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

by Carlo Rovelli

But Einstein realizes that energy and mass are two facets of the same entity, just as the electric and magnetic fields are two facets of the same field, and as space and time are two facets of the one thing, spacetime.

We would have realized that saying “here and now” makes sense, but that saying “now” to designate events “happening now” throughout the universe makes no sense.

It is like asking whether our galaxy is “above or below” the galaxy of Andromeda: a question that makes no sense, because “above” or “below” has meaning on the surface of Earth, not in the universe. There isn’t an “up” and a “down” in the universe. Similarly, there isn’t always a “before” and an “after” between two events in the universe.

But despite the outcry provoked by the steps forward made by Einstein in 1905, we are not yet at his masterpiece. Einstein’s triumph is the second theory of relativity, the theory of general relativity, published ten years later, when he was thirty-five. The theory of general relativity is the most beautiful theory produced by physics, and the first of the pillars of quantum gravity. It is at the heart of the narrative of this book. Here the real magic of twentieth-century physics begins.

But something troubles him: special relativity does not square with what was known about gravity.

There must be a gravitational field, and some equations analogous to Maxwell’s, capable of describing how “Faraday’s gravitational lines” move. In the first years of the twentieth century, this is clear to any sufficiently reasonable person; that is to say, only to Albert Einstein.

This space had to be a large, empty container, a rigid box for the universe. An immense scaffolding in which objects run in straight lines, until a force causes them to curve. But what is this “space,” which contains the world, made of? What is space?

To us the idea of space seems natural, but it is our familiarity with Newtonian physics that makes it so. If you think about it, empty space is not part of our experience.

Thus, Einstein addresses not one but two problems. First, how can we describe the gravitational field? Second, what is Newton’s space?

And it’s here that Einstein’s extraordinary stroke of genius occurs, one of the greatest flights in the history of human thinking: what if the gravitational field turned out actually to be Newton’s mysterious space? What if Newton’s space was nothing more than the gravitational field? This extremely simple, beautiful, brilliant idea is the theory of general relativity.

It is a momentous simplification of the world. Space is no longer different from matter. It is one of the “material” components of the world, akin to the electromagnetic field. It is a real entity that undulates, fluctuates, bends, and contorts.

We are not contained within an invisible, rigid scaffolding: we are immersed in a gigantic, flexible mollusk (the metaphor is Einstein’s). The sun bends space around itself, and Earth does not circle around it drawn by a mysterious distant force but runs straight in a space that inclines, like a bead that rolls in a funnel: there are no mysterious forces generated by the center of the funnel; it is the curved nature of the funnel wall that guides the rotation of the bead. Planets circle around the sun, and things fall, because space around them is curved (figure 3.6).

Riemann’s result was that the properties of a curved space (or spacetime) in any dimension are described by a particular mathematical object, which we now call “Riemann curvature,” and indicate with the letter R. If you think of a landscape of plains, hills, and mountains, the curvature R of the surface is zero in the plains, which are flat—“without curvature”—and different from zero where there are valleys and hills; it is at its maximum where there are pointed peaks of mountains, that is to say, where the ground is least flat, or most “curved.” Using Riemann’s theory, it is possible to describe the shape of curved spaces in three or four dimensions.

In words: spacetime curves more where there is matter. That is it.

But it is not only space that curves: time does too. Einstein predicts that time on Earth passes more quickly at higher altitude, and more slowly at lower altitude. This is measured, and also proves to be the case. Today we have extremely precise clocks, in many laboratories, and it is possible to measure this strange effect even for a difference in altitude of just a few centimeters. Place a watch on the floor and another on a table: the one on the floor registers less passing of time than the one on the table. Why? Because time is not universal and fixed; it is something that expands and shrinks, according to the vicinity of masses: Earth, like all masses, distorts spacetime, slowing down time in its vicinity. Only slightly—but two twins who have lived respectively at sea level and in the mountains will find that, when they meet up again, one will have aged more than the other (figure 3.8).

