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

by Livio, Mario

  Arthur Eddington was another strong supporter of the cosmological constant. So much so, in fact, that at one point he declared defiantly, “Return to the earlier view [without the cosmological constant] is unthinkable. I would as soon think of reverting to Newtonian theory as of dropping the cosmical constant.” The main rationale for Eddington’s advocacy was that he thought that the repulsive gravity was the true explanation for the observed expansion of the universe. In his words:

  There are only two ways of accounting for large receding velocities of the nebulae: (1) they have been produced by an outward directed force as we have supposed, or (2) as large or larger velocities have existed from the beginning of the present order of things. Several rival explanations of the recession of the nebulae, which do not accept it as evidence of a repulsive force, have been put forward. These necessarily adopt the second alternative, and postulate that the large velocities have existed from the beginning. This might be true; but it can scarcely be called an explanation of the large velocities.

  In other words, Eddington recognized that even without the cosmological constant, general relativity allowed for an expanding universe solution. However, this solution had to assume that the cosmos started with large velocities, without providing an explanation for those particular initial conditions. The inflationary model—the idea that the universe underwent a stupendous expansion when it was only a fraction of a second old—was born out of a similar dissatisfaction with having to rely on specific initial conditions as a cause for observed cosmic effects. For instance, inflation is assumed to have puffed up the universe’s fabric so much that it flattened the cosmic geometry. At the same time, inflation is believed to have been the agent that took quantum fluctuations of subatomic size in the density of matter and inflated them to cosmological scales. These were the density enhancements that later became the seeds for the formation of cosmic structure.

  As I have noted already in chapter 9, Hoyle’s steady state model of 1948 did reproduce some of the features of inflationary cosmology. The field term that Hoyle had introduced into Einstein’s equations for the continuous creation of matter acted in many ways like a cosmological constant. In particular, it caused the universe to expand exponentially. Consequently, steady state cosmology helped keep some form of the cosmological repulsive factor in vogue for another fifteen years or so.

  When astronomer and longtime Hoyle supporter William McCrea came to summarize the then-prevailing ideas about the cosmological constant in 1971, he distinguished presciently between two possibilities: Either general relativity is a complete, self-consistent theory, or general relativity should be regarded only as one part of a more comprehensive “theory of everything” that describes the cosmos and all phenomena within it. In the first case, McCrea noted, the cosmological constant becomes a nuisance, since its value cannot be determined from within the theory itself. In the second, he argued insightfully, the value of the cosmological constant may be fixed through the connection between general relativity and other relevant branches of physics. As we shall soon see, physicists are trying to understand the nature of the cosmological constant precisely through their efforts to unify the large and the small—general relativity with quantum mechanics.

  CHAPTER 11

  OUT OF EMPTY SPACE

  If we admit that ether is to some degree condensable and extensible, and believe that it extends through all space, then we must conclude that there is no mutual gravitation between its parts, and cannot believe that it is gravitationally attracted by the sun or the earth or any ponderable matter; that is to say, we must believe ether to be a substance outside the law of universal gravitation.

  —LORD KELVIN

  The cosmological constant introduced into the physics vocabulary a repulsive gravitational force that is proportional to distance and acts over and above the ordinary gravitational attraction between masses. As with so many other physical concepts, Newton was the first to consider the effects of a similar force. In his celebrated Principia, he discussed, in addition to the normal force of gravity, a force that “increases in a simple ratio of the distance.” Newton was able to show that for this type of force, as with gravity, one could treat spherical masses as if all the mass was concentrated at their centers. What he did not do, however, was to fully examine the problem for the case in which the two forces act in tandem. Newton might have paid more attention to this scenario had he realized, or taken more seriously, the fact that his law of gravitation could not easily be applied to the universe as a whole. If one attempts to calculate the gravitational force at any point in a cosmos of infinite extent and uniform density, the computation does not yield any definite value. The situation is a bit like trying to calculate the sum of the infinite sequence 1–1+1–1+1–1 . . . The result depends on where you stop.

  Toward the end of the nineteenth century, a few physicists attempted to find a way out of this conundrum. They suggested solutions ranging from small modifications to Newton’s law of gravitation, to the introduction of more exotic concepts such as negative masses. The ubiquitous Lord Kelvin proposed, for instance, that the ether—the stuff then presumed to permeate all space—does not gravitate at all. (See his quote at the beginning of this chapter.) Eventually, all these early endeavors culminated in Einstein’s theory of general relativity and the subsequent augmentation of its equations by the cosmological constant. As we have seen, however, Einstein repudiated this term later, and except for its short-lived reincarnation as part of Hoyle’s steady state cosmology, it was essentially banished from the theory for a few decades. Astronomical observations of the late 1960s provided the impetus for the next rise of this phoenix from its ashes. Astronomers seemed to find an excess in the counts of quasars clustered around an epoch of about ten billion years ago. This overdensity could be explained if the size of the universe somehow lingered for a while around the dimensions it had at that time—about one-third of its current extent. Indeed, a few astrophysicists showed that such cosmic loitering could be obtained in Lemaître’s model, since that involved (through its employment of the cosmological constant) a leisurely coasting, quasi-static phase. Even though this particular model did not survive for long, it did draw attention to one potential interpretation of the cosmological constant: that of the energy density of empty space. This idea is so fundamental, and yet so mind boggling that it deserves some explanation.

