einstein’s chagrin, the original equations admitted an empty space-time as a solution. He was therefore happy to discover that the static universe turned out not to need any boundary conditions at all, since it was finite and curved on itself like the surface of a sphere, with no boundaries whatsoever. A light ray in this universe came back to its point of origin before starting a new circuit. In this philosophical sense, Einstein, like Plato long before him, always recoiled from the open ended—that which philosopher Georg Wilhelm Hegel referred to as “bad infinity.”

  I realize that readers who may be a bit rusty on their general relativity would welcome a refresher course, so here is a very brief review of the core principles involved.

  Warped Space-Time

  In his theory of special relativity, which preceded his articulation of general relativity, Einstein disposed of Newton’s notion of an absolute or universal time, one that all clocks would supposedly measure. Newton’s goal was to present absolute time and absolute space symmetrically. In that spirit, he stated, “Absolute, true and mathematical time, of itself, and from its own nature, flows equally without relation to anything external.” By making the central theme of special relativity the postulate that all observers should measure the same speed for light, no matter how fast or in which direction they are moving, Einstein had to pay the price of forever linking space and time together into one interwoven entity called space-time. Numerous experiments have since confirmed the fact that the time intervals measured by two observers moving relative to each other do not agree. Most recently, by comparing two optical atomic clocks connected through an optical fiber, researchers at the National Institute of Standards and Technology managed in 2010 to observe this effect of “time dilation” even for relative speeds as low as twenty-two miles per hour!

  Given the central role of light (more generally, electromagnetic radiation) in the theory, special relativity was tailored to agree with the laws that describe electricity and magnetism. Indeed, Einstein entitled his 1905 paper that presented the theory “On the Electrodynamics of Moving Bodies.” However, as early as in 1907, he was becoming aware of the fact that spe
cial relativity was incompatible with Newton’s gravity. Newton’s gravitational force was supposed to act instantaneously across all space. The implication was that, for instance, when our Milky Way galaxy and the Andromeda galaxy will collide a few billion years from now, the change in the gravitational field due to the redistribution of mass would be felt simultaneously throughout the entire cosmos. This condition would manifestly conflict with special relativity, since it would mean that information can travel faster than light—impermissible in special relativity. Moreover, the mere concept of worldwide simultaneity would require the existence of the very universal time that special relativity carefully invalidated. While Einstein would not have used this particular example in 1907 because he was unaware of it, he fully understood the principle. To overcome these difficulties—and, in particular, to also allow his theory to apply to accelerated motion—Einstein embarked on a rather winding path that involved many missteps, but one that eventually led him to general relativity.

  General relativity is still considered by many to be the most ingenious physical theory ever articulated. The famous physicist Richard Feynman confessed once, “I still can’t see how he thought of it.” The theory was based largely on two profound insights: (1) the equivalence between gravity and acceleration, and (2) the transformation of the role of space-time from that of a passive spectator to that of a major player in the drama of universal dynamics. First, by contemplating the experience of a person who is free-falling in the gravitational field of the Earth, Einstein realized that acceleration and gravity are essentially indistinguishable. A person living inside a closed elevator on Earth, with the elevator accelerating upward continuously, may think that she lives in a place that has a stronger gravity—a bathroom scale will certainly record a weight that is higher than her normal weight. Similarly, astronauts in the space shuttle were experiencing “weightlessness” because both they and the shuttle were undergoing the same acceleration relative to the Earth. In his Kyoto Lecture in 1922, an impromptu speech to students and faculty members, Einstein described how the idea came to him: “I was sitting in a chair in the patent office in 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.”

  Einstein’s second idea turned Newton’s gravity on its ear. Gravity is not some mysterious force that acts across space, Einstein contended. Rather, mass and energy warp space-time in the same way that a person standing on a trampoline causes it to sag. Einstein defined gravity as the curvature of space-time. That is, planets move along the shortest paths in the curved space-time created by the Sun, just as a golf ball follows the undulation of the green, or a Jeep negotiates the dunes of the Sahara Desert. Light does not travel in straight lines, either, but curves in the warped neighborhood of large masses. Figure 32 shows a letter written by Einstein in 1913, as he was developing the theory. In the letter, addressed to the American astronomer George Ellery Hale, Einstein explained the bending of light in a gravitational field and the Sun’s deflection of light from a distant star. This crucial prediction was first tested in 1919 during an eclipse of the Sun. The person who organized the observations (in Brazil and on Principe Island in the Gulf of Guinea) was Arthur Eddington, and the deviations recorded by his team and by the expedition headed by the Irish astronomer Andrew Crommelin (of about 1.98 and 1.61 seconds of arc) were consistent, within the estimated observational errors, with Einstein’s prediction of 1.74 seconds of arc. (Newtonian gravity predicted half that.) Time is “curved” as well in general relativity: Clocks that are near massive bodies tick more slowly than clocks that are far away from them. Experiments have confirmed this effect, which is also taken into account routinely by GPS satellites.

