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.

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, 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.