Our Mathematical Universe

by Max Tegmark

  Figure 2.2: Every time we humans have managed to zoom out to larger scales, we’ve discovered that everything we knew was part of something greater: our homeland is part of a planet (left), which is part of a solar system, which is part of a galaxy (middle left), which is part of a cosmic pattern of galactic clustering (middle right), which is part of our observable Universe (right), which may be part of one or more levels of parallel universes.

  There’s no better guarantee of failure than convincing yourself that success is impossible, and therefore never even trying. In hindsight, many of the great breakthroughs in physics could have happened earlier, because the necessary tools already existed. The ice-hockey equivalent would be missing an open goal because you mistakenly think your stick is broken. In the chapters ahead, I’m going to share with you striking examples of how such confidence failures were finally overcome by Isaac Newton, Alexander Friedmann, George Gamow and Hugh Everett. In that spirit, this quote by physics Nobel laureate Steven Weinberg resonates with me: “This is often the way it is in physics—our mistake is not that we take our theories too seriously, but that we do not take them seriously enough.”

  Let’s first explore how to figure out the size of the Earth and the distances to the Moon, the Sun, stars and galaxies. I personally find it to be one of the most flavorful detective stories ever, and arguably the birth of modern science, so I’m eager to share it with you as an appetizer before the main course: the latest breakthroughs in cosmology. As you’ll see, the first four examples involve nothing more complicated than some measurements of angles. They also illustrate the importance of letting yourself be puzzled by seemingly everyday observations, since they may turn out to be crucial clues.

  Figure 2.3: During a lunar eclipse, the Moon passes through the shadow cast by Earth (as seen above). Over two millennia ago, Aristarchos of Samos compared the size of the Moon to the size of the Earth’s shadow during a lunar eclipse to correctly deduce that the Moon is about four times smaller than the Earth. (Time-lapse photography by Anthony Ayiomamitis)

  The Size of Earth

  As soon as sailing caught on, people noticed that when ships departed over the horizon, their hulls disappeared before their sails. This gave them the idea that the surface of the ocean was curved and that Earth was spherical, just as the Sun and Moon appeared to be. Ancient Greeks also found direct evidence of this by noticing that Earth cast a rounded shadow on the Moon during a lunar eclipse, as you can see in Figure 2.3. Although it’s easy to estimate the size of Earth from the ship-sail business.1 Eratosthenes obtained a much more accurate measurement over 2,200 years ago by making clever use of angles. He knew that the Sun was straight overhead in the Egyptian city of Syene at noon on the summer solstice, but that it was 7.2 degrees south of straight overhead in Alexandria, located 794 kilometers farther north. He therefore concluded that traveling 794 kilometers corresponded to going 7.2 degrees out of the 360 degrees all around Earth’s circumference, so that the circumference must be about 794 km × 360°/7.2° ≈ 39,700 km, which is remarkably close to the modern value of 40,000 km.

  Amusingly, Christopher Columbus totally bungled this by relying on subsequent less-accurate calculations and confusing Arabic miles with Italian miles, concluding that he needed to sail only 3,700 km to reach the Orient when the true value was 19,600 km. He clearly wouldn’t have gotten his trip funded if he’d done his math right, and he clearly wouldn’t have survived if America hadn’t existed, so sometimes being lucky is more important than being right.

  * * *

  1Earth’s radius is approximately d2/2h, where d is the greatest distance at which you can see a sail of height h from sea level.

  Distance to the Moon

  Eclipses have inspired fear, awe and myths throughout the ages. Indeed, while stranded on Jamaica, Columbus managed to intimidate natives by predicting the lunar eclipse of February 29, 1504. However, lunar eclipses also reveal a beautiful clue to the size of our cosmos. Over two millennia ago, Aristarchos of Samos noticed what you can see for yourself in Figure 2.3: when Earth gets between the Sun and the Moon and causes a lunar eclipse, the shadow that Earth casts on the Moon has a curved edge—and Earth’s round shadow is a few times larger than the Moon. Aristarchos also realized that this shadow is slightly smaller than Earth itself, because Earth is smaller than the Sun, but correctly accounted for this complication and concluded that the Moon is about 3.7 times smaller than Earth. Since Erathostenes had already figured out the size of Earth, Aristarchos simply divided it by 3.7 and got the size of the Moon! To me, this was the moment when our human imagination finally got off the ground and started conquering space. Countless people before Aristarchos had looked at the Moon and wondered how big it was, but he was the first to figure it out. And he managed to do it with mental power rather than rocket power.

  One scientific breakthrough often enables another, and in this case, the size of the Moon immediately revealed its distance. Please hold your hand up at arm’s length and check which things around you can be blocked from view by your pinkie. Your little finger covers an angle of about one degree, which is about double what you need to cover the Moon—make sure to check this for yourself the next time you do some lunar observing. For an object to cover half a degree, its distance from you needs to be about 115 times its size, so if you’re looking out your airplane window and can cover a 50-meter (Olympic-size) swimming pool with half your pinkie, you’ll know that your altitude is 115 × 50 m = 6 km. In the exact same way, Aristarchos calculated the distance to the Moon to be 115 times its size, which came out to be about 30 times the diameter of Earth.

