Brilliant Blunders: From Darwin to Einstein - Colossal Mistakes by Great Scientists That Changed Our Understanding of Life and the Universe (36 page)

Figure 36

Even during Einstein’s lifetime, there were a few scientists who did not want to give up on the cosmological constant.
Physicist Richard C. Tolman, for instance, wrote to Einstein in 1931, “A definite assignment of Λ = 0, in the absence of experimental determination of its magnitude, seems arbitrary and not necessarily correct.” Lemaître, in addition to his general sentiment that Λ should not be rejected only because it had been introduced for the wrong reasons, had two other main motivations for wanting to keep the cosmological constant alive. First, it offered a potential solution to the discrepancy between the perceived young age of the universe (as Hubble’s observations seemed to imply) and geological timescales. In some of Lemaître’s models,
a universe with a cosmological constant could linger for a long time in a coasting state, thus prolonging the age of the cosmos. Lemaître’s second reason for championing Λ had to do with his ideas about the formation of galaxies. He conjectured that regions of higher density would be amplified and grow into protogalaxies during that coasting phase.
While this particular idea was shown not to work in the late 1960s, it did help to keep the cosmological constant on the back burner for a while.

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.

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

T
he 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.

By definition, mathematical equations are expressions or propositions asserting the equality of two quantities. Einstein’s most famous equation,
E
=
mc
²
, 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, this 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.