Relativity & Quantum: The Physics Revolution - Reality Is Not What It Seems

Okay, so, like, let's talk about the start of the revolution, right? I mean, physics in the 20th century *completely* changed how we saw the universe after Newton. And, honestly, a lot of our modern tech is built on these changes. So, basically, our deeper understanding of the world rests on two main theories: general relativity and quantum mechanics. And get this, both of them *really* force us to rethink, like, seriously rethink our old ideas about the world, you know? Time and space in relativity, and then matter and energy in quantum theory.

So, in this part, I’m gonna, like, really dive into these two theories and try to explain what they're all about, show you how revolutionary they were. And, you know, that's where the amazingness of 20th-century physics begins. Really, getting into this stuff, understanding it, it's like, it's a total adventure.

These two theories, relativity and quantum mechanics, well, they're, like, the foundation for building a quantum theory of gravity, which we're still working on, and, yeah, they're totally crucial for moving forward.

Alright, let's talk about Albert. So, Einstein's dad, he built power plants in Italy, which is kinda cool. When Einstein was a kid, Maxwell's equations, they'd only been around for a few decades, but Italy was already deep into the Industrial Revolution, and those turbines and transformers his dad made, they were based on those equations. So, you could see, like, how powerful this new physics was.

Now, Albert, he was, like, a rebel, you know? His parents sent him to high school in Germany, but he just thought the whole system was way too rigid, too focused on, like, the military. He just couldn't stand the authority, so he dropped out. He went to Pavia, Italy, with his parents and, well, just kinda hung out for a while. Then he went to Switzerland to study, but he didn’t actually get into Zurich Polytechnic at first. And after college, he couldn't find a research job, so, um, he got a job at the patent office in Bern, just to be with his girlfriend.

Now, that job, it didn’t exactly need someone with a physics degree, but it gave him, like, tons of time to think and work independently, which is what he'd been doing since he was a kid. He'd be reading Euclid's "Elements," Kant's "Critique of Pure Reason," not the stuff they were teaching in school. You know, you can't get anywhere new by just following everyone else.

So, when he was twenty-five, Einstein sent three papers to the "Annalen der Physik" – and each one was, like, Nobel Prize-worthy. Each one was, like, a pillar for understanding the world. The first one, I mentioned it before, the young Albert calculated the size of atoms, and, after, like, twenty-three centuries, proved that Democritus was right: that matter *is* made of tiny particles.

The second paper, the most famous one, well, that's where he introduced relativity. And that's what we're going to be talking about here.

Actually, there are two kinds of relativity. What Einstein sent in that envelope was the paper explaining the first one, what we now call "special relativity." Before we get to his *most* important theory, general relativity, we have to talk about special relativity first because it explains, like, the structure of time and space.

Special relativity is, like, super subtle and conceptually difficult, maybe even harder to understand than general relativity. So, if you get to these next few pages and it feels, like, you're in deep water, don’t give up. This theory was the first one to show us that Newton’s view of the world wasn’t just missing something; it needed to be, like, totally changed, in a way that went against common sense. It was the first *real* leap, changing how we, like, instinctively think about the world.

So, Newton and Maxwell’s theories, they seemed to, like, contradict each other in a subtle way. Maxwell's equations give you a speed, the speed of light. But Newton's mechanics, it doesn't really work with constant speeds because his equations involve acceleration, not speed. In Newtonian physics, speed can only be relative to something else. Galileo said the Earth is moving relative to the Sun, even if we don’t feel it, because what we usually call "speed" is the speed "relative to the Earth." So, we say speed is relative, meaning it doesn't make sense to talk about the speed of an object by itself; it's always the speed of one object relative to another. That's the physics that students learned in the 19th century, and still learn today. But if that's true, what's light's speed relative to, according to Maxwell's equations?

One idea was that there's this, like, universal substance, and the speed of light is relative to *that*. But Maxwell's theory didn't seem to have anything to do with this substance. And experiments at the end of the 19th century trying to measure the Earth’s speed relative to this, you know, supposed substance, they all failed.

Einstein said that no experiment really helped him; he found the right path just by thinking about this big contradiction between Maxwell’s equations and Newtonian mechanics. He asked himself if there was a way to make Newton and Galileo's main discovery agree with Maxwell's theory.