  Distance to the Sun and the Planets

  So what about the Sun? Try blocking it with your pinky and you’ll see that it covers about the same angle as the Moon, about half a degree. It’s clearly farther away than the Moon, since the Moon (just barely) blocks it from view during a total solar eclipse, but how much farther away? That depends on its size: for example, if it were three times the size of the Moon, it would need to be three times as far away to cover the same angle.

  Aristarchos of Samos was on a roll back in his time, and cleverly answered this question as well. He realized that the Sun, the Moon and Earth formed the three corners of a right triangle during “quarter Moon,” when we see exactly half the Earth-facing lunar surface illuminated by sunlight (see Figure 2.4), and he estimated that the angle between the Moon and the Sun was about 87 degrees at this time. So he knew both the shape of the triangle and the length of the Earth–Moon edge, and was able to use trigonometry to figure out the length of the Earth–Sun edge, that is, the distance between the Earth and the Sun. His conclusion was that the Sun was about twenty times farther away than the Moon and therefore twenty times bigger than the Moon. In other words, the Sun was huge: over five times bigger than Earth in diameter. This insight prompted Aristarchos to propose the heliocentric hypothesis long before Nicolaus Copernicus: he felt that it made more sense for Earth to be orbiting the much larger Sun than vice versa.

  Figure 2.4: By measuring the angle between the quarter moon and the Sun, Aristarchos was able to estimate our distance from the Sun. (This drawing isn’t to scale; the Sun is over one hundred times larger than Earth and about four hundred times as distant as the Moon.)

  This tale is both inspiring and cautionary, teaching us about both the importance of cleverness and the importance of quantifying uncertainties in our measurements. The ancient Greeks were less adept at the second part, and Aristarchos was unfortunately no exception. It turned out to be quite difficult to tell precisely when the Moon was 50% illuminated, and the correct Moon–Sun angle at that time isn’t 87 degrees but about 89.85 degrees, extremely close to a right angle. This makes the triangle in Figure 2.4 very long and skinny: in fact, the Sun is almost 20 times farther away than Aristarchos estimated, and about 109 times larger than Earth in diameter—so you could fit over a million Earths inside the volume of the Sun. Unfortunately, this glaring mistake wasn’t corrected unt
il almost two thousand years later, so when Copernicus came along and figured out the size and shape of our Solar System with further geometric ingenuity, he got the shapes and relative sizes right for all the planetary orbits, but the overall scale of his Solar System model was about twenty times too small—that’s like confusing a real house with a doll house.

  Distance to the Stars

  But what about the stars? How far away are they? And what are they? Personally, I think this is one of the greatest “cold case” detective stories ever. Figuring out the distances to the Moon and the Sun was impressive, but at least there was some information to use as clues: they change their sky positions in interesting ways, and they have shapes and angular sizes that we can measure. But a star seems totally hopeless! It looks like a faint white dot. You look at it more carefully and see … still just a faint white dot, with no discernible shape or size, merely a point of light. And the stars never seem to move across the sky, except for the apparent overall rotation of whole patterns of stars, which we know to be a mere illusion caused by the fact that Earth is rotating.

  Some ancients speculated that the stars were small holes in a black sphere through which distant light shone through. The Italian astronomer Giordano Bruno suggested that they were instead objects like our Sun, just much farther away, perhaps with their own planets and civilizations—this didn’t go down too well with the Catholic Church, which had him burned at the stake in 1600.

  In 1608, a sudden glimmer of hope: the telescope is invented! Galileo Galilei quickly improves the design, looks at stars through his ever-improving telescopes, and sees … just white dots again. Back to square one. I have fond memories of playing “Twinkle, Twinkle, Little Star” on my grandma Signe’s piano as a kid. As recently as 1806, when this song was first published, the line “How I wonder what you are” still resonated with many people, and nobody could honestly claim to really know the answers.

  If stars are really distant suns as Bruno suggested, then they must be dramatically farther away than our Sun to look so faint. But how much farther? That depends on how luminous they really are, which we’d also like to know. Thirty-two years after the song was published, the German mathematician and astronomer Friedrich Bessel finally achieved a breakthrough in this detective case. Please hold your thumb up at arm’s length and alternate closing your left and right eyes a few times. Do you see how your thumb appears to jump left and right by a certain angle relative to background objects? Now move your thumb closer to your eyes, and you’ll see this jump angle growing. Astronomers call this jump angle the parallax, and you can clearly use it to figure out how far away your thumb is. In fact, you needn’t worry about doing the math, since your brain does it for you so effortlessly that you don’t even notice—this fact that your two eyes measure different angles to objects depending on their distance is the very essence of how your brain’s depth-perception system works to provide you with three-dimensional vision.

