Our Mathematical Universe

by Max Tegmark

  Figure 3.1: A cannon ball (D) fired faster than 11.2 kilometers per second escapes from Earth never to return (ignoring air resistance). If fired slightly more slowly (C), it instead enters an elliptical orbit around Earth. If fired horizontally at 7.9 kilometers per second (B), its orbit will be perfectly circular, and if fired at lower speeds (A), it eventually crashes into the ground.

  Suddenly, the pieces of the puzzle started falling into place. By combining his law of gravity with mathematical laws of motion he formulated, Newton was able to explain not merely the motion of the Moon, but also the motions of the planets around the Sun: Newton was even able to mathematically derive the fact that general orbits are ellipses rather than circles, which to Kepler had been just a mysterious, unexplained fact.

  Like most great breakthroughs in physics, Newton’s laws answered way more questions than those that prompted the discovery. For example, they explained tides: the gravitational pull from the Moon and the Sun is greater on the seawater closer to them, causing water to slosh around as Earth rotates. Newton’s laws also showed that the total amount of energy is conserved (in physics, we use the word conserved to mean “preserved” and “unchanging”), so if energy appears somewhere, it can’t have been created from nothing, but must have come from somewhere else. Tides dissipate lots of energy (some of which can be recovered by tidal power plants), but where is all this energy coming from? In large part from Earth’s rotation, which is slowed down by tidal friction: if you ever feel that there aren’t enough hours in a day, just wait 200 million years, and days will be twenty-five hours long!

  This means that friction affects even planetary motion, which kills the idea of an eternal solar system: Earth must have spun faster in the past, and you can calculate that the Earth-Moon system in its present form can’t be more than 4 to 5 billion years old, or else Earth would once have spun so fast that centrifugal forces would have torn it apart. Finally, a first clue to the origin of our Solar System: we have an estimate for the time of the crime!

  Newton’s breakthrough empowered our human minds to conquer space: he showed that we could first discover physical laws by making experiments down here on the ground, then extrapolate these laws to explain what was happening in the heavens. Although Newton first applied this idea only to motion and gravity, the concept spread like wildfire and was gradually applied to other topics such as light, gases, liquids, solids, electricity and magnetism. People boldly extrapolated not only to the macrocosmos of space, but also to the microcosmos, finding that many properties of gases and other substances could be explained by applying Newton’s laws of motion to the atoms that they were made of. The scientific revolution had begun. It ushered in both the Industrial Revolution and the information age. This progress in turn enabled us to create powerful computers that could help further advance science, solving our equations of physics and calculating answers to many interesting physics questions that had previously stumped us.

  We can make use of the laws of physics in several different ways. Often we wish to use knowledge of the present to predict the future, as with weather forecasts. However, the equations can be solved equally well in reverse, using knowledge of the present to reveal the past—such as reconstructing the exact details of the eclipse Columbus witnessed on Jamaica. A third is to imagine a hypothetical situation and use our physics equations to calculate how it will change over time—as when we simulate the launch of a rocket to Mars and figure out whether it will arrive at the desired destination. This third approach has produced new clues about the origin of our Solar System.

  Imagine a large cloud of gas in outer space: what will happen to it over time? The laws of physics predict a battle between two forces that will seal its fate: its gravity will try to crush it while its pressure will try to blow it apart. If gravity starts gaining the upper hand, compressing the cloud, it will get hotter (this is why my bike pump heats up with use), which in turn boosts its pressure, halting gravity’s advance. The cloud can remain stable for a long time while gravity and pressure balance each other out, but this uneasy truce is eventually upset. Because it’s hot, the gas cloud glows, radiating away some of the heat energy that gave it pressure. This allows gravity to compress the cloud further, and so on. By plugging the laws of gravity and gas physics into our computers, we can simulate this hypothetical battle in detail to see what happens. Eventually, the densest part of the cloud gets so hot and dense that it turns into a fusion reactor: hydrogen atoms are fused into helium, while intense gravity prevents it all from blowing apart. A star has been born. The outer parts of the nascent star are hot enough to shine intensely, and this starlight begins to blow away the rest of the gas cloud, bringing the newborn star into sight of our telescopes.

  Rewind. Replay. As the gas cloud gradually contracts, any slight rotation of the cloud gets amplified, just as a figure skater spins faster when she pulls her arms closer to her body. The centrifugal forces from this ever-faster rotation prevents gravity from crushing the gas cloud down to a point—instead, it’s crushed into a pizza shape, just as when the pizza baker near my old elementary school spun his dough to flatten it out. The main ingredients of all such cosmic pizzas are hydrogen and helium gas, but if the ingredient list also contains heavier atoms such as carbon, oxygen, and silicon, then while the center of this gas pizza forms a star, the outer parts may clump into other colder objects, planets, which become revealed once the newborn star blows away the rest of the pizza dough. Since all the spin (or angular momentum, as we physicists call it) comes from the rotation of the original cloud, it’s no surprise that all planets in our Solar System are orbiting around the Sun in the same direction (counterclockwise if you’re looking down at the North Pole), which is also the same direction that the Sun itself rotates roughly once per month.

