7

In the spring of 1937, when Bohr browsed through the latest issue of the weekly journal Nature, his eyes stopped at a note signed P. A. M. Dirac. Gamow, who happened to be visiting the Danish physicist, recalled that a perplexed Bohr turned towards him and said, “Look what happens to people when they get married.” What caused Bohr’s consternation was Dirac’s unorthodox proposal, based on no less unorthodox numerological reasons, of a new picture of the universe and its evolution in time. Newton’s gravitational constant might not be constant, Dirac suggested, but decreased slowly in cosmic time.

Until that time, Dirac had worked almost exclusively in quantum theory and shown no interest in other branches of physics, such as astrophysics and cosmology. However, he was well acquainted with the recent advances in these fields and thus prepared to make his unexpected move from the mathematical theory of the micro-cosmos to the much less precise and more speculative theory of the macro-cosmos at its largest possible scale, the universe.

The mainstream cosmology that emerged in the 1930s built on Einstein’s general theory of relativity, which, as first shown by the Belgian physicist Georges Lemaître in 1927, offered an explanation of the observationally-based expansion of the universe. Four years later Lemaître (who was also a Catholic priest) went a step further, arguing that the expansion had started a finite time ago when all matter in the universe was squeezed together in a primordial “atom” of unimaginably high density. Contrary to accepted wisdom, the universe could thus be ascribed a definite age. Lemaître’s hypothesis of an exploding universe was the first version of what eventually became known as the Big Bang theory, a name that was coined in 1948. Dirac, who had met Lemaître and discussed his hypothesis with him, was among the very few physicists and astronomers in the 1930s who accepted the audacious idea of a finite-age exploding universe. His 1937 Nature note was based on the assumption that some two billion years ago at the origin of time “all the spiral nebulae were shot out from a small region of space or perhaps from a point.”

The other element in Dirac’s cosmological hypothesis related to the constants of nature that enter and define the fundamental laws of physics. Examples are Planck’s constant h, the gravitational constant G, the speed of light c, and the elementary charge e. Their numerical values can be measured but not be derived from theory, as Dirac knew from his work with the magnetic monopole. The values of the constants evidently depend on the chosen system of units and are in this respect arbitrary. Light moves at the speed of roughly 300,000 km per second, but if we use as a length unit the distance of the Earth from the Sun, the figure shrinks to 0.002. The dimensionless combinations of natural constants are not subject to this kind of arbitrariness because they include no units. For example, the ratio of the proton’s mass to the mass of the electron is 1836, a pure number. The fine-structure constant α = 0.0073 is another example.

While Eddington and a few other physicists had previously considered numbers of this kind and proposed to connect them, Dirac was interested in the exceedingly large numbers. One of these dimensionless numbers, well known at the time, is the ratio between the electrical and the gravitational forces acting between a proton and an electron. Since gravity and the electrical force vary with distance in exactly the same way, the ratio is independent of the distance between the two particles. It is a huge number, about 10³⁹. Dirac then considered the age of the universe, not in years but in the much smaller and more fundamental units of “atomic time,” as given by a combination of the speed of light and the electron’s mass and charge (more precisely e²/mc³). An atomic time unit corresponds to the time it takes light to pass an atomic nucleus. The result again becomes 10³⁹, at least approximately. What a curious coincidence! Or could it be more than just a coincidence? Dirac was convinced that the near equality of the two numbers expressed one of nature’s deep secrets.

From this conviction—and it was nothing but—Dirac concluded that the gravitational constant must be related to the age of the universe. More specifically, G must vary inversely with the time elapsed since the Big Bang, which we can write as G ∼ 1/t. In the past, gravity was thus greater than today and in the future it will become smaller. Although the variation is very slow, what matters is that G is no longer a true constant as had been taken for granted since the days of Newton, and been taken over by Einstein’s theory of gravitation. It was a radical hypothesis indeed. And this was not all, for applying similar numerological arguments Dirac further suggested that the number N of elementary particles in the visible universe must increase with the age—in this case as N ∼ t². As the universe expands and grows older, particles will be created continuously and spontaneously, apparently contradicting the fundamental law of energy conservation.