  If your eyes were farther apart, you’d have better depth perception at large distances. In astronomy, we can use this same parallax trick and pretend that we’re giants with eyes 300 billion meters apart, which is the diameter of Earth’s orbit around the Sun. We can do this because we can compare telescopic photographs taken six months apart, when Earth is on opposite sides of the Sun. Doing this, Bessel noticed that while most stars appeared in the exact same positions in both of his pictures, one particular star didn’t: a star that went by the obscure name 61 Cygni. Instead, it had moved by a tiny angle, revealing its distance to be almost a million times that to the Sun—a distance so huge that it would take eleven years for its starlight to reach us, whereas sunlight gets here in just eight minutes.

  Before long, many more stars had their parallax measured, so many of these mysterious white dots now had distances! If you watch a car drive away at night, the brightness of its taillights drops as the inverse square of its distance (twice as far means four times dimmer). Now that Bessel knew the distance to 61 Cygni, he used this inverse-square law to figure out how luminous it was. His answer was a luminosity in the same ballpark as that of the Sun, suggesting that the late Giordano Bruno had been right after all!

  Around the same time, there was a second major break in the case using a totally different approach. In 1814, the German optician Joseph von Fraunhofer invented a device called a spectrograph, which let him separate white light into the rainbow of colors from which it’s made up, and measure them in exquisite detail. He discovered mysterious dark lines in the rainbow (see Figure 2.5), and that the detailed positions of these lines within the spectrum of colors depended on what the light source was made of, constituting a type of spectral fingerprint. During the following decades, such spectra were measured and cataloged for many common substances. You can use this information to pull a great party trick, impressing your friends by telling them what’s glowing in their lantern just from analyzing its light, without ever going near it. Sensationally, the spectrum of sunlight revealed that the Sun, this mysterious fiery orb in the sky, contained elements well known from Earth, such as hydrogen. Moreover, when starlight from a telescope was observed through a spectroscope, it revealed that stars are made of roughly the same mixture of gases as the Sun! This clinched it in favor of Bruno: stars are distant suns, similar in both their energy output and contents. So in a brief few decades, stars had gone from being inscrutable white dots to being giant balls of hot gas whose chemical composition we could measure.

  Figure 2.5: The rainbow spotted by my son Alexander leads not to a pot of gold, but to a goldmine of information about how atoms and stars work. As we’ll explore in Chapter 7, the relative intensities of the various colors are explained by light being made of particles (photons), and the positions and strengths of the many dark lines can all be calculated from the Schrödinger equation of quantum mechanics.

  A spectrum is a goldmine of astronomical information, and every time you think you’ve milked it for all it’s worth, you find more clues encoded in it. For starters, a spectrum lets you measure the temperature of an object without touching it with a thermometer. You know without touching that a piece of metal glowing white is hotter than one glowing red, and similarly that a whitish star is hotter than a reddish star; with a spectrograph, you can determine the temperatures quite accurately. As a surprise bonus, this information now reveals the star’s size, much like figuring out a word in a crossword puzzle can reveal another word. The trick is that the temperature tells you how much light emerges from each square meter of the star’s surface. Since you can calculate the total amount of light radiated by the star (from its distance and apparent brightness), you now know how many square meters of surface area the star must have, and therefore how big it is.

  As if this weren’t enough, the spectrum of a star also contains hidden clues about its motion, which slightly shifts the frequency (color) of the light through the so-called Doppler effect, the effect that makes the pitch fall in the vroooooooom of a passing car: the frequency is higher when the car is moving toward you, then lower as it moves away from you. Unlike our Sun, most stars are in stable pair relationships with a companion star, and the two partners dance around each other in a regular orbit. We can often detect this dance through the Doppler effect, which causes the spectral lines of the stars to move back and forth once per orbit. The magnitude of the shift reveals the speed of motion, and by looking at the two stars, we can sometimes measure how far apart they are. Combining this information allows us to pull another major stunt: we can weigh the stars without putting them on a gigantic bathroom scale, using Newton’s laws of motion and gravitation to calculate how massive they must be to have the observed orbits. In some cases, such Doppler shifts have also revealed that planets orbit a star. If the planet moves in front of the star, the slight dip in the star’s brightness reveals the size of the planet, and the slight change in spectral lines can reveal whether the planet has an atmosphere and what it’s made of. And spectra are the gift that just keeps giv
ing. For example, by measuring the width of spectral lines for a star of a given temperature, we can measure its gas pressure. And by measuring the extent to which spectral lines split into two or more nearby lines, we can measure how strong the magnetism is at the star’s surface.

  In conclusion, the only information we have about stars is in their faint light that reaches us, but through clever detective work, we can decode this light into information about their distance, size, mass, composition, temperature, pressure, magnetism and any solar system they may host. That our human minds have deduced all this from seemingly inscrutable white dots is a feat that I think would have made even the great detectives Sherlock Holmes and Hercule Poirot proud!

  Distance to the Galaxies

  When my grandma Signe passed away at age 102, I spent some time reflecting on her life, and it struck me that she grew up in a different universe. When she went off to college, our known Universe was simply our Solar System and a swarm of stars around it. She and her friends probably thought of these stars as incredibly distant, with light taking several years to arrive from the closest ones and thousands of years from the farthest ones known. All of which we nowadays consider merely our cosmic backyard.