Reality Is Not What It Seems: The Journey to Quantum Gravity

by Carlo Rovelli

Without the notion of fields introduced by Faraday, without the spectacular power of mathematics, without the geometry of Gauss and Riemann, this “certain physics” would have remained incomprehensible. Empowered by new conceptual tools and by mathematics, Einstein writes the equations that describe Democritus’s void and finds for its “certain physics” a colorful and amazing world where universes explode, space collapses into bottomless holes, time slows down in the vicinity of a planet, and the boundless expanses of interstellar space ripple and sway like the surface of the sea. . . . All this sounds like a tale told by an idiot, full of sound and fury, signifying nothing. And yet, instead, it is a glance toward reality. Or better, a glimpse of reality, a little less veiled than our blurred and banal everyday view of it. A reality that seems to be made of the same stuff our dreams are made of but that is nevertheless more real than our clouded daily dreaming.

Hilbert, a gentleman, never questioned Einstein’s victory, even though he was working on very similar equations at the time. In fact, he left a gentle and beautiful phrase, which captures perfectly Einstein’s difficult relationship with mathematics. And which captures, perhaps, the difficult relationship that exists generally between the whole of physics and mathematics. The math that was necessary to formulate the theory was geometry in four dimensions, and Hilbert writes: Any youngster on the streets of Göttingen understands geometry in four dimensions better than Einstein.* And yet, it was Einstein who completed the task. Why? Because Einstein had a unique capacity to imagine how the world might be constructed, to “see” it in his mind. The equations, for him, came afterward; they were the language with which to make concrete his visions of reality. For Einstein, the theory of general relativity is not a collection of equations: it is a mental image of the world arduously translated into equations.

In the end, a degree of tension between Hilbert and Einstein did develop. A few days before Einstein made his successful equation public, Hilbert had sent an article to a periodical that showed just how close he had come to the same solution—and even today historians of science are faced with doubts when trying to evaluate the respective contributions of these two giants. At some point their relations cooled; Einstein feared that Hilbert, more senior and powerful, would seek to attribute to himself too much of the merit for the construction of the theory. But Hilbert never claimed to be the first to discover general relativity, and in a world such as that of science—where often, too frequently, disputes over precedence become poisonous—the two gave a truly wonderful example of wisdom, clearing the field of all negative tension. Einstein writes a marvelous letter to Hilbert, summarizing the profound sense of the shared course they had taken: There was a moment in which something like an irritation came between us, the origin of which I no longer want to analyze. I have fought against the bitterness which it provoked in me, and have succeeded completely in doing so. I again think of you with unclouded friendship, and I ask you to do the same for me. It is really a pity if companions such as we are, who have managed to forge a path aside from the pettiness of this world, could find anything other than joy in each other’s company.3

Now, think about it; for someone who has learned from his teacher that the form of the surface of our planet is such that by walking always in a straight line we return to the point we started from, it is perhaps not so difficult to take the next obvious step and imagine that the form of the entire universe is such that, flying always in a straight line, we return to the same point of departure: a 3-sphere is a space in which “two winged knights who could fly in opposite directions would meet up on the other side.” In technical terms: the description of the geometry of Earth offered by Brunetto Latini in Li tresor is given in terms of intrinsic geometry (seen from the inside) rather than extrinsic (seen from the outside), and this is exactly the one that is suitable to generalize the notion of “sphere” from two dimensions to three. The best way of describing a 3-sphere is not to try to “see it from the outside” but rather to describe what happens when moving within it. The method developed by Gauss to describe curved surfaces, and generalized by Riemann to describe the curvature of spaces in three or more dimensions, basically amounts to Brunetto Latini’s way. That is to say, the idea is to describe a curved space not as “seen from the outside,” stating how it curves in an external space, but instead in terms of what may be experienced by somebody within that space, who is moving and always remaining within it. For instance, the surface of an ordinary sphere, as Brunetto o...

We do not know with certainty where the electron will appear, but we can compute the probability that it will appear here or there. This is a radical change from Newton’s theory, where it is possible, in principle, to predict the future with certainty. Quantum mechanics brings probability to the heart of the evolution of things. This indeterminacy is the third cornerstone of quantum mechanics: the discovery that chance operates at the atomic level. While Newton’s physics allows for the prediction of the future with exactitude, if we have sufficient information about the initial data and if we can make the calculations, quantum mechanics allows us to calculate only the probability of an event. This absence of determinism at small scale is intrinsic to nature. An electron is not obliged by nature to move toward the right or the left—it does so by chance. The apparent determinism of the macroscopic world is due only to the fact that the microscopic randomness cancels out on average, leaving only fluctuations too minute for us to perceive in everyday life.

When Einstein died, his greatest rival, Bohr, found for him words of moving admiration. When, a few years later, Bohr in turn died, someone took a photograph of the blackboard in his study. There’s a drawing on it. It represents the “box of light” of Einstein’s thought experiment. To the very last, the desire to debate, to understand more. To the very last, doubt. This permanent doubt, the deep source of science.

The bounce of a collapsing star can be very fast, if watched from down there. But—remember—time passes much slower there than outside. Seen from the outside, the process of the bounce can take billions of years. After this time, we can see the black hole explode. In the end, basically, this is what a black hole is: a shortcut to the distant future.

But the infinities of quantum field theory follow from an assumption at the basis of theory: the infinite divisibility of space. For example, to calculate the probabilities of a process, we sum up—as Feynman has taught us—all the ways in which the process could unfold, and these are infinite, because they can happen in any one of the infinite points of a spatial continuum. This is why the result can be infinite. When quantum gravity is taken into account, these infinities also disappear. The reason is clear: space is not infinitely divisible, there are no infinite points; there are no infinite things to add up. The granular discrete structure of space resolves the difficulties of the quantum theory of fields, eliminating the infinities by which it is afflicted. This is a tremendous result: on the one hand, taking quantum mechanics into account resolves the problems generated by the infinities of Einstein’s theory of gravity, that is to say, the singularities. On the other, taking gravity into account solves the problems generated by quantum field theory, that is to say, the divergences. Far from being contradictory, as they at first seemed, the two theories each offer the solution to the problems posed by the other!