  This explanation of our Solar System’s origins is now supported not only by theoretical calculations, but also by telescope observations of many other solar systems “caught in the act” of the birth process in various stages. Our Galaxy contains vast numbers of giant molecular clouds, gas clouds containing molecules that help them radiate away their heat, cool and contract, and we can see new stars being born in many of them. In some cases, we can even see baby stars with their pizzalike protoplanetary discs of gas still largely intact around them. The recent discovery of vast numbers of solar systems around other stars has given astronomers a wealth of new clues with which to refine our understanding of how our Solar System formed.

  If this birth process is what happened to form our Solar System, then when exactly did it happen? Just over a century ago, it was still widely believed that the Sun may have formed as recently as 20 million years ago, because if you waited much longer, the loss of energy radiated away as sunshine would have caused gravity to compress it to a much smaller size than we observe. Similarly, it was calculated that if one waited much longer than that, most of Earth’s inner heat (manifested as volcanoes and geothermal vents) would cool away.

  The mystery of what keeps the Sun warm wasn’t solved until the 1930s when nuclear fusion was discovered. But before then, the 1896 discovery of radioactivity demolished the old estimates of Earth’s age and also provided a great method for making better ones. The most common isotope of uranium atoms spontaneously decays into thorium and other lighter atoms at such a rate that half of the atoms have fallen apart after 4.47 billion years. Such radioactive decays generate enough heat to keep Earth’s core nice and toasty for billions of years, explaining why Earth is so warm even if it’s way older than 20 million years. Moreover, by measuring what fraction of the uranium atoms in a rock have decayed, you can determine the age of the rock, and in this way, some rocks from the Jack Hills of western Australia have been found to be over 4.404 billion years old. The record age for meteorites is 4.56 billion years, suggesting that both our planet and the rest of our Solar System formed in the ballpark of 4.5 billion years ago—in good agreement with those much cruder estimates from tides.

  In sum
mary, discovering and using laws of physics has given us humans a qualitative and quantitative answer to one of our ancestors’ greatest questions: How and when was our Solar System created?

  Where Did the Galaxies Come From?

  So we’ve pushed the frontier of our knowledge back to 4.5 billion years ago, when our Solar System was formed by the gravitational collapse of a giant molecular cloud. But as Philip’s classmate asked: Where did the giant molecular cloud come from?

  Galaxy Formation

  Armed with telescopes, pencils and computers, astronomers have discovered a convincing resolution to this mystery as well, although important details still remain to be filled in. Basically, the same battle between gravity and pressure that formed our pizza-shaped Solar System repeats itself on a vastly larger scale, compressing a much larger region of gas into a pizza shape millions to trillions of times heavier than the Sun. This collapse turns out to be quite unstable, so it doesn’t lead to a solar system on steroids with a single mega-star surrounded by mega-planets. Instead, it fragments into countless smaller gas clouds that form separate solar systems: thus, a galaxy has been born. Our Solar System is one of hundreds of billions in one of these pizza-shaped galaxies, the Milky Way, and we orbit around it once every couple of hundred million years, about halfway from the center (see Figure 2.2).

  Galaxies sometimes collide with one another in huge cosmic traffic accidents. This isn’t quite as bad as it sounds, as their stars mostly miss each other; in the end, gravity merges most of the stars into a new, larger galaxy. Both the Milky Way and our nearest big neighbor, Andromeda, are pizza-shaped galaxies, usually called spiral galaxies because of their beautiful spiral arm structure, which you can see in Figure 2.2. When two spiral galaxies collide, the result looks really messy at first, then settles into a roundish blob of stars known as an elliptical galaxy. This is our fate, since we’re heading for a collision with Andromeda in a few billion years—we don’t know if our descendants will call their home “Milkomeda,” but we’re pretty sure it will be an elliptical galaxy, because telescopes have imaged many other similar collisions in various stages, and the results roughly match our theoretical predictions.

  If today’s galaxies have been built up by mergers of smaller ones, then how small were the first ones? This quest to push our knowledge frontier backward in time was the topic of the very first research project I ever got really stuck on. A key part of my calculation was to figure out how chemical reactions in the gas produced molecules that could in turn reduce the gas pressure by radiating away heat energy. But every time I thought my calculations were done, I discovered that the molecule formulas I’d been using were wrong in some major way, invalidating all my conclusions and forcing me to start over. Four years after my grad school thesis advisor, Joe Silk, first got me started on this, I was so frustrated that I considered printing a custom-designed T-shirt saying I HATE MOLECULES, with my nemesis, the hydrogen molecule, crossed out by a big red stripe as on a no-smoking sign. Then luck intervened: after moving to Munich to do a postdoc, I met a friendly undergrad named Tom Abel who’d just completed a truly encyclopedic calculation of all the molecule formulas I needed. He joined our team of coauthors, and twenty-four hours later, we were done. We predicted that the very first galaxies weighed “only” about a million times as much as our Sun; we lucked out, since this finding basically agrees with the much more sophisticated computer simulations that Tom is making nowadays as a professor at Stanford.