  Figure 32

  Einstein’s pivotal premise in general relativity was a truly revolutionary idea: What we perceive as the force of gravity is merely a manifestation of the fact that mass and energy cause space-time to warp. In this sense, Einstein was closer, at least in spirit, to the geometrical (rather than dynamical) views of the astronomers of ancient Greece than to Newton and his emphasis on forces. Instead of being a rigid and fixed background, space-time can flex, curve, and stretch in response to the presence of matter and energy, and those warps, in turn, cause matter to move the way it does. As the influential physicist John Archibald Wheeler once put it, “Matter tells space-time how to curve, and space-time tells matter how to move.” Matter and energy become eternal partners to space and time.

  By introducing general relativity, Einstein dazzlingly solved the problem of the faster-than-light propagation of the force of gravity—the predicament that bedeviled Newton’s theory. In general relativity, the speed of transmission boils down to how fast ripples in the fabric of space-time can travel from one point to another. Einstein showed that such warps and swells—the geometrical manifestation of gravity—travel precisely at the speed of light. In other words, changes in the gravitation field cannot be transmitted instantaneously.

  What’s in a Word?

  As happy as Einstein might have been with the cosmological constant and his static universe, this satisfaction was soon to evaporate, since new scientific discoveries rendered the concept of a static universe untenable. First, there were a few theoretical disappointments, the earliest of which hit almost immediately. Just one month after the publication of Einstein’s cosmological paper, his colleague and friend Willem de Sitter found a solution to Einstein’s equations with no matter at all. A cosmos devoid of matter was clearly in contradiction to Einstein’s aspiration to connect the geometry of the universe to its mass and energy content. On the other hand, de Sitter himself was quite pleased, since he objected to the introduction of the cosmological constant from day one. In a letter to Einstein dated March 20, 1917, he argued that lambda may have been desirable philosophically but not physically. He was troubled in particular by the fact that he thought that the value of the cosmological constant could not be determined empirically. At that instant, Einstein himself was still keeping an open mind to all options. In his reply to de Sitter, on April 14, 1917, he prophetically wrote a beautiful paragraph, very reminiscent of Darwin’s famous “In the distant future . . . light will be thrown on the origin of man” (see chapter 2):

  In any case, one thing stands. The general theory of relativity allows the inclusion of Λgμv [the cosmological term] in the field equations. One day, our actual knowledge of the composition of the fixed star sky, the apparent motions of fixed stars, and the position of spectral lines as a function of distance, will probably have come far enough for us to be able to decide empirically the question of whether or not Λ vanishes. Conviction is a good motive, but a bad judge!

  As we shall see in the next chapter, Einstein predicted precisely what astronomers would achieve eighty-one years later. But in 1917, the setbacks just kept coming. Even though de Sitter’s model appeared at first blush to be static, that proved to be an illusion. Later work by physicists Felix Klein and Hermann Weyl showed that when test bodies were inserted into it, they were not at rest—rather, they flew away from one another.

  The second theoretical blow came from Aleksandr Friedmann. As I noted earlier, Friedmann showed in 1922 that Einstein’s equations (with or without the cosmological term) allowed for nonstatic solutions, in which the universe either expanded or contracted. This prompted the disappointed Einstein to write in 1923 to his friend Weyl, “If there is no quasi-static world, then away with the cosmological term.” But the most serious challenge was observational. As we have seen in chapter 9, Lemaître (tentatively) and Hubble (unequivocally) showed in the late 1920s that the universe is, in fact, not static—it is expanding. Einstein realized the implications immediately. In an expanding universe, the attractive force of gravity merely slows the expansion. Following Hubble’s discovery, therefore, he had to admit that there was no longer a need for an intricate balancing act
between attraction and repulsion; consequently, the cosmological constant could be removed from the equations. In a paper published in 1931, he formally abandoned the term, since “the theory of relativity seems to satisfy Hubble’s new results more naturally . . . without the Λ term.” Then, in 1932, in a paper Einstein published together with de Sitter, the authors concluded: “Historically the term containing the ‘cosmological constant’ Λ was introduced into the field equations in order to enable us to account theoretically for the existence of a finite mean density in a static universe. It now appears that in the dynamical case this end can be reached without the introduction of Λ.”

  Einstein was aware of the fact that without the cosmological constant, Hubble’s measured rate of expansion produced an age for the universe that was uncomfortably short compared with estimated stellar ages, but he was initially of the opinion that the problem might be with the latter. The largest contribution to the error in the observationally determined cosmic expansion rate was corrected only in the 1960s, but uncertainties of a factor of about two in the rate continued to linger until the advent of the Hubble Space Telescope. Surprisingly, however, the banished cosmological constant did return with a bang in 1998.

  You’ll notice that the language used by Einstein and de Sitter regarding the cosmological constant is benign; they note merely that in an expanding universe, it is not needed. Yet, if you read almost any account of the history of the cosmological constant, you will invariably find the story that Einstein denounced the introduction of this constant into his equations as his “biggest blunder.” Did Einstein actually say this, and if so, why?