Our Mathematical Universe

by Max Tegmark

  If there were astronomers at her college, they’d have argued about the so-called nebulae, diffuse cloudlike objects in the night sky, some with beautiful spiral shapes like those in van Gogh’s famous painting Starry Night. What were these things? Many astronomers dismissed them as boring gas clouds between the stars, but some had a more radical idea: that they were “island universes,” which we today call galaxies: enormous groups of stars so far away that they couldn’t be seen individually with our telescopes, appearing instead as a nebulous haze. To settle this controversy, astronomers needed to measure the distance to some nebulae. But how?

  The parallax technique, which had worked so well for nearby stars, failed for the nebulae: they were so far away that their parallax angles were too small to detect. How else can you measure large distances? If you look at a distant lightbulb with a telescope and notice that it has “100 watts” printed on it, you’re all set: you simply use the inverse-square law to calculate how far away it must be to look as bright as it does. Astronomers call such useful objects of known luminosity standard candles. Using the above-mentioned detective methods, astronomers had unfortunately discovered that stars are anything but standard, some shining a million times more brightly than the Sun and others a thousandfold more faintly. However, if you could observe a star and see that it had “4 × 1026 watts” written on it (that would be the correct label for our Sun), you’d have your standard candle and could calculate its distance just as for the lightbulb. Fortunately, nature has provided us with a particular type of stars this helpful, called Cepheid variables. Their luminosity oscillates over time as they pulsate in size, and Harvard astronomer Henrietta Swan Leavitt discovered in 1912 that their pulsation rate acts like a watt meter: the more days there are between successive pulses, the more watts of light are radiated.

  These Cepheid stars also have the advantage of being bright enough to see at vast distances (some can shine 100,000 times brighter than our Sun), and the American astronomer Edwin Hubble discovered several of them in the so-called Andromeda nebula—a Moon-size haze that you can see with your naked eye if you’re far from city lights. Using the recently completed Hooker telescope in California (its 2.5-meter mirror was the largest in the world), he measured their pulsation rates, used Leavitt’s formula to figure out how luminous they were, compared that with how bright they appeared, and calculated their distances. When he announced his answer at a 1925 conference, jaws dropped: he argued that Andromeda was a galaxy about a million light-years away, a thousandfold farther than most stars my grandma saw in her night sky! We now know that the Andromeda galaxy is even more distant than Hubble estimated, about three million light-years from us, so Hubble inadvertently continued the tradition of accidental underestimation from Aristarchos and Copernicus.

  In the years that followed, Hubble and other astronomers went on to discover ever more distant galaxies, expanding our horizons from millions to billions of light-years away from us, and we’ll push into the trillions and beyond in Chapter 5.

  What Is Space?

  So, as that kindergartner asked: does space go on forever? We can approach this question in two ways: observationally and theoretically. So far in this chapter, we’ve done the former, exploring how clever measurements have gradually revealed ever more distant regions of space, with no end in sight. However, lots of progress has been made on the theoretical front as well. First of all, how could space not go on forever? As I discussed with those kindergartners, it would be pretty weird if we reached a sign like the one in Figure 2.6, warning that we’d reached the end of space. I remember thinking about this when I was a kid: what would there be beyond the sign? To me, worrying about reaching the end of space sounded as silly as ancient seafarers worrying about falling off the end of the Earth. I therefore concluded that space simply had to go on forever and be infinite, based on pure logic. Indeed, using logical reasoning back in ancient Greece, Euclid realized that geometry was really mathematics, and that infinite 3-D space could be described with the same rigor as other mathematical structures such as sets of numbers. He developed this beautiful mathematical theory of infinite 3-D space and its geometric properties, and this was widely viewed as the only logically possible way that our physical space could be.

