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

by Carlo Rovelli

it is also due to the fact that Dante was writing well before Newton convinced everyone that the infinite space of the cosmos was the flat one of Euclidean geometry. Dante was free of the restraints upon our intuition we have as a result of our Newtonian schooling.

In Li tresor, Brunetto explains in detail the fact that Earth is round. But he does so, curiously to the eyes of a modern reader, in terms of “intrinsic” rather than “extrinsic” geometry. That is to say, he does not write: “Earth is like an orange,” as Earth would look if seen from the outside, but writes instead: “Two knights who could gallop sufficiently far in opposite directions would meet up on the other side.”

In other words, he adopts an internal, not an external point of view: the perspective of someone who walks Earth, not of someone who looks at it from afar. At first glance it might seem like a pointless, complicated way of explaining that Earth is a ball.

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.”

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.

Einstein’s spacetime is not curved in the sense that it curves “in an external space.” It is curved in the sense that its intrinsic geometry—that is to say, the web of distances between its points, which can be observed by staying within it—is not the geometry of a flat space.

Imagine that you are an ant on the floor of the Baptistery, and are able to walk in any direction; regardless of which direction you follow to climb the wall, you would reach the ceiling at the same point of light surrounded by angels: the point of light and its angels both “surround” and “are surrounded by” the rest of the decorated interior of the Baptistery (figure 3.16

I believe that he may have been inspired by the Baptistery, not only by Coppo di Marcovaldo’s Inferno but also by its overall architecture, for his vision of the cosmos. The Paradiso reproduces its structure remarkably precisely, including the nine circles of angels and the point of light, just translating the structure from two to three dimensions.

In the end, Dante does no more than mount the pieces that were already existing into a coherent architectural whole, which follows the suggestive architecture of the Baptistery, and resolves the ancient problem of the borders of the universe.

Dante’s 3-sphere is only an intuition within a dream. Einstein’s 3-sphere has mathematical form, and follows from the theory’s equations. The effect is different. Dante moves us deeply, touching the sources of our emotions. Einstein opens a road toward the unsolved mysteries of our universe.

Einstein tries to insert the idea of the 3-sphere into his equations. Here he encounters a problem. He is convinced that the universe is fixed and immutable, but his equations tell him that this is not possible.

Einstein does not believe what his own equations are telling him.* He modifies his equations, trying to avoid the expansion. It is for this reason that he adds the term Λgab in the equation as noted earlier. But it is a further mistake: the added term is correct, but it does not change the fact that the equation predicts that the universe must be expanding.

Fourteen billion years ago, the universe was concentrated almost to a single, furiously hot point. From there it expanded in a colossal cosmic explosion, and here the term “cosmic” is not used in any rhetorical sense—it is literally a cosmic explosion. This is the “Big Bang.”

the leap made by Einstein is unparalleled: spacetime is a field; the world is made only of fields and particles; space and time are not something else, something different from the rest of nature: they are just a field among the others (figure 3.17).

The two pillars of twentieth-century physics—general relativity and quantum mechanics—could not be more different from each other. General relativity is a compact jewel: conceived by a single mind, based on combining previous theories, it is a simple and coherent vision of gravity, space, and time.

Quantum mechanics, or quantum theory, on the other hand, emerges from experiments in the course of a long gestation over a quarter of a century, to which many have contributed; achieves unequaled experimental success; and leads to applications that have transformed our everyday lives (the computer on which I write, for instance); but more than a century after its birth, it remains shrouded in obscurity and incomprehensibility.

In 1900, the German physicist Max Planck tries to compute the amount of electromagnetic waves in equilibrium in a hot box. To obtain a formula reproducing the experimental results, he ends up using a trick that does not appear to make much sense: he assumes that the energy of the electric field is distributed in “quanta,” that is to say, in small packets, little bricks of energy.

For waves of frequency ν, every quantum, or every packet, has energy: E = hν

The idea that energy could be made up of finite packets is at odds with everything that was known at the time: energy was considered something that could vary in a continuous manner, and there was no reason to treat it as if it were made up of grains. For example, the energy of a pendulum measures the amplitude of the swing. There seems to be no reason for a pendulum to oscillate only with certain determined amplitudes and not others.

In the article, Einstein argues that light truly is made up of small grains, particles of light. He considers a phenomenon that had been recently observed: the photoelectric effect. There are substances that generate a weak electric current when struck by light. That is to say, they emit electrons when light shines on them.

light carries energy (it warms us, for example), and its energy makes the electrons “jump out” of their atoms; it gives them a push.

it seems reasonable to expect that if the energy of light is scarce, namely if the light is dim, the phenomenon would not take place; and that it would take place when the energy is sufficient, namely when the light is bright. But it isn’t like this.

the phenomenon only happens if the frequency of light is high and does not happen if the frequency is low. That is to say, it happens or doesn’t happen depending on the color of light (...