And that's how Einstein made this, like, mind-blowing discovery. To understand it, imagine all the past, present, and future events (relative to right now, while you're reading this) arranged in a diagram.

Einstein's discovery was that, uh, that diagram is wrong. It should really look like *this* instead.

Between the past and the future of an event, like, right now, when you're reading this, there’s this, like, "intermediate region," a "stretched present," a region that's, like, neither past nor future. That's what special relativity discovered.

This, like, intermediate region that's neither past nor future? It's there, it’s a, it's a very short amount of time, it depends on where the event is happening relative to you, like in the diagram. The further away the event is, the longer this stretched present lasts. Where you are, like, a few feet from your face, this neither-past-nor-future region lasts only nanoseconds, practically zero. That's much less time than we can notice. On the other side of the ocean, it lasts a thousandth of a second, still way below the threshold of what we can perceive. But on the Moon, this stretched present lasts a few seconds, and on Mars, it's, like, a quarter of an hour! So, we can say that, right now on Mars, there are events that have already happened and events that haven't happened yet, and then this, like, fifteen minutes where things are neither in the past nor in the future.

These events are elsewhere. We never realize it because, around us, this "elsewhere" is so short, we can't perceive it. But it's real.

That’s why it’s hard to have a smooth conversation between Earth and Mars. Say I'm on Mars and you're on Earth. I ask you a question, and you answer as soon as you hear it, but your answer doesn't reach me until a quarter of an hour after I asked the question. That quarter of an hour is neither in the past nor in the future relative to when you answered. The important thing that Einstein realized about nature is that this quarter of an hour is unavoidable. We can’t get rid of it. It’s woven into the fabric of space-time. We can't shorten it, just like we can't send a letter to the past.

It's weird, but that's how the world is. It's just as weird as people in Sydney being upside down, but that’s how it is. Once you get used to the facts, they become normal and obvious. It's the structure of space and time that makes it this way.

So, it doesn't make sense to say that something is "happening" on Mars, because "now" doesn't exist, you know?

So, Einstein realized that "absolute simultaneity" doesn't exist. There isn't a "now" that’s happening throughout the universe. What happens in the universe can’t be described as a series of "nows" one after the other. It has a more complicated structure, as we saw in the diagram. That diagram shows space-time in physics: a set of past and future events, and also events that are neither past nor future. These events don't happen in an instant; they last for a certain amount of time.

In Andromeda, this stretched present lasts two million years. Anything that happened in those two million years is neither in the past nor in the future for us. If some advanced, friendly civilization in Andromeda decides to send a fleet of spaceships to visit us, it doesn't make sense to ask if the fleet has left "now." The only thing that makes sense is when we receive the first signal from the fleet, and from *that* moment on, not before, the fleet is in our past.

This structure of space-time that young Einstein discovered in 1905, well, it has real consequences. The fact that time and space are so closely linked, it meant that Newton's mechanics had to be cleverly rebuilt, which Einstein did quickly in 1905 and 1906. One of the first results was that just like space and time fused into the unified concept of space-time, electric and magnetic fields also merged into a single thing, what we now call the electromagnetic field. And Maxwell’s complex equations that described these two fields, well, they became much simpler when you used this new language.

There was another, huge consequence. In this new mechanics, "energy" and "mass" became one thing, just like time and space, and electric and magnetic fields. Before 1905, there were two seemingly rock-solid laws: the conservation of mass and the conservation of energy. The first one had been proven by chemists, mass doesn’t change in chemical reactions. The second one, the conservation of energy, it came straight from Newton's equations, and it was seen as one of the most solid laws. But Einstein realized that energy and mass are two sides of the same coin, just like electric and magnetic fields are two aspects of the same field, and space and time are two sides of the same thing, space-time. So, mass by itself isn't conserved. Energy, as we understood it then, isn't conserved either. One can be transformed into the other, and there’s only *one* conservation law, not two. What’s conserved is the total mass and energy, not either of them individually. There had to be some process that could turn energy into mass, or mass into energy.