  Our Universe Could Be Expanding

  We’ve seen that Earth’s grand drama—generation upon generation of organisms being born, interacting and dying—had a beginning about 4.5 billion years ago. Moreover, we’ve discovered that this is all part of a much grander drama, where generation upon generation of galaxies are born, interact and eventually die in a cosmic ecosystem of sorts. So could there be a third level in this dramaturgy, whereby even universes are created and die? In particular, is there any indication that our Universe itself had some sort of beginning? If so, what happened, and when?

  Why don’t the galaxies fall down? The answer to this question triggered the next push of our knowledge frontier backward in time. We saw that the Moon doesn’t fall down because it’s orbiting us at high speed. Our Universe is teeming with galaxies in all directions, and it’s pretty obvious that this same explanation doesn’t work for them. They’re not all orbiting around us. If our Universe has been eternal and essentially static, so that distant galaxies aren’t moving much, then why don’t they eventually fall toward us just as the Moon would do if you stopped it in its orbit, held it still and dropped it?

  Back in Newton’s day, people of course didn’t know about galaxies. But if they, as Giordano Bruno, contemplated an infinite static universe uniformly filled with stars, then they had at least a half-baked excuse not to worry about why it didn’t come crashing down on us: Newton’s laws showed that each star would feel a strong (in fact, infinite) force pulling on it equally hard in each and every direction, so you could argue that these opposing forces would cancel each other out and the stars would all stay put.

  In 1915, this excuse was refuted by Albert Einstein’s new theory of gravity, the general theory of relativity. Einstein himself realized that a static infinite universe uniformly filled with matter didn’t obey his new gravity equations. So what did he do? Surely, he’d learned the key lesson from Newton to boldly extrapolate, figuring out what sort of universe did obey his equations, and then asking whether there were observations that could test whether we inhabit such a universe. I find it ironic that even Einstein, one of the most creative scientists ever, whose trademark was questioning unquestioned assumptions and authorities, failed to question the most important authority of all: himself, and his prejudice that we live in an eternal unchanging universe. Instead, in what he later described as his greatest blunder, he changed his equations by adding an extra term that allowed our Universe to be static and eternal. In a double irony, it now seems as if this extra term is really there in the form of the cosmic dark energy we’ll discuss later, but with a different value that doesn’t make our Universe static.

  The person who finally had the confidence and ability to really listen to Einstein’s equations was the Russian physicist and mathematician Alexander Friedmann. He solved them for the most general case of a universe uniformly filled with matter, and discovered something shocking: most of the solutions were not static, but changing over time! Einstein’s static solution wasn’t merely an exception to typical behavior, but it was unstable, so that an almost static universe couldn’t remain that way for long. Just as Newton’s work showed that the natural state of the Solar System is to be in motion (Earth and the Moon can’t just sit still forever), Friedmann’s work showed that the natural state of our entire Universe is to be in motion.

  But what sort of motion, precisely? Friedmann discovered that the most natural state of affairs was to find yourself in a universe that’s either expanding or contracting. If it’s expanding, that means that all distant objects are moving away from one another, like chocolate chips in a rising muffin (Figure 3.2). In that case, everything must have been closer together in the past. Indeed, in Friedmann’s simplest solutions for an expanding universe, there was a particular time in the past when everything we can observe today was in the same place, creating an infinite density. In other words, our Universe had a beginning, and this cosmic birth was a cataclysmic explosion of something infinitely dense. The Big Bang was born.

  The response to Friedmann’s Big Bang was a deafening silence. Although his paper was published in one of Germany’s most prestigious physics journals and was discussed by Einstein and others, it ended up largely ignored and had essentially no impact whatsoever on the prevailing worldview at the time. Ignoring great insights is a venerable tradition in cosmology (and indeed in science more generally): we’ve already discussed the heliocentrism of Aristarchos and the distant solar systems of Bruno, and we will encounter many more examples in the
pages and chapters ahead. In Friedmann’s case, I think part of the reason he was ignored was that he was ahead of his time: in 1922 the known Universe was essentially our Milky Way Galaxy (actually, just the limited part of it that we could see), and our Galaxy is not expanding, with its hundreds of billions of stars bound into orbits by its gravitational attraction. This is the answer to question 9 on our list from the last chapter: Is the Milky Way expanding? Friedmann’s expansion applies only on scales so large that we can ignore the clumping of matter into galaxies and galaxy clusters. We can see in Figure 2.2 that the distribution of galaxies gets rather smooth and uniform on huge scales such as 100 million light-years, implying that Friedmann’s homogeneous-universe solutions apply and that all galaxies separated by such large distances should be moving away from each other. But as we discussed earlier, Hubble didn’t establish that galaxies even existed until 1925, three years later! Now time was finally ripe for Friedmann. Unfortunately time was also up for him: he died of typhoid fever that same year, only thirty-seven years old.

  Figure 3.2: Distant galaxies recede from one another like chocolate chips in a rising muffin (left): from the vantage point of any one of them, all others are moving straight away with a speed proportional to their distance. But if we think of space as stretching as the muffin dough does, then the galaxies aren’t moving relative to space, and space simply has all its distances stretched uniformly (right), as if we relabeled the tick marks on our rulers from millimeters to centimeters.