But the predictions of the theory go well beyond these minute effects. Stars burn as long as they have available hydrogen—their fuel—then die out. The remaining material is no longer supported by the pressure of the heat and collapses under its own weight. When this happens to a large enough star, the weight is so strong that matter is squashed down to an enormous degree, and space curves so intensely as to plunge down into an actual hole. A black hole.

This rich and complex range of phenomena—bending of rays of light, modification of Newton’s force, slowing down of clocks, black holes, gravitational waves, expansion of the universe, the Big Bang—all of this follows from understanding that space is not a dull, fixed container but possesses its own dynamic, its own “physics,” just like the matter and the other fields it contains. Democritus himself would have smiled with pleasure, had he been able to see that his idea of space would turn out to have such an impressive future. It is true that he termed it “non-being,” but what he meant by “being” (δέν) was matter; and he wrote that his “non-being,” the void, nevertheless “has a certain physics [ϕÚσιν] and a substantiality of its own.”* How right he was.

And all this is only the result of an elementary intuition: that spacetime and the gravitational field are one and the same thing. And a simple equation that I can’t resist copying out here, even if most of my readers will certainly not be able to decipher it. I do so anyway, in the hope that they might be able to catch a glimpse of its beautiful simplicity: Rab – ½ Rgab + Λgab = 8πG Tab In 1915, the equation was simpler still, because the term “+ Λgab,” which Einstein added two years later (and which I discuss in the next section) did not yet exist.* Rab depends on Riemann’s curvature, and together with ½ Rgab, represents the curvature of spacetime; Tab stands for the energy of matter; G is the same constant that Newton had found, the constant that determines the strength of the force of gravity.

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.

Two absurd alternatives—the absurdity of an infinite space, and the absurdity of a universe with a fixed border—didn’t seem to leave any reasonable choice between them. But Einstein finds a third way: the universe can be finite and at the same time have no boundary.

On the surface of Earth, if I were to keep walking in a straight line, I would not advance ad infinitum: I would eventually get back to the point I started from. Our universe could be made in the same way: if I leave in a spacecraft and journey always in the same direction, I fly around the universe and eventually end up back on Earth. A three-dimensional space of this kind, finite but without boundary, is called a “3-sphere.”

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.”

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.

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.

There is a way of understanding the curvature of space from within it, and without looking at it from outside, that is important for what follows. Imagine you are at the North Pole and walk southward until you reach the equator, carrying with you an arrow pointing ahead. Once you reach the equator, turn to the left without changing the direction of the arrow. The arrow still points south, which is now to your right. Advance a little toward the east along the equator and then turn again toward the north—again without changing the direction of the arrow, which will now be pointing behind you. When you reach the North Pole again, you have executed a closed circuit—a “loop,” as it is termed—and the arrow does not point in the same direction as when you started out (figure 3.14). The angle through which the arrow has turned in the course of the loop measures the curvature.

Whether or not such influence occurred, I believe that this example demonstrates how great science and great poetry are both visionary, and may even arrive at the same intuitions. Our culture is foolish to keep science and poetry separated: they are two tools to open our eyes to the complexity and beauty of the world.

But let’s return to 1917, when 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. It isn’t difficult to understand why. Everything attracts, therefore the only way for a finite universe not to collapse on itself is to be expanding: just as the only way to prevent a football from falling to the ground is to kick it upward. It either goes up or falls down—it can’t stay still, halfway up in the air.

A few years later, Einstein is forced to give up: it is his theory that is right, not his reservations about it.

Astronomers realize that all galaxies are indeed moving away from us.

But 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).

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. The size of the packets, he assumes, depends on the frequency (that is, the color) of the electromagnetic waves. For waves of frequency ν, every quantum, or every packet, has energy: E = hν This formula is the first of quantum mechanics; h is a novel constant, which today we call the “Planck constant.” It fixes how much energy there is in each packet of energy, for radiation of frequency (color) ν. The constant h determines the scale of all quantum phenomena.

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.