  In the 1800s, however, the mathematicians Carl Friedrich Gauss, János Bolyai and Nikolai Lobachevsky all discovered that there were other logical possibilities for uniform 3-D space, and Bolyai excitedly wrote to his father: “Out of nothing I have created a strange new universe.” These new spaces obey different rules: for example, they no longer have to be infinite like the space Euclid envisioned, and the angles in a triangle no longer have to add up to 180 degrees as Euclid’s formula stipulates. Imagine drawing a triangle on each of the 2-D surfaces of the 3-D shapes in Figure 2.7: its three angles will add up to more than 180 degrees for the sphere (left), exactly 180 degrees for the cylinder (middle), and less than 180 degrees for the hyperboloid (right). Moreover, the 2-D surface of the sphere is finite even though it lacks any sort of edge.

  Figure 2.6: It’s hard to imagine how space could be finite. If it could end, then what would lie beyond?

  This example shows that surfaces can break Euclid’s geometry rules if they aren’t flat. However, Gauss and the others had a more radical insight: a space can be curved all by itself, even if it isn’t the surface of anything! Suppose you’re a blind ant and want to figure out which one of the three surfaces in Figure 2.7 you’re walking around on. You feel like you’re effectively living in a 2-D space, because you have no access to the third dimension (away from your surface), but this won’t thwart your detective work: you can still define a straight line (as the shortest path between two points), so you simply sum the three angles of a triangle. For example, if you get 270 degrees, you exclaim: “Aha! It’s more than 180 degrees so I’m on the sphere!” To further impress your ant friends, you can even figure out how far you’ll need to walk in a straight line before returning to where you started. In other words, all the usual geometry business of points, lines, angles, curvature and so on can be rigorously defined by referring only to what’s in your 2-D space, without making any reference to a third dimension. This means that mathematicians can rigorously define a curved 2-D surface even if no third dimension exists: a curved 2-D space all by itself, which isn’t the surface of anything.

  Figure 2.7: If you draw triangles on these surfaces, their angles will add up to more than 180 degrees (left), exactly 180 degrees (middle) and less than 180 degrees (right), respectively. Einstein taught us that these three options are possible for triangles in our 3-D physical space as well.

  To most people, this mathematical discovery of non-Euclidean spaces probably seemed like little more than esoteric mathematical abstraction, of no practical relevance to our physical world. But then Einstein came along with his theory of general relativity, which effectively says: “We’re the ants!” Einstein’s theory allows our 3-D space to be curved—even without it having any hidden fourth dimension for it to curve within. So the question of what kind of space we inhabit can’t be settled from pure logic alone, as some Euclid fans had hoped. It can only be resolved by performing measurements—such as making a huge triangle in space (with light rays as edges, say) and checking whether the angles add up to 180 degrees. In Chapter 4, I’ll tell you about how my colleagues and I have had fun doing precisely this; the answer turns out to be about 180 degrees for universe-sized triangles, but significantly more than 180 degrees if a neutron star or a black hole fills up much of the triangle, so the shape of our physical space is more complicated than the three simple options illustrated in Figure 2.7.

  Returning to that kindergartner’s question, we see that Einstein’s theory allows space to be finite in a way that isn’t silly as in Figure 2.6: it can be finite by being curved. For example, if our 3-D space is curved like the surface of a 4-D hypersphere, then if we could travel as far as we wanted in a straight line, we’d eventually return home from the opposite dir
ection. We wouldn’t fall off the edge of our 3-D space because it has no edge, just as the ant in Figure 2.7 encounters no edge when crawling around the sphere.

  Indeed, Einstein allows our 3-D space to be finite even if it isn’t curved! The cylinder in Figure 2.7 is flat rather than curved in the mathematical sense: if you draw a triangle on a paper cylinder, its angles will sum to 180 degrees. To see this, simply cut the triangle out with a pair of scissors, and note that you can lay it flat on a table; you couldn’t do this with a paper sphere or hyperboloid without the paper tearing or crumpling. However, although the cylinder in Figure 2.7 therefore looks flat to an ant walking on a small patch of it, the cylinder nonetheless connects back on itself: the ant can return home after walking in a horizontal straight line. Mathematicians call the connectedness of a space its topology. They’ve defined flat spaces that connect back on themselves in all their dimensions, and call such a space a torus. A 2-D torus has the same topology as the surface of a bagel or a traditional donut (the kind with a hole in it). Einstein allows the possibility that the physical space we inhabit is a 3-D torus, in which case it’s both flat and finite. Or it could be infinite.