Not afraid of generalizing, in his paper of 1937 Dirac proposed what he called the Fundamental Principle, better known as the Large Numbers Hypothesis (LNH), a term he first used in 1972. According to one version of the principle, when two very large numbers of the order 10³⁹ or its square 10⁷⁸ occur in nature, they must be related by a simple mathematical relation. The following year Dirac developed his hypothesis into a proper cosmological theory, although he decided to forget about matter creation and keep only the G(t) hypothesis, meaning the decrease of G according to G ∼ 1/t.

Whether in cosmology or some other branch of science, a scientific theory must have consequences that can be tested either directly or indirectly. Dirac’s theory had, at least, two consequences, which distinguished it from other models of the universe. First, it followed directly from the G(t) hypothesis, that relative to its current value, G should decrease at a rate of only 10^-10 per year, or one part in ten billion years. This was a testable consequence if perhaps more in principle than in practice. The second consequence was more problematic. Dirac’s model accounted for the expansion, but at a slow rate that corresponded to an embarrassingly small age of the universe—just 700 million years, or considerably less than the age of the Earth! One might think that this calculation alone ruled out the model, but according to Dirac, it might be possible to account for the glaring contradiction between theory and fact.

Dirac knew that his cosmological theory was methodologically unorthodox and empirically weak. He realized that the G(t) hypothesis was incompatible with Einstein’s general theory of relativity, which he greatly admired. Nonetheless, his faith in the LNH overshadowed whatever doubts he might have had. In a lecture in Edinburgh in early 1939, he imagined that the mysteries of the universe might ultimately find their explanation in terms of the large whole numbers such as 10³⁹. “Might it not be,” he suggested, “that all present events correspond to properties of this large number, and, more generally, that the whole history of the universe corresponds to properties of the whole sequence of natural numbers?” Not only might the law of gravity change with time, but all the laws of physics might turn out to be evolutionary: “At the beginning of time the laws of Nature were probably very different from what they are now. Thus, we should consider the laws of Nature as continually changing with the epoch, instead of as holding uniformly throughout all space-time.”

As if this were not radical enough, Dirac went on to speculate that the laws of nature might also depend on position in space, meaning, for example, that the laws of electromagnetism valid in the Andromeda Galaxy are not quite the same as the laws we know here on Earth. His reason for the unusual suggestion was that one could in this way “preserve the beautiful idea of relativity that there is a fundamental similarity between space and time.” But it was just a speculation. Dirac was wise enough not to return to it.

Dirac’s cosmological theory of 1938, at first, attracted little scientific interest among physicists and even less so among astronomers. Eddington dismissed it as “unnecessarily complicated and fantastic.” With only five citations in the period between 1938 and 1947, it was unsuccessful, at least as seen from a sociological perspective. Dirac could not possibly have foreseen the rich and diverse literature his theory would eventually give rise to.

From a more philosophical perspective, it was heavily criticized by Herbert Dingle, a British astrophysicist and philosopher of science, who accused Dirac of having betrayed the true spirit of science. “Instead of the induction of principles from phenomena,” he thundered, “we are given a pseudo-science of invertebrate cosmythology, and invited to commit suicide to avoid the need of dying.” In a subsequent round of discussion in the journal Nature, Dirac sought to avoid philosophical issues and merely restated the main points of his theory.

Uncomfortable with public debate, he somewhat lamely answered that his LNH built on the constants of nature as given by observation, and, for this reason, lived up to the standards of empirical science. Dingle disagreed. He deplored that the great Dirac had fallen victim to rationalist intoxication: “I cited Prof. Dirac’s letter not as a source of infection but as an example of the bacteria which can flourish in the poisoned atmosphere; in a pure environment it would not have come to birth, and we should still have the old, incomparable Dirac.”