Einstein quickly calculated how much energy you could get by turning a gram of matter into energy, and that's where we get the famous formula E = mc². Since the speed of light, c, is a *huge* number, c² is even bigger, so the energy you get from turning a gram of matter into energy is, like, enormous, as big as millions of bombs exploding all at once. Enough energy to light up a city or power a country's factories for months, or, on the other hand, enough to destroy a city like Hiroshima and kill hundreds of thousands of people in a second.

This theoretical result from young Einstein brought humanity into a new era: the nuclear era, full of new possibilities and new dangers. Thanks to this young rebel’s unconventional mind, we have the tools to bring light to ten billion Earth families in the future, to travel to other planets, or to hurt each other and destroy the Earth. It’s up to us, what we choose, what leaders we trust.

Today, Einstein's structure of space-time is completely understood. It's been tested in labs again and again, and it holds up. Our understanding of time and space isn't the same as it was in Newton's time. Space doesn’t exist separately from time. In that extended space, there's no special part you can call "space now." Our intuitive understanding of the present, that all events are "now" happening in the universe, that’s just something we believe because we're ignorant, because we can't perceive those tiny time intervals. It's an illogical conclusion based on our limited experience.

It’s like thinking the Earth is flat, that's an illusion, we imagine the Earth is flat because our senses are limited, because we're short-sighted. If we lived on a tiny planet a few miles across, like the Little Prince, we’d easily realize we’re living on a sphere. If our brains and senses were sharper, if we could easily perceive nanoseconds, we wouldn't have this idea of a universal "now." We'd easily realize there’s an intermediate region between the past and the future. We’d realize that saying "here and now" makes sense, but that taking "now" as something shared by the whole universe, that doesn't make sense. Just like asking if our galaxy is "above or below" Andromeda doesn't make sense because "up" and "down" only make sense on the Earth's surface, not in the universe. There's no "up" or "down" in the universe. In the same way, there’s no "before" or "after" for two events in the universe. The structure of time and space all woven together, that’s what physicists call "space-time."

When Einstein’s paper came out, everything was clear, and it was a huge shock to the physics world. The obvious conflict between Maxwell's equations and Newtonian physics was well known, but no one knew how to fix it. Einstein’s solution was so simple, it blew everyone away. There’s a story about a serious professor at the University of Krakow who came running out of his office, waving Einstein’s paper and shouting, "A new Archimedes has been born!"

Even though Einstein’s leap in 1905 was amazing, we haven't even talked about his real masterpiece yet. His greatest achievement was the second theory of relativity, general relativity, which he published ten years later, when he was thirty-five.

General relativity is, like, the most beautiful theory physicists have ever created, and it’s the first pillar of quantum gravity, which is, like, the core of this whole thing. That's where the *real* magic of 20th-century physics begins.

After publishing special relativity, Einstein became a well-known physicist, and he got invited to a lot of universities. But one thing kept bothering him: special relativity didn’t really mesh with the theory of gravity. He realized this while writing reviews of his theory, and he wanted to figure out if Newton’s, you know, *big* theory of universal gravitation should be reconsidered to fit with relativity.

The origin of the problem is easy to understand. Newton had explained why things fall and why planets orbit, and he imagined this force of attraction between all things, "gravity." But he never really understood *how* this force could attract distant objects without anything in between. Like we’ve seen, Newton himself suspected that something was missing from the idea of forces acting between objects without touching. If the Earth is attracting the Moon, there had to be something between them that could carry this force. And after a couple centuries, Faraday found the answer, not for gravity, but for the electromagnetic force: the field. The electromagnetic field can carry electromagnetic forces.

So, it made sense that gravity had to have its own field, like Faraday’s lines of force. By analogy, the force of gravity between the Sun and the Earth, or between the Earth and falling objects, was clearly also due to a field, the gravitational field. The solution that Faraday and Maxwell found for what carries forces had to apply to gravity as well as to the electric force. There had to be a gravitational field and equations similar to Maxwell’s equations that could describe the movement of Faraday’s lines of gravitational force. This was clear to anyone smart enough at the beginning of the 20th century. Which, of course, meant it was only clear to Albert Einstein.