Einstein uses Planck’s idea of the packets of energy, with a size that depends upon frequency, and realizes that if these packets are real, the phenomenon can be explained. It isn’t difficult to understand why. Imagine that the light arrives in the form of grains of energy. An electron will be swept out of its atom if the individual grain hitting it ha...

what determines whether your car will be dented is not the total quantity of hail that falls but the size of the individual hailstones.

In the same way, even if light is intense, which amounts to saying that there is a great deal of light packets, the electrons would not be extracted from their atoms if the individual grains of light are too small, that is, if the frequency of light is too low.

At first, Einstein’s idea that light could be granular is regarded by his colleagues as no more than youthful waywardness. Everyone commends him for his theory of relativity, but everybody judges the notion of photons to be outlandish.

the first building block of this theory has been established: there exists a fundamental granularity in all things, including light.

Experiments had shown that an atom is like a small solar system: the mass is concentrated in a heavy central nucleus, around which light electrons revolve, more or less like the planets around the sun.

Since color is the frequency of light, light is emitted by substances at certain fixed frequencies. The set of the frequencies that characterizes a given substance is known as the “spectrum” of this substance. A “spectrum” is a collection of fine lines of different hues, in which the light emitted by a given substance is decomposed (for instance, by a prism).

Spectra of numerous substances had been studied and catalogued in many laboratories at the turn of the century, and nobody knew how to explain why each substance had this or that spectrum.

Color is the speed at which Faraday’s lines vibrate, and this is determined by the vibrations of the electric charges that emit light.

studying spectra, we can understand how electrons mo...

in Newton’s mechanics, an electron can revolve around its nucleus at any speed, and hence emit light at any frequency. But then why does the light emitted by an atom not contain all colors, rather than just a few particular ones?

Bohr finds a tentative solution, by way of a strange hypothesis. He realizes that everything could be explained if the energy of electrons in atoms could assume only certain “quantized” values.

Once again, the key is a granularity, but now not for the energy of light but rather for the energy of the electrons in the atom. It begins to become clear that granularity is something widespread in nature.

Bohr makes the hypothesis that electrons can exist only at certain “special” distances from the nucleus, that is, only on certain particular orbits, the scale of which is determined by Planck’s constant h. And that electrons can “leap” between one orbit with the permitted energy to another. These are the famous “quantum leaps.”

since only certain orbits are allowed, it follows that only certain frequencies are emitted.

With these assumptions (outlandish, but simple) Bohr manages to compute the spectra of all atoms, and even to accurately predict spectra not yet observed. The experimental success of this simple model is astonishing.

Werner Heisenberg is twenty-five years old when he writes the equations of quantum mechanics, the same age as Einstein was when he wrote his three major articles.

Heisenberg sees a figure pass by. Actually, he does not see him pass: he sees him appear beneath a lamp, then disappear into the dark before reappearing beneath another lamp, and then vanishing back into the dark again. And so on, from pool of light to pool of light, until he eventually disappears altogether into the night.

What if, effectively, electrons could vanish and reappear? What if these were the mysterious “quantum leaps” that appeared to underlie the structure of the atomic spectra? What if, between one interaction with something, and another with something else, the electron could literally be nowhere.

What if the electron could be something that only manifests itself when it interacts, when it collides with something else; and that between one interaction and another, it had no precise position?

Heisenberg returns home gripped by feverish emotion and plunges into calculations. He emerges, sometime later, with a disconcerting theory: a fundamental description of the movement of particles, in which they are not described by their position at every moment, but only by their position at a particular instant—the instants in which they interact with something else.

Electrons don’t always exist. They exist when they interact. They materialize in a place when they collide with something else. The “quantum leaps” from one orbit to another constitute their way of being real: an electron is a combination of leaps from one interaction to another.

Instead of writing the position and velocity of the electron, Heisenberg writes tables of numbers (technically called “matrices”). He multiplies and divides tables of numbers representing possible interactions of the electron. And, as if from the magical abacus of a magus, the results correspond exactly with what was observed.

Despite his scientific stature, Dirac is much less well-known than Einstein. This is due, in part, to the rarefied abstraction of his science; and partly due to his disconcerting character.

For him the world is not made of things; it’s constituted of an abstract mathematical structure that shows us how things appear, and how they behave when manifesting themselves.

Dirac’s quantum mechanics is the mathematical theory used today by any engineer, chemist, or molecular biologist. In it, every object is defined by an abstract space,* and has no property in itself, apart from those that are unchanging, such as mass.