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

by Carlo Rovelli

The fact that space and time are intimately connected, as in figure 3.2, implies a subtle restructuring of Newton’s mechanics, which Einstein rapidly completes in 1905 and 1906.

the electric field and the magnetic fields fuse together in the same way, merging into a single entity that today we call the “electromagnetic field.”

The concepts of “energy” and “mass” get combined in the same way as time and space, electric and magnetic fields are fused together, in the new mechanics.

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. This implies that mass, by itself, is not conserved; and energy—as it was conceived at the time—is not independently conserved, either.

What is conserved is the sum of mass and energy, not each separately. Processes must exist that transform energy into mass, or mass into energy.

Since the speed of light, c, is a very large number, and c2 an even greater number, the energy obtained transforming one gram of mass is enormous; it is the energy of millions of bombs exploding at the same time—enough energy to illuminate a city and power the industries of a country for months, or, conversely, capable of destroying in a second hundreds of thousands of human beings in a city such as Hiroshima.

“Space” does not exist independently from time. In the expanded space of figure 3.2, there is no particular slice having a better claim than others to be called “space now.”

The present is like the flatness of Earth: an illusion. We imagined a flat Earth because of the limitations of our senses, because we cannot see much beyond our own noses. Had we lived on an asteroid of a few kilometers in diameter, like Antoine de Saint-Exupéry’s Little Prince, we would have easily realized we were on a sphere.

Einstein’s triumph is the second theory of relativity, the theory of general relativity, published ten years later, when he was thirty-five.

It’s clear, at this stage, to any reasonable person, that the force of gravity must have its Faraday lines as well. It’s clear also, by analogy, that the force of attraction between the sun and Earth, or between Earth and falling objects, must be attributed to a “field”—in this case a “gravitational field.”

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.

Instead of simply inventing the mathematical form of the gravitational field and seeking to devise the equations for it, Einstein fishes out the other unresolved question in the furthest depths of Newton’s theory and combines the two questions.

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.

If between two things there is nothing, Aristotle reasoned, then there is nothing. How can there be at the same time something (space) and nothing? What is this “empty space” within which particles move? Is it something, or is it nothing? If it is nothing, it doesn’t exist, and we can do without it.

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?

The world is not made up of space + particles + electromagnetic field + gravitational field. The world is made up of particles + fields, and nothing else; there is no need to add space as an extra ingredient.

But unlike Newton’s space, which is flat and fixed, the gravitational field, by virtue of being a field, is something that moves and undulates, subject to equations—like Maxwell’s field, like Faraday’s lines.

We are not contained within an invisible, rigid scaffolding: we are immersed in a gigantic, flexible mollusk (the metaphor is Einstein’s).

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).

A little more precisely, what curves is not space but spacetime—the spacetime that, ten years previously, Einstein himself had shown to be a structured whole rather than a succession of instants.

Einstein’s only problem was to find the equations to make it concrete. How to describe this bending of spacetime? And here Einstein is lucky: the problem had already been solved by the mathematicians.

The student, Bernhard Riemann, produced a ponderous doctoral thesis of the kind that seems completely useless. 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. The equation is the analogue of the Maxwell equations, but for gravity rather than electricity.

Even up to the beginning of the 1980s, almost nobody took the majority of these fantastical predictions entirely seriously. And yet, one after another, they have all been verified by experience.

To begin with, Einstein recalculates the effect of a mass like the sun on the curvature of the space that surrounds it, and the effect of this curvature on the movements of the planets. He finds the movements of the planets as predicted by Kepler’s and Newton’s equations, but not exactly: in the vicinity of the sun, the effect of the curvature of space is stronger than the effect of Newton’s force.

He finds a difference: the point of the orbit of Mercury closest to the sun moves every year 0.43 seconds of arc more than that predicted by Newton’s theory. It is a small difference, but within the scope of what astronomers were able to measure, and comparing the predictions with the observations of astronomers, the verdict is unequivocal: Mercury follows the trajectory predicted by Einstein, not the one predicted by Newton.

Einstein’s equation, then, describes how space curves very close to a star. Due to this curvature, light deviates. Einstein predicts that the sun causes light to curve around it.

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.

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.

The distances between meridians are shorter the more northerly you are; therefore it is better to head northward, to shorten the route

believe it or not, a ball thrown upward falls downward for the same reason: it “gains time” moving higher up, because time passes at a different speed up there. In both cases, airplane and ball follow a straight trajectory in a space (or spacetime) that is curved (figure 3.10).

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.

One of these black holes, with a mass a million times greater than the sun, is located at the center of our galaxy—we can observe stars orbiting around it. Some, passing too close, are destroyed by its violent gravity.

Further still, the theory predicts that space ripples like the surface of the sea, and that these ripples are waves similar to the electromagnetic ones that make television possible. The effects of these “gravitational waves” can be observed in the sky on binary stars: they radiate such waves, losing energy and slowly falling toward each other.

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. . .

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.

Hilbert immediately understood that Einstein was in the process of making a major discovery, grasped the idea, and tried to overtake Einstein—and be the first to write the correct equations of the new theory Einstein was slowly building. The sprint to the finish line between the two giants was a nail-biting affair, eventually decided by a matter of just a few days.

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.

To imagine a curved space in four dimensions is more complicated, because in our habitual perception, we do not have an intuition of a “larger space” within which spacetime can curve.

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.

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.

Einstein finds a third way: the universe can be finite and at the same time have no boundary. How? Just as the surface of Earth is not infinite, but does not have a boundary either, where it “ends.”

And in the theory of general relativity, of course, three-dimensional space can also be curved. Consequently, our univ...

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 di...

Notice that an inhabitant of the Southern Hemisphere is in a certain sense “surrounded” by the Northern Hemisphere, since in whichever direction he goes to exit his hemisphere, he will always arrive in the other one. But the contrary is obviously true as well: each hemisphere “surrounds,” and is surrounded by, the other.

When we leave one ball, we enter into the other, just as when we leave one of the disks, in the representation of the globe, we enter into the other disk. Each ball “surrounds” and is surrounded by the other. Einstein’s idea is that space could be a 3-sphere: something with a finite volume (the sum of the volume of the two balls), but without borders.

There is one more observation that I would like to make about Einstein’s 3-sphere. However incredible it might seem, the same idea had already been conceived by another genius, from an entirely different cultural universe: Dante Alighieri, Italy’s greatest poet. In the Paradiso, the third part of his major poem, the Divine Comedy (La Divina Commedia), Dante offers a grandiose vision of the medieval world, calqued on the world of Aristotle, with the spherical Earth at its center, surrounded by the celestial spheres

he looks even higher—and what does he see? He sees a point of light surrounded by immense spheres of angels, that is to say, by another immense ball that, in his words “surrounds and is at the same time surrounded by” the sphere of our universe! Here are Dante’s verses from Canto XXVII of the Paradiso: Questa altra parte dell’Universo d’un cerchio lui comprende si come questo li altri: “This other part of the universe surrounds the first like the first surrounds the others.”

The first to notice that the Paradiso describes the universe as a 3-sphere was an American mathematician, Mark Peterson, in 1979. In general, scholars of Dante are not very familiar with 3-spheres.