  In summary, the space we live in might go on forever and it might not—both possibilities are perfectly reasonable according the best theory we have for the nature of space, Einstein’s general relativity. So which way is it? We’ll return to this fascinating question in Chapters 4 and 5, finding evidence that space is truly infinite after all. But our pursuit of the kindergartner’s deep question raises another one: what is space, really? Although we all start our lives thinking about space as something physical, forming the very fabric of our material world, we’ve now seen how mathematicians talk of spaces as being mathematical things. To them, studying space is the same as studying geometry, and geometry is just part of mathematics. One could indeed argue that space is a mathematical object, in the sense that its only intrinsic properties are mathematical properties—properties such as dimensionality, curvature and topology. We’ll push this argument much further in Chapter 10, arguing that in a well-defined sense, our entire physical reality is a purely mathematical object.

  We’ve spent this chapter exploring our place in space, revealing a vastly larger Universe than our ancestors were aware of. To really understand what’s going on at the greatest distances we can observe with our telescopes, however, it’s not enough to explore only our place in space. We also need to explore our place in time. That’s our battle cry for the next chapter.

  THE BOTTOM LINE

  • Over and over again, we humans have realized that our physical reality is vastly larger than we’d imagined, and that everything we knew of was part of an even grander structure: a planet, a solar system, a galaxy, a galaxy supercluster, etc.

  • Einstein’s theory of general relativity allows for the possibility that space goes on forever.

  • It also allows the alternative option where space is finite without having an end, so that if you could travel far and fast enough, you’d return home from the opposite direction.

  • The very fabric of our physical world, space itself, could be a purely mathematical object in the sense that its only intrinsic properties are mathematical properties—numbers such as dimensionality, curvature and topology.

  3

  Our Place in Time

  Real knowledge is to know the extent of one’s ignorance.

  —Confucius

  The highest form of ignorance is when you reject something you don’t know anything about.

  —Wayne Dyer

  Where does our Solar System come from? My son Philip got into a heated discussion about this question when he was in second grade, which went something like this:

  “I think it was made by God,” a girl in his class said.

  “But my dad said it was made by a giant molecular cloud,” Philip interjected.

  “But where did the giant molecular cloud come from?” another boy asked.

  “Maybe God made the giant molecular cloud, and then the giant molecular cloud made our Solar System,” said the girl.

  I bet that as long as people have walked the Earth, they’ve gazed into the night sky and wondered where everything comes from. Just as in times past, there are things we know and things we don’t. We know lots about here and now, and also quite a bit about events close in space and time, such as what’s right behind us and what we ate for breakfast. Farther away and longer ago, we eventually hit the frontier of our knowledge, where our ignorance begins. In the last chapter, we saw how human ingenuity gradually pushed this knowledge frontier outward in space, expanding our realm of the known to incorporate our entire planet, our Solar System, our Galaxy, and even billions of light-years of space in all directions. Let’s now launch a second intellectual expedition, and explore how we humans have gradually pushed this frontier backward in time.

  Why doesn’t the Moon fall down? The answer to this question triggered our first push.

  Where Did Our Solar System Come From?

  As recently as four hundred years ago, this question still seemed rather hopeless. We just saw how ingenious detective work revealed the locations of the key parts visible to the naked eye: the Sun, the Moon, Mercury, Venus, Mars, Saturn and Jupiter. Diligent sleuthing by Nicolaus Copernicus, Tycho Brahe, Johannes Kepler and others also revealed the motions of these objects: our Solar System was found to be reminiscent of a clockwork, with its parts moving in precise orbits over and over again, seemingly forever. There was no indication whatsoever that the clockwork would stop one day, or that it had started at any particular time in the past. But was it really eternal? If not, where did it come from? We were still clueless.