Philosophy apart, the impossibly low age of Dirac’s universe made it unpalatable to all astronomers and physicists (except Dirac himself, of course). The question of varying gravity was different, for it could be evaluated independently of the cosmological model and confronted with empirical data. In 1948, the Hungarian-American nuclear physicist Edward Teller pointed out that if G varied as predicted, it would have severe consequences for the history of the Earth. For the first time, paleontology was used to test a theory of the universe. This was possible because the luminosity of the Sun and its distance from the Earth depend on the value of G. The two effects taken together control the surface temperature of the Earth. If G were larger in the past, the Earth’s climate would have been considerably warmer than it is today. Based on a rough calculation, Teller inferred that on Dirac’s hypothesis the temperature in the Cambrian era, some 300 million years ago, would have been slightly above the boiling point of water. The result obviously conflicted with paleontological evidence of a rich marine life in the Cambrian, including trilobites and much more. Teller felt forced to conclude that Dirac’s hypothesis was probably wrong. Most scientists agreed.

It is uncertain how Dirac reacted to Teller’s objection, but apparently, he was unable to come up with a satisfactory answer to it. For this reason, among others, he chose to stay out of cosmology for no less than 35 years, returning to the subject only in 1973. Yet, during this long period of silence, he maintained his belief in the G(t) hypothesis, which he felt was a necessary consequence of the beautiful LNH. The predicted rate of decrease depended on the so-called Hubble time, which Dirac had originally and in accordance with astronomical observations taken to be about two billion years. The Hubble time is proportional to and of the same order as the age of the universe, the precise relationship between the two quantities depending on the chosen cosmological model. In the mid-1950s, it turned out that the Hubble time and therefore also the age of the universe was much larger than assumed.

The new and better estimate gave a rate of decrease of G that made Teller’s palaeoclimatic argument less serious. It would still be very warm in the Cambrian period, but not so warm that life could not survive. Dirac discussed the new situation in correspondence with Gamow, arguing that his pet hypothesis was still viable. In a letter of 1961, he wrote: “It was a difficulty with my varying gravitational constant that the time scale appeared too short, but I always believed the idea was essentially correct. Now that the difficulty is removed, of course I believe more than ever.” Dirac maintained this belief to the end of his life.

Gamow valued Dirac’s LNH but not his varying-G hypothesis, which he thought contradicted empirical knowledge. The climate in the past was far from the only problem, for there were astrophysical and geophysical problems as well. For example, calculations showed that, with a greater G in the past, the Sun would burn too fast and run out of hydrogen fuel within a period of just two billion years. We know this did not happen! When confronted with this and other difficulties Dirac sought to evade them, either by utilizing ad hoc hypotheses or simply by denying that the alleged difficulties were real. Gamow tried patiently to make him change his mind, but his empirical and methodological arguments fell on deaf ears. In a 1967 letter, Gamow wrote that he had “healthy respect for observation and experiment in my brain,” indirectly suggesting that the brain of his friend Dirac did not possess the same healthy respect.

It is quite clear from the Dirac-Gamow correspondence that Dirac at no time seriously considered to abandon the G(t) hypothesis. He had, once and for all, decided that it followed uniquely from the LNH and, therefore, was correct. Dirac’s dogmatic defense of the hypothesis not only featured prominently in his private correspondence, but the same kind of attitude also appeared in his publications on cosmology in the 1970s.

Most physicists and astronomers ignored Dirac’s cosmological ideas, but not all did. During the long period when Dirac remained silent, the G(t) hypothesis was advocated and further developed by Jordan in Germany, who in a general way shared many of the research interests of his colleague in Cambridge. Jordan was an early convert to Lemaître’s exploding universe, which he endorsed in a book published in 1936. When Dirac suggested his cosmology-based LNH, Jordan eagerly took it up and developed it in his own way. In 1952, he rightly pointed out that, “I am the only one who has been ready to take Dirac’s world model seriously … and to consider its more precise formulation.” He described Dirac’s LNH and the resulting G(t) hypothesis as “one of the great insights of our time,” a view not shared by other physicists. Jordan went considerably further than Dirac by applying the hypothesis not only to astrophysics and cosmology, but also to geophysics.

From about 1955 to 1970, Dirac’s hypothesis of weakening gravitation was referred to in the geological literature more frequently than in papers and books on cosmology. In the 1965 edition of his authoritative textbook Principles of Geophysics, Arthur Holmes, a highly reputed British geologist, referred favorably to the idea that “the universal constant of gravitation, G, may have decreased with time, as inferred by P. A. M. Dirac in 1938.” This was the period in which Alfred Wegener’s controversial hypothesis of drifting continents was revived and turned into the modern theory of plate tectonics. In this process, some earth scientists suggested, the Earth was expanding, thereby making the continents spread apart. Jordan and several others argued that the mechanism causing the expansion was Dirac’s decreasing gravity. In 1971, Jordan published a monograph titled The Expanding Earth, characteristically subtitled Some Consequences of Dirac’s Gravitation Hypothesis.