  From the Largest to the Smallest Scales

  By definition, mathematical equations are expressions or propositions asserting the equality of two quantities. Einstein’s most famous equation, E = mc2, for instance, expresses the fact that the energy associated with a given mass (on the left-hand side of the equality sign) is equal to the product of that mass and the square of the speed of light (on the right-hand side). Einstein’s original equation of general relativity was of the following form: It had on its left-hand side a term describing the curvature of space, and on the right-hand side a term specifying the distribution of mass and energy (multiplied by Newton’s constant denoting the strength of the gravitational force). This was a clear manifestation of the essence of general relativity: Matter and energy (right-hand side) determine the geometry of space-time (left-hand side), which is the expression of gravity. When he introduced the cosmological constant, Einstein added it on the left-hand side (multiplied by a quantity that defines distances), since he thought of it as yet another geometrical property of space-time. However, if one moves this term to the right-hand side, it acquires a whole new physical meaning. Instead of describing the geometry, the cosmological term is now part of the cosmic energy budget. The characteristics of this new form of energy, however, are different from those of the energy associated with matter and radiation in two important ways. First, while the density of matter (both ordinary and the one called “dark,” which does not emit light) decreases as the universe expands, the density of the energy corresponding to the cosmological constant remains eternally constant. And if that is not strange enough, thi
s new form of energy has a negative pressure!

  Negative pressure sucks. This is not a joke; positive pressure, like that exerted by a compressed regular gas, pushes outward. Negative pressure, on the other hand, sucks inward instead of pushing outward. This property turns out to be crucial, since in general relativity, in addition to mass and energy, pressure is also a source of gravity—it applies its own gravitational force. Moreover, whereas positive pressure generates an attractive force of gravity, negative pressure contributes a repulsive gravitational force (a feature that probably makes Newton turn in his grave). This was precisely the attribute of the cosmological constant that Einstein had used in his attempt to keep the universe static. The basic symmetry of general relativity, that the laws of nature should make the same predictions in different frames of reference, implies that only the vacuum—literally empty space—can have an energy density that does not dilute upon expansion. Indeed, how can empty space dilute any further? But energy of the vacuum? Why does empty space have any energy at all? Isn’t empty space simply “nothing”?

  Not in the weird world of quantum mechanics. When one enters the subatomic realm, the vacuum is far from being nothing. In fact, it is a frenzy of virtual (in the sense that they cannot be observed directly) pairs of particles and antiparticles that pop in and out of existence on fleetingly short timescales. Consequently, even empty space can be endowed with an energy density and, concomitantly, can be a source of gravity. This is an entirely different physical interpretation from the one originally suggested by Einstein. Einstein regarded his cosmological constant as a potential peculiarity of space-time—describing the universe on its cosmic largest scales. The identification of the cosmological constant with the energy of empty space, even though mathematically equivalent, intimately relates it to the smallest subatomic scales—the province of quantum mechanics. McCrea’s observation in 1971 that one could perhaps determine the value of the cosmological constant from physics outside classical general relativity proved to be truly visionary.

  I should note that Einstein himself made one interesting attempt to connect the cosmological constant to elementary particles. In what could be regarded as his first foray into the arena of trying to unify gravity and electromagnetism, Einstein proposed in 1919 that perhaps electrically charged particles are being held together by gravitational forces. This led him to an electromagnetic constraint on the value of the cosmological constant. Apart, however, from one additional short note on the subject in 1927, Einstein never returned to this topic.

  The idea that the vacuum is not empty but, rather, could contain a vast amount of energy is not really new. It was first proposed by the German physical chemist Walther Nernst in 1916, but since he was interested primarily in chemistry, Nernst did not consider the implications of his idea for cosmology. The practitioners of quantum mechanics in the 1920s, Wolfgang Pauli in particular, did discuss the fact that in the quantum domain the lowest possible energy of any field is not zero. This so-called zero-point energy is a consequence of the wavelike nature of quantum mechanical systems, which causes them to undergo jittery fluctuations even in their ground state. However, even Pauli’s conclusions did not propagate into cosmological considerations. The first person to specifically connect the cosmological constant to the energy of empty space was Lemaître. In a paper published in 1934, not long after he had met with Einstein, Lemaître wrote, “Everything happens as though the energy in vacuo would be different from zero.” He then went on to say that the energy density of the vacuum must be associated with a negative pressure, and that “this is essentially the meaning of the cosmological constant Λ.” Figure 37 shows Einstein and Lemaître meeting in Pasadena in January 1933.