  For the man-made clockworks for sale at the time, the laws that governed the motion of their cogwheels, springs and other parts were so well understood that one could make predictions about both the future and the past. One could predict that a clock would keep ticking at a steady rate, and also that, because of friction, it would eventually stop unless it was wound up. By studying it carefully, you might conclude that it must have been wound up within the last month, say. If there were similarly precise laws that described and explained celestial motions, then might they, too, involve some frictionlike effects that would eventually alter our Solar System, and that might also give clues to when and how it formed?

  The answer seemed to be a resounding no. Down on the ground, we’d developed a fairly good understanding of how things move through space, from hurled stones to rocks launched by Roman catapults to iron balls fired from cannons. But whatever laws governed heavenly objects seemed to be different from the laws governing things down here on the ground. For example, what about the Moon? If it’s some kind of giant rock in the sky, why doesn’t it fall down like ordinary rocks do? The classic answer was that the Moon was a heavenly object, and heavenly objects simply play by different rules. Like being immune to gravity and not falling down. Some went further and offered an explanation: heavenly objects are this way because they’re perfect. They have perfectly spherical shapes because the sphere is the perfect shape; they move in circular orbits because circles are also perfect; and falling down would be about as far from perfect as it gets. On Earth, imperfection abounds: friction slows things down, fires burn out and people die. In the heavens, on the other hand, the motions appear frictionless, the Sun doesn’t burn out and there’s no end in sight.

  This perfect reputation of the heavens didn’t hold up to closer scrutiny, however. By analyzing the measurements of Tycho Brahe, Johannes Kepler established that planetary motions weren’t circles but ellipses, which are elongated and arguably less-perfect versions of circles. Through his telescopes, Galileo saw that the Sun had its perfection tarnished by ugly black spots. And that the Moon wasn’t a perfect sphere but what looked like a place, complete with mountains and giant craters. So why didn’t it fall down?

  Isaac Newton finally answered this question by exploring an idea that was as simple as it was radi
cal: that heavenly objects obey the same laws as objects here on Earth. Sure, the Moon doesn’t fall down like a dropped rock, but might it be possible to throw an ordinary rock in such a way that it doesn’t fall down either? Newton knew that Earth rocks fall toward Earth rather than toward the much more massive Sun, and concluded that this must be because the Sun was much farther away and the gravitational attraction of an object weakens with distance. So could you hurl a rock upward so fast that it escaped Earth’s gravitational pull before this pull had time to reverse the rock’s motion? Newton personally couldn’t do it, but realized that a hypothetical supercannon should do the trick, provided that it could give the rock enough speed. As you can see in Figure 3.1, this means that the fate of a horizontally fired cannon ball depends on its speed: it crashes into the ground only if its speed is below some magic value. If you keep firing balls with ever higher speeds, they’ll travel farther and farther before landing, until you reach the magical speed where they keep their height over the ground exactly constant and never land, merely orbiting Earth in a circle—just like the Moon! Since he knew the strength of gravity near Earth’s surface from experiments with falling rocks, apples, etc., he was able to calculate what this magic speed was: a roaring 7.9 kilometers per second. Assuming that the Moon really was obeying the same laws as a cannon ball, he could similarly predict what speed it needed to have to be in a circular orbit—all that was missing was a rule for how much weaker Earth’s gravity was out there where the Moon was. Moreover, since the Moon took one month to travel around a circle whose circumference Aristarchos had figured out, Newton already knew its speed: about 1 kilometer per second, the same as for an M16 rifle bullet. Now he made a remarkable discovery: if he assumed that the force of gravity weakened like the inverse square of the distance from the center of Earth, then this magical speed that would give the Moon a circular orbit exactly matched its measured speed! He had discovered the law of gravity and found it to be universal, applying not merely here on Earth, but in the heavens as well.