Dirac was aware of the expanding Earth hypothesis and its connection to his own G(t) hypothesis, but he had little interest in geology and, with a few exceptions, took no part in the discussion. And perhaps wisely so, for by the early 1970s the expanding Earth theory was abandoned by the large majority of earth scientists who instead joined the new paradigm of plate tectonics. The hypothesis of a varying G was unorthodox, but about 1970, it attracted nonetheless considerable interest and was taken up by other scientists. One of them was Fred Hoyle, the famous Cambridge physicist and cosmologist, who used to be one of the very few Ph.D. students supervised by Dirac. In the early 1970s, Hoyle suggested a new theory of gravitation, which led to the same result as Dirac’s hypothesis—a slowly decreasing gravitational constant. However, Hoyle’s theory did not rely on the LNH.

In 1973, Dirac finally returned to his cosmological G(t) theory, although now in its original formulation that included matter creation increasing with the square of cosmic time. He vaguely described the spontaneous creation of matter as “a new physical process, a kind of radioactivity, which is quite different from all the observed radioactivity.” There were two possibilities for matter creation, he explained. Either new matter might be created uniformly throughout the universe (“additive creation”) and therefore mostly in intergalactic regions; or it might be created at places where there already was a high concentration of matter, as in stars and planets (“multiplicative creation”). Dirac generally preferred multiplicative creation because it clashed less violently with general relativity. Whether creation occurred in one way or the other, G would decrease slowly at about the same rate as in his old theory. On the positive side, the age problem that had plagued his original theory disappeared, for Dirac’s model of the 1970s implied an age of the universe as high as 18 billion years.

In a rare comment on the geologists’ discussions about the history of the Earth, Dirac supported the idea of an expanding Earth, which he ascribed to matter creation and not to decreasing gravity: “According to the present theory the earth must have been expanding during geological times, owing to the continual creation of new matter inside it. The observed drifting apart of the continents supports this view.” Dirac realized that the matter creation hypothesis was problematic in several respects, but although he noted the problems, he apparently did not take them very seriously. For example, in 1973, he pointed out that if atoms had multiplied during geological time, it was hard to understand why it did not have an effect on the growth of crystals. This, he said, “might lead to insuperable difficulties.” After this comment, he gave up on the problem. Again, in which form was the new matter created? How was it that the continuous creation process mysteriously reproduced the chemical composition of matter already existing? Dirac provided no answer.

The new theory was not very successful. Not only was it based on the empirically unsupported hypothesis of gravitation decreasing in time, but it was also unable to explain the cosmic microwave background without special and rather implausible assumptions. The new Big Bang model of the universe that arose in the 1960s rested on two empirical foundations. The one was the successful explanation of the background radiation as a fossil of the Big Bang, and the other was the explanation of the large amount of helium in the universe. Dirac’s G(t) cosmology could offer no satisfactory answer to either of the phenomena, which naturally caused cosmologists in the new mainstream tradition to look at it with suspicion. After having examined Dirac’s theory, one of the mainstream cosmologists at Yale University, Gary Steigman, concluded: “The application of the LNH to physics and cosmology is fraught with ambiguity.” What he meant was that this kind of cosmology should not have been taken seriously.

Dirac listened to the criticism, but it did not sway him. He was convinced that, in the end, the LNH and the G(t) hypothesis derived from it would prove right, that is, be in agreement with experimental evidence. The crucial test would consist of measurements of the gravitational constant, which either proved or disproved that G varied at the predicted rate. Dozens of measurements and arguments of this kind were discussed in the 1960s and 1970s, but they all turned out to be as inconclusive as Teller’s old argument based on the climate of the ancient Earth. Precise astronomical measurements that could directly detect a variation in the gravitational force were needed.