  Figure 37

  As perceptive as Lemaître’s comments were, the subject lay dormant for more than three decades until a brief revival of interest in the cosmological constant attracted the attention of the versatile Jewish Belarusian physicist Yakov Zeldovich. In 1967 Zeldovich made the first genuine attempt to calculate the contribution of vacuum jitters to the value of the cosmological constant. Unfortunately, along the way, he made some ad hoc assumptions without articulating his reasoning. In particular, Zeldovich assumed that most of the zero-point energies somehow cancel out, leaving only the gravitational interaction between the virtual particles in the vacuum. Even with this unjustified omission, the value that he obtained was totally unacceptable; it was about a billion times larger than the energy density of all the matter and radiation in the observable universe.

  More recent attempts to estimate the energy of empty space have only exacerbated the problem, producing values that are much, much higher—so high, in fact, that they cannot be considered anything but absurd. For instance, physicists first assumed naïvely that they could sum up the zero-point energies up to the scale where our theory of gravity breaks down. That is, to that point at which the universe was so small that one needs to have a quantum theory of gravity (a theory that does not exist currently). In other words, the hypothesis was that the cosmological constant should correspond to the cosmic density when the universe was only a tiny fraction of a second old, even before the masses of the subatomic particles were imprinted. However, when particle physicists carried out that estimate, it resulted in a value that was about 123 orders of magnitude (1 followed by 123 zeros) greater than the combined cosmic energy density in matter and radiation. The ludicrous discrepancy prompted physics Nobel laureate Steven Weinberg to dub it “the worst failure of an order-of-magnitude estimate in the history of science.” Obviously, if the energy density of empty space were truly that high, not only would galaxies and stars not have existed but also the enormous repulsion would have instantly torn apart even atoms and nuclei. In a desperate attempt to correct the guesstimate, physicists used symmetry principles to conjecture that adding up the zero-point energies should be cut off at some lower energy. Dismally, even though the revised estimate resulted in a considerably lower value, the energy was still some 53 orders of magnitude too high.

  Faced with this crisis, some physicists resorted to believing that a yet-undiscovered mechanism somehow completely cancels out all the different contributions to the energy of the vacuum, to produce a value of exactly zero for the cosmological constant. You’ll recognize that mathematically speaking, this is precisely equivalent to Einstein’s simple removal of the cosmological constant from his equations. Assuming that the cosmological constant vanishes means that the repulsive term need not be included in the equation. The reasoning, however, was completely different. Hubble’s discovery of the cosmic expansion quickly subverted Einstein’s original motivation for introducing the cosmological constant. Even so, many physicists regarded as unjustified the assignment of the specific value of zero to lambda for the mere sake of brevity or as remedy to a “bad conscience.” In its modern guise as the energy of empty space, on the other hand, the cosmological constant appears to be obligatory from the perspective of quantum mechanics, unless all the different quantum fluctuations somehow conspire to add up to zero. This inconclusive, frustrating situation lasted until 1998, when new astronomical observations turned the entire subject into what is arguably the most challenging problem facing physics today.

  The Accelerating Universe

  Since Hubble’s observations in the late 1920s, we knew that we live in an expanding universe. Einstein’s theory of general relativity provided the natural interpretation of Hubble’s findings: The expansion is a stretching of the fabric of space-time itself. The distance between any two galaxies increases just as the distance between any two paper chads glued to the surface of a spherical balloon would increase if the balloon were inflated. However, in the same way that the Earth’s gravity slows down the motion of any object thrown upward, one would anticipate that the cosmic expansion should be slowing, due to the mutual gravitational attraction of all the matter and energy within the universe. But in 1998 two teams of astronomers, working independently, discovered that the cosmic expansion is
not slowing down; in fact, over the past six billion years, it has been speeding up! One team, the Supernova Cosmology Project, was led by Saul Perlmutter of the Lawrence Berkeley National Laboratory, and the other, the High-Z Supernova Search Team, was led by Brian Schmidt of Mount Stromlo and Siding Spring Observatory and Adam Riess of the Space Telescope Science Institute and the Johns Hopkins University.

  The discovery of accelerating expansion came as a shock initially, since it implied that some form of repulsive force—of the type expected from the cosmological constant—propels the universe’s expansion to speed up. To reach their surprising conclusion, the astronomers relied on observations of very bright stellar explosions known as Type 1a supernovae. These exploding stars are so luminous (at maximum light, they may outshine their entire host galaxies) that they can be detected (and the evolution of their brightness followed) more than halfway across the observable universe. In addition, what makes Type 1a supernovae particularly suitable for this type of study is the fact that they are excellent standard candles: Their intrinsic luminosities at peak light are nearly the same, and the small deviations from uniformity that exist can be calibrated empirically. Since the observed brightness of a light source is inversely proportional to the square of its distance—an object that is three times farther than another is nine times dimmer—knowledge of the intrinsic luminosity combined with measurement of the apparent one allows for a reliable determination of the source’s distance.