Chapter 18
In This Chapter
Defining the boundary of the universe
Predicting the shape of space
Recognizing the expansion of the universe
A fter Einstein completed “his most beautiful discovery,” the field equation of the general theory of relativity (see Chapter 12), he began to use it as a tool to discover nature’s secrets — to find out how the world works. The first secret he tackled was the most basic one of all: the nature of the universe. He wanted to know how the universe was made and how it works. When he was done, he’d created a new area in physics. He’d also made a big mistake.
In this chapter, I introduce Einstein’s equations for the universe and the early models that were developed with them. Uncharacteristically, Einstein was afraid of his own equations and introduced what he called his greatest blunder.
Looking for the Edge of the Universe
In February of 1917, Einstein submitted a paper to the Prussian Academy of Sciences presenting his model of the universe. His model was in sharp contrast to the previous attempts based on Isaac Newton’s physics.
Deconstructing Newton
Einstein began his paper by listing all the things that weren’t right about Newton’s universe. Newton constructed an unchanging universe that extended to infinity and was filled with stars that felt each other’s gravity according to his universal law of gravitation (see Chapter 4).
Einstein and others pointed out that gravity wouldn’t allow Newton’s universe to remain unchanged. Also, the idea of an infinite universe was problematic, and eventually, scientists were able to show that it couldn’t exist. An infinite universe would collapse.
But a finite universe is also problematic. If the universe doesn’t extend forever, if space has an end, then the next question is: What’s on the other side? What lies beyond the edge of the universe? Answering with “another universe” doesn’t cut it. You can simply keep asking the same questions. Is the other universe infinite, or does it have an end? And so on.
If a universe has an edge, you could travel to the edge of the universe and stick your hand out. Do you extend the universe by doing that? Are you creating more universe?
These questions about a universe based on Newton’s physics have never been answered. Newton’s mechanics showed us how the solar system works, how stars move and live within galaxies, and how galaxies interact with each other. Scientists use his mechanics even today with great success to calculate satellite and space station orbits, and to send spaceships to Mars. But you can’t use it to figure out how the entire universe works or whether it extends forever or has an end.
Reconciling “finite” with “unbounded”
Einstein’s model, as outlined in his paper to the Academy (“Cosmological observations on the general theory of relativity”), presented an unchanging universe that has no boundaries but is finite in size. Einstein built his universe with the following properties:
Except for local differences, everything in the universe is made of the same stuff.
The universe looks the same in all directions and from every place (in a large scale, with local differences).
The universe is unbounded and finite.
The universe is static, which means that it has existed forever and will continue to exist forever in the same general form. There was no beginning and there will be no end, but there are local changes.
Einstein found out very quickly that this last assumption was incorrect. Except for that, Einstein’s model is, in very general terms, close to what we use today.
The first property of Einstein’s model of the universe makes life easier. If you assume that the universe changes from place to place, your model becomes extremely complicated. Although scientists are now talking about strange forms of matter that they’ve discovered, according to observations over the last 100 years, at the large scale, the universe appears to be made of the same stuff. Take a look at Chapter 19 for more information.
The second property has been corroborated recently (see Chapter 14). Matter in the universe, in the form of galaxies and clusters of galaxies, seems to be distributed the same way in all directions.
At first glance, the third property seems like a contradiction. Einstein says that the universe doesn’t extend forever, but that it doesn’t have an edge. Einstein didn’t just come up with this property out of the blue. His field equation required it. He told his friend Michele Besso that “in gravitation I am now looking for the limiting conditions in the infinite. Surely it is interesting to reflect to what extent there exists a finite world, i.e. a world of naturally measured finite extension, in which all inertia is really relative.”
The universe that Einstein’s field equations gave him has a finite extension in space. The three dimensions of space are closed, similarly to the way the two dimensions of the surface of a sphere are closed. Imagine a two-dimensional universe where two-dimensional beings live (see Figure 18-1). There are stars and planets in this universe. The stars are discs made out of hot gases that emit light and heat. The planets are much smaller dark discs, and they orbit some of these stars. Everything takes place on the surface of the sphere. There is no “up” or “down” in this universe.
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Figure 18-1: A two-dimensional universe. |
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The surface of this two-dimensional universe is finite and unbounded. A two-dimensional being traveling in this universe will move along the surface of the sphere, never encountering an edge. Eventually, the traveler will end up at the starting position. Because there are no edges to this universe, the traveler can’t stick her flat, two-dimensional hand out. (She can’t stick it up, because there is no up.) She is forever stuck on the surface.
Einstein’s model of the universe is the three-dimensional version of this spherical universe. Our space is unbounded but has a finite size. Einstein’s universe doesn’t have the problems that Newton’s universe had. It doesn’t extend forever, but you can’t travel to the edge and stick your hand out. There is no edge. There is no other side.
Calculating the Curvature of the Universe
How would you go about discovering the shape of the universe? It’s easier to think about in a fictional two-dimensional universe.
A two-dimensional example
Imagine that Roundworld is one of the inhabited planets in orbit around a yellow star in this two-dimensional universe. Roundworld is a very flat world, and its inhabitants had always thought that their entire universe was flat, like a disc. Scientists had made very precise measurements of different geometrical shapes, and everything worked out according to the theorems of plane geometry, which everyone learned in school. For example, the sum of the angles of a triangle was always 180 degrees.
Over the years, scientists argued whether their flat universe extended forever, with infinite stars and planets, or whether it was finite, with an edge. But if it was finite, what was beyond the edge?
About 90 years ago, a famous scientist discovered that their universe is not flat but curved. Roundworld is still flat, but the entire universe is actually spherical. If one of the Roundworlders could travel for millions of years along a straight line, he would eventually come back around, having traveled all the way around the universe. He would end up at the starting point.
The scientist proposed making a huge triangle with laser beams sent out from spacecraft that had been launched for such a mission several years in advance. When the angles of the triangle formed with the laser beams were measured, the sum of these angles came out to be larger than 180 degrees (see Figure 18-2). The universe is curved, the scientist concluded. It’s a sphere. Everyone was astonished but happy that they had discovered the shape of the universe. But they couldn’t visualize it. They knew about curved lines, like circles, but didn’t have the concept of a curved surface. Curved in what direction? They didn’t have up or down. But their physicists could see it in their equations.
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Figure 18-2: The Roundworld scientists measured the angles of a large triangle in space. |
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A four-dimensional cylinder
Einstein didn’t propose sending out spacecraft with laser beams to measure the curvature of the universe. He performed a thought experiment.
Imagine a spinning disc or wheel. You know from the geometry we all learned in school that if you measure the circumference of the disc and divide it by its diameter, you get the value π. According to special relativity, however, you’d measure a smaller value for the circumference when the wheel is spinning, because the length along the direction of motion is contracted (see Chapter 9). The wheel is in motion relative to you, and you measure a smaller length in the moving frame (see Figure 18-3). The diameter doesn’t change because it’s at right angles to the direction of spin, and only lengths along the direction of motion are shortened.
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Figure 18-3: If you measure the circumference of a wheel that is spinning at relativistic speeds, you get a smaller value. |
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The shortening of the circumference relative to the wheel’s diameter is due to the curvature of space. To see why, draw a wheel on a flat sheet of paper. Cut out a