Based on observations of this kind, Thomas Van Flandern, an American astronomer, announced in 1975 that he had found strong evidence for a gravitational constant decreasing in accordance with Dirac’s prediction. Dirac, who was in contact with Van Flandern, thought for a while that his hypothesis had now finally been confirmed. However, Van Flandern’s sensational announcement turned out to be premature. The optimistic conclusion was soon challenged by more precise data from the Viking landers on Mars. Based on more than 1,000 measurements of the Earth-Mars distance between 1976 and 1982, astrophysicist Ronald Hellings and his collaborators got results that squarely contradicted Dirac’s variation prediction. Commenting on the paper by the American team, New Scientist spelled out its findings in plain words: “Gravity does not vary in time.”

Eighty-one-year-old Dirac was informed of this conclusion, but it did not shake his belief that G decreased in time in agreement with the LNH. He believed that the experiments, not the theory, were flawed. As Dirac recalled when Einstein proposed in 1905 that, based on his theory of relativity, electrons gained mass when moving at high speed, experiments at first proved his prediction wrong. But Einstein insisted that the theory was correct and that the experimental result was unreliable. He was right, for improved techniques changed the verdict from refuted to confirmed. Dirac thought that history would repeat itself, and his theories would be found to be correct as well.

But history rarely repeats itself in exactly the same way, and, in this case, it did not. Today it is known that if G varies, it does so at a rate much slower than Dirac predicted. By the very nature of things—or rather by the inevitable error margins of all measurements—it can never be proved experimentally that G does not change at all.

So, whether Dirac liked it or not, the hypothesis that he so firmly believed in for more than 30 years was just wrong. And yet, his work on the LNH and its consequences was more than just a failed excursion into cosmythology, a chapter in the history of physical thought that should better be forgotten. This work gave rise to much research in the constants of nature and their possible variation in time, areas of research that are still cultivated by physicists. Even though Dirac’s G(t) is dead, his LNH is not. It is a principle that continues to be explored by some physicists. Dirac’s paper of 1937, in which he introduced the LNH concept, has received a total of 1,360 citations in the scientific literature. More than half of the citations were made after his death.

One reason for the continued appeal of the LNH is related to the recent interest in the so-called anthropic principle, the general claim that the constants of nature and the laws of physics are “designed” to allow the existence of intelligent carbon-based life in the universe. The roots of the much-discussed anthropic principle go back to the early 1960s, when the American physicist Robert Dicke reconsidered the meaning of Dirac’s LNH, arguing that the large numbers 10³⁹ and 10⁷⁸ were conditioned by biological factors. He thought that Dirac was wrong in implicitly assuming that the epoch of humans was random. According to Dicke, the present value of the age of the universe—that is, 10³⁹ in atomic time units—should be understood as a consequence of the existence of habitable planets with human life. Dicke’s argument generated a characteristically brief reply from Dirac, who pointed out that if Dicke were right, humans could exist only for a limited period: “With my assumption,” he noted, “they could exist indefinitely in the future and life need never end. There is no decisive argument for deciding between these assumptions. I prefer the one that allows the possibility of endless life.”

This was more than just a casual remark. Dirac sincerely believed that human life in the universe could never come to an end, or, rather, that it must not come to an end. Far from being a deduction from science, it was an “article of faith” for him, as he wrote in private notes from 1933. It was “an assumption that I must make for my peace of mind.” To Dirac, as to most believers in eternal life, it was not only a question of preserving life, but also of securing an “endless chain of progress.” It was not a new theme in science, for none other than Charles Darwin had said the same when faced with the consequences of the “heat death” as predicted by the physicists’ law of irreversible entropy increase. It was, Darwin said, an “intolerable thought” that humans and “all other sentient beings are doomed to complete annihilation after such long-continued slow progress.” It is unlikely that Dirac knew of Darwin’s autobiography, but he definitely agreed with the emotional desire of the illustrious naturalist.

At about the time of Dirac’s death, the theme of eternal life was taken up by a few physicists and astronomers who have developed it into the branch of science sometimes known as “physical eschatology.” Dirac would probably not have found the work of the new physical eschatologists attractive. For him, endless life was an emotional desideratum, not something that needed scientific justification.