In Einstein's dad's power plants, the electromagnetic field could push the rotors, and Einstein had been fascinated by that since he was young. He started working on the gravitational field, looking for the math that could describe it. He thought hard about the problem and took ten years to solve it. Ten years of intense work, trying, failing, getting confused, having brilliant ideas and wrong ones, publishing articles with incorrect equations and even more mistakes and stress. Finally, in 1915, he finished the paper with the full answer and called it "General Relativity." His masterpiece was born. Lev Landau, the Soviet Union's most brilliant theoretical physicist, called it "the most beautiful theory."

It’s not hard to see why it's beautiful. Einstein not only created the math for the gravitational field, and wrote the equations that describe it, but he also explored the other *deepest* unsolved mystery in Newton's theory and combined the two.

Newton came back to Democritus's idea that objects move in space. This space had to be a huge empty container, a solid box that could hold the universe, with a huge scaffold where objects move in straight lines until an outside force makes them change direction. But what was this "space" that held the world made of? What *is* space?

The concept of space seems natural to us, but that’s just because we're so used to Newtonian physics. If you think about it carefully, empty space isn't really our intuitive experience. For two thousand years, from Aristotle to Descartes, Democritus's idea that space is a special thing separate from objects wasn’t taken for granted. For Aristotle and Descartes, objects had extension, it was one of their properties. If there were no objects to be extended, there’d be no extension. I can pour the water out of a glass, and then the air fills it. Have you ever seen a *really* empty glass?

Aristotle explained that if there’s nothing between two objects, then there’s nothing. How could there be something (space) that’s also nothing? What exactly *is* the space that particles move in? Is it a thing, or is it nothing? If it’s nothing, then it doesn’t exist, and we could do without it. If it’s something, then the only thing it does is just sit there and do nothing, right?

Since ancient times, the idea of empty space swaying between existence and non-existence has bothered thinkers. Democritus himself made empty space the base of his atomic world, but he didn't really explain the problem. He said that empty space was something "between being and non-being": "Democritus supposed the full and the empty, calling one being and the other non-being," Simplicius commented. Atoms exist, space doesn't exist. But it's an existing non-existence. It couldn't be harder to understand.

Newton brought back Democritus’s idea of space. He said space was God's sensorium, trying to solve the problem that way. No one really understood what Newton’s "God’s sensorium" meant, and maybe Newton didn't either. Einstein definitely didn't believe in God, (with or without a sensorium), except as a joking hypothesis. He thought Newton’s explanation of the nature of space was completely unbelievable.

Newton fought hard to overcome the resistance of scientists and philosophers to bringing back Democritus's idea of space. At first no one took it seriously. Criticism slowly died down only when his equations showed their power and always predicted the right results. But the doubts about the logic of Newton’s idea of space never stopped, and Einstein, who read philosophy, knew this well. Ernst Mach, a philosopher whom Einstein admired, pointed out the conceptual problems with Newton's idea of space, and Mach himself didn’t believe in the existence of atoms, by the way.

So, Einstein asked not one problem, but two. First, how do we describe the gravitational field? Second, what *is* Newton's space?

And here’s where Einstein showed his amazing genius, one of the most brilliant moments in the history of thought: What if the gravitational field *is* Newton's mysterious space? What if Newton's space is just the gravitational field? This incredibly simple, beautiful, and brilliant idea is general relativity.

The world isn’t made of space, particles, electromagnetic fields, and gravitational fields. It’s only made of particles and fields, and nothing else. There’s no need to add space as an extra ingredient. Newton's space *is* the gravitational field. Or, to say the same thing in another way, the gravitational field *is* space.

But unlike Newton’s flat, still space, since the gravitational field is a field, it moves and undulates, and it follows certain equations, like Maxwell’s fields and Faraday’s lines of force.

It's a huge simplification of the world. Space isn't separate from matter anymore. It's also a material component of the world, like the electromagnetic field. It's a real thing that can wave, ripple, bend, and twist.

We’re not held inside an invisible solid scaffolding. We’re inside a huge, moving mollusk (Einstein’s metaphor). The Sun curves the space around it. The Earth doesn’t orbit the Sun because of some mysterious attraction at a distance. It moves in a straight line in curved space. Like a bead rolling around in a funnel, there’s no mysterious force from the center of the funnel; it’s the curved shape of the funnel walls that makes the bead rotate. Planets orbit the Sun, and objects fall, because the space around them is curved.