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 has a great deal of energy. What matters is the energy of each grain, not the number of grains. If, as in Planck’s hypothesis, the energy of each grain is determined by frequency, the phenomenon will occur only if frequency is sufficiently high, that is to say if the individual grains of energy are sufficiently large, independently from the total amount of energy that’s around. It is like when it hails: what determines whether your car will be dented is not the total quantity of hail that falls but the size of the individual hailstones.

Light falls on a surface like a gentle hail shower. To comprehend how light may be simultaneously an electromagnetic wave and a swarm of photons will require the entire construction of quantum mechanics. But the first building block of this theory has been established: there exists a fundamental granularity in all things, including light.

Color is the speed at which Faraday’s lines vibrate, and this is determined by the vibrations of the electric charges that emit light. These charges are the electrons that move inside the atoms. Therefore, studying spectra, we can understand how electrons move around nuclei.

In fact, the whole thing seemed impossible, because 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? Why are atomic spectra not a continuum of colors, instead of just a few separate lines?

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.

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.” The frequency at which the electron moves on these orbits determines the frequency of the emitted light, and since only certain orbits are allowed, it follows that only certain frequencies are emitted.

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? What if always having a precise position is something that is acquired only if one is substantial enough—large and heavy like the man that passed by a little while ago, like a ghost in the dark, and then disappeared into the night . . . ?

This is a second cornerstone of quantum mechanics, its hardest key: the relational aspect of things. 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. When nothing disturbs it, an electron does not exist in any place. 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. These are the first fundamental equations of quantum mechanics. From here on, these equations will do nothing but work, work, work. Up until now, incredible as it may seem, they have never failed.

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. Its position and velocity, its angular momentum and its electrical potential only acquire reality when it collides—interacts—with another object. It is not just its position that is undefined, as Heisenberg had recognized: no variable of the object is defined between one interaction and the next. The relational aspect of the theory becomes universal.

The theory also gives information on which value of the spectrum will manifest itself in the next interaction, but only in the form of probabilities. 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.

Dirac’s quantum mechanics thus allows us to do two things. First, to calculate which values a physical variable may assume. This is called “calculation of the spectrum of a variable”; it captures the granular nature of things. When an object (atom, electromagnetic field, molecule, pendulum, stone, star, and the like) interacts with something else, the values computed are those that its variables can assume in the interaction (relationism). The second thing that Dirac’s quantum mechanics allows us to do is to compute the probability that this or that value of a variable appears at next interaction. This is called “calculation of an amplitude of transition.” Probability expresses the third feature of the theory: indeterminacy, the fact that it does not give unique predictions, only probabilistic ones.

This is Dirac’s quantum mechanics: a recipe for calculating the spectra of the variables, and a recipe for calculating the probability that one or another value in the spectrum appears during an interaction. That’s it. What happens between one interaction and the next is not mentioned in the theory. It does not exist.

For a certain number of decades, it was as if it were Christmas every day for physicists: for every new problem, there was an answer that followed from the equations of quantum mechanics, and it was always the correct answer. One example of this will suffice.

Not only are the particles in a certain sense diffused in space like fields, but the fields interact like particles. The notions of fields and particles, separated by Faraday and Maxwell, end up merging in quantum mechanics.

The way this happens in the theory is elegant: the equation of Dirac determines the values a variable can take. Applied to the energy of Faraday’s lines, they tell us that this energy can take on only certain values and not others. Since the energy of the electromagnetic field can take on only certain values, the field behaves like a set of packets of energy. These are precisely the quanta of energy introduced by Planck and Einstein thirty years earlier. The circle closes, and the story is complete. The equations of the theory, written by Dirac, account for the granular nature of light, which Planck and Einstein had intuited.

The general form of quantum theory compatible with special relativity is thus called “quantum field theory,” and forms the basis of today’s particle physics. Particles are quanta of a field, just as photons are quanta of light. All fields display a granular structure in their interactions.

During the course of the twentieth century, the list of fundamental fields was repeatedly updated, and today we have a theory called the “Standard Model” of elementary particles, which describes almost all we see, with the exception of gravity, in the context of quantum field theory.* The development of this model occupied physicists for a good part of the last century, and represents in itself a wonderful adventure of discovery.