To be more precise, what curves isn’t space, it's space-time, the space-time that Einstein had proved ten years earlier wasn’t a series of instants, but a structured whole.

The idea was there, and now Einstein had to find the equations that could make the idea solid. How could he describe the curve of space-time? Einstein was lucky: mathematicians had already solved the problem.

Carl Friedrich Gauss, the prince of mathematics, he'd already done the math to describe curved surfaces, like the surface of a mountain, or like. Then he got a brilliant student to generalize this math to curved spaces in three dimensions or more. This student, Bernhard Riemann, wrote a long doctoral thesis that seemed completely useless.

Riemann’s result was that the properties of a curved space (or space-time) in any number of dimensions can be described by a certain mathematical object, which we call the Riemann curvature, and which we represent with the letter R. So, the curvature R is zero on a flat surface, like a plain. It’s not zero where there are valleys and hills, and it has the greatest value at the top of a mountain, where it’s most uneven or most curved. Using Riemann’s theory, you can describe the shape of curved spaces in three or four dimensions.

After a lot of effort, and asking for help from friends who were better at math than he was, Einstein learned Riemann’s math, and he wrote an equation where R is proportional to the energy of the matter. That is, where there’s matter, space is more curved. That was the answer, an equation similar to Maxwell’s equations, but for gravity instead of the electric force. The equation is only half a line long, it’s that simple. An insight, that space can curve, turned into an equation.

But the equation leads to a rich universe. This amazing theory has led to a series of dreamlike predictions that sounded like the ravings of a madman but that have all been proven true. Even at the beginning of the 1980s, almost no one took these predictions seriously, and then, one by one, experiments have confirmed them. Let’s look at a few.

At first, Einstein recalculated how objects like the Sun curve the space around them, and how this curve affects the movement of planets. He found that planets move in ways that are basically the same as what Kepler and Newton’s equations predict, but not exactly. Near the Sun, the curve in space has a stronger effect than Newton’s force. Einstein calculated the movement of Mercury, since it’s the closest planet to the Sun, so the difference between his and Newton’s theory is greatest. He found a difference: the point in Mercury’s orbit that’s closest to the Sun moves 0.43 seconds of arc per year more than Newton’s theory predicted. That’s a very small difference, but it’s big enough for astronomers to see. By comparing the two predictions with what astronomers had seen, the conclusion was clear: Mercury moves in the way that Einstein predicted, not the way Newton predicted. Mercury, the messenger of the gods, followed Einstein, not Newton.

Einstein’s equations describe how space curves near stars. Because of this curve, light can bend. Einstein predicted that the Sun would bend the light around it. Experiments were done in 1919, and the bending of light was measured, and it matched the prediction perfectly.

But not only does space curve, time does too. Einstein predicted that time passes faster higher up on the Earth, and slower closer to the ground. And it’s true. Now there are extremely accurate clocks in a lot of labs, and they can measure this strange effect even if there’s only a few inches of difference in height. Put one clock on the floor and another one on a table, and the clock on the floor will measure less time than the one on the table. Why? Because time isn't uniform and still. It stretches or shrinks depending on how close it is to matter. The Earth, like other matter, curves space-time and slows down time near it, even if it's only a little. But two twins who live near the sea and in the mountains will find that when they meet again, one of them is older than the other.

Okay, believe it or not, the fact that a ball thrown up into the air falls down is due to the same thing, it goes higher up to "add time," because time passes at a different speed there. In both cases, the paths of airplanes and balls are straight lines in curved space.

But the theory predicts far more than just these tiny effects. Stars burn as long as they have enough hydrogen for fuel, and then they slowly stop. When the pressure caused by the heat can't support the remaining matter, it collapses under its own weight. When a star that’s big enough does this, the matter gets crushed to the point that space bends so much that it creates a hole, which is called a black hole.

When I was in college, black holes were seen as unbelievable predictions of this weird theory. Now hundreds of them have been observed and are being studied in detail by astronomers. One of them, a million times the mass of the Sun, is right in the middle of our galaxy. We can watch stars orbiting it, and some of them are destroyed by its terrible gravity because they get too close.

Also, the theory predicts that space can ripple like the surface of the sea, waves similar to the electromagnetic waves in TVs. You can see these "gravitational waves" in the skies with double stars that emit gravitational waves, lose energy, and slowly get closer to each other.

Gravitational waves created by two black holes were directly seen by antennas on Earth, and the announcement in 2016 made the world stop and listen again. Einstein's weird theory had been confirmed again.

And, the theory predicts that the universe is expanding, and that it was born in a Big Bang 14 billion years ago, a topic that I'll talk about more later on.

All these rich and complex things, the bending of light, the correction of Newton’s gravity, the slowing down of clocks, black holes, gravitational waves, the expanding universe, the Big Bang, they all come from understanding that space isn't just a still container. It has its own dynamics and "physics," like the matter and fields that are contained in it. Democritus would have smiled to see his idea of space have such a big future. He called space "non-being" and used "being" for matter. For "non-being, the void" he thought "there is a physics and substance of its own." How right he was.

Without the idea of fields that Faraday introduced, without the power of math, without the geometry of Gauss and Riemann, this "special physics" would still be incomprehensible. With the help of new concepts and math, Einstein wrote the equations that describe Democritus's void. His "special physics" discovered a colorful and amazing world where the universe is expanding, space is collapsing into bottomless holes, time slows down near planets, and the endless interstellar space ripples like the surface of the sea...

All this is like a tale told by an idiot, full of sound and fury, signifying nothing. But, it’s a glimpse toward reality. Or rather, a glimpse of reality, clearer than our usual dim and blurry vision. Reality seems to be made of the same stuff as our dreams, but it's more real than our cloudy dreams.

And all this came from one basic intuition: that space-time and the gravitational field are the same thing.

I can’t resist writing the simple equation here, even if most of my readers can’t understand it. But I hope they can at least see its beautiful simplicity:

Rab, which depends on the Riemann curvature, it indicates how curved space-time is; Tab represents the energy of the matter; and G is the constant that Newton discovered: the constant that determines the strength of gravity.

And that’s how a new vision and a new equation were born.

Before we keep talking about physics, I want to stop and talk about math for a second. Einstein wasn't a great mathematician. He himself said he had a hard time with math. He wrote back to a nine-year-old girl named Barbara who asked him about her troubles with math. It sounds like a joke, but Einstein wasn't kidding. He needed help with math. He needed students and friends like Marcel Grossmann to patiently explain the math to him. But his intuition as a physicist was amazing.

In the last year he was constructing the theory, Einstein realized he was competing with one of the greatest mathematicians, David Hilbert. Einstein gave a lecture in Göttingen, and Hilbert was there. Hilbert immediately realized that Einstein was about to make a big discovery. He grasped the main points and tried to go beyond Einstein to be the first to write the equations of the new theory that Einstein was slowly putting together. The race to the finish line between the two giants was intense, and it would be decided in just a few days. Einstein gave a public lecture almost every week in Berlin, each time suggesting a different equation, terrified that Hilbert would find the answer before he did, and each time the equation was wrong. Finally, at the last minute, just barely ahead of Hilbert, Einstein found the right equation and won the race.

Hilbert was a gentleman, and even though he had written very similar equations at the same time, he never questioned Einstein’s victory. In fact, he left a beautiful quote that precisely describes Einstein’s difficulties with math, and maybe even the difficulties that exist generally between physics and math. Hilbert wrote:

"Every young person in the streets of Göttingen understands more about four-dimensional geometry than Einstein. But it was Einstein who did the work."

Why was that? Because Einstein had this unique ability to imagine how the world is made, to "see" it in his mind, and then the equations followed. The equations were the language that expressed his insight into reality. For Einstein, general relativity wasn't just a pile of equations. It was a mental picture of the world that had been hard-won and then translated into equations.

The idea behind the theory is that space-time curves. If space-time only had two dimensions, and we lived on a plane, it would be easy to imagine what "physical space curving" means. It would mean that the physical space where we live isn’t like a flat table, but like the surface of hills and valleys. But the world where we live has more than two dimensions, it has three. And really, when we add time, it has four dimensions. Imagining curved four-dimensional space is more complicated, because we can’t experience the "larger space" where space-time curves in our everyday lives. But Einstein could easily imagine the soft, crushable, stretchable, twistable universe where we live. It was thanks to this clear imagination that Einstein was the first one to finish the theory.

In the end, there was a certain amount of tension in the relationship between Hilbert and Einstein. Hilbert sent an article to a journal, showing that he was very close to the same answer, just a few days before Einstein published the correct equations, and even today, historians of science are hesitant to evaluate the contributions of the two scientific giants individually. But at a certain point, the tension between them eased. Einstein had feared that Hilbert, older and more authoritative than he was, would give himself more credit for constructing the theory, but Hilbert never claimed to have been the first to discover general relativity.

Einstein wrote a wonderful letter to Hilbert that summarized the important meaning of what they had done together:

"There’s been a little unpleasantness between us, the causes of which I don’t want to analyze. I’ve been fighting the pain it caused, and I’ve completely won. I feel toward you in the old friendly way, and I’m asking you to feel the same way toward me. It’s really too bad if two real friends, who have been able to somewhat free themselves from this contemptible world, aren’t able to appreciate each other."

So, two years after publishing the equations, Einstein decided to use them to describe the space of the entire universe, to study the largest scales of the cosmos, and here he had another amazing idea.

For thousands of years, humans had been asking themselves if the universe is finite or infinite. Each hypothesis ran into tough problems. An infinite universe didn't seem reasonable: for example, if the universe is infinite, somewhere there has to be a reader like you reading the same book (infinity is so vast that there aren't enough combinations of atoms for objects to be different). In fact, there wouldn’t just be one, but an infinite number of readers exactly like you. But if the universe has limits, what's at the boundary? If there’s nothing on the other side, what does the boundary even mean? In the 6th century BC, Archytas, a Pythagorean philosopher from Taranto, wrote:

"If I found myself on the edge of the farthest heavens, where the fixed stars are, could I stretch my hand or my staff out beyond the heavens? It would be absurd not to be able to do so, but if I can, then there’s something outside, whether it’s matter or space. And I can get further in that way, until I reach the edge, continually asking the same question, if there will always be space where I can stretch my staff."

The two absurd alternatives, an infinite space and a universe with a fixed boundary, both seemed unreasonable.

But Einstein found a third way: The universe could be finite, but at the same time without boundaries. How could that be? Like the surface of the Earth, it isn’t infinite, but it doesn’t have boundaries. It happens naturally if things can curve. In general relativity, three-dimensional space can curve, so our universe can be finite but without a boundary.

On the surface of the Earth, if I travel in a straight line, I don't keep going forever. I come back to where I started. The universe could be constructed in the same way: if I got in a spaceship and kept going in the same direction, I would circle the universe and eventually return to Earth. A three-dimensional space that is finite but without boundaries like this is called a three-dimensional sphere.

To understand the geometry of a three-dimensional sphere, let’s go back to an ordinary sphere, the surface of a ball. To show the Earth’s surface from an airplane, we can draw the continents in two disks.

Residents of the southern hemisphere are in a certain sense "surrounded" by the northern hemisphere, because no matter which way he tries to leave his hemisphere, he ends up in the other. It's the same the other way around: each hemisphere surrounds the other and is surrounded by the other. A three-dimensional sphere can be represented in a similar way, but with one more dimension: two balls completely attached together along their surface.

Leaving one sphere, you enter the other, just like leaving one disk that represents the Earth and entering the other. Each sphere surrounds the other and is surrounded by the other. Einstein’s idea was that space could be a three-dimensional sphere, finite in volume but without any boundary. The idea of the three-dimensional sphere was the solution that Einstein proposed in 1917 to the problem of the boundaries of the universe. This paper launched modern cosmology, the study of the entire visible universe at its largest scales. The discovery of the expansion of the universe, the Big Bang theory, the question of the origin of the universe, and many other discoveries came from this.

I can also make one more observation about Einstein’s three-dimensional sphere. Incredible as it may seem, the same idea had already been conceived by another genius, from a very different cultural background, Dante Alighieri. In the third part of his great poem, The Divine Comedy, Dante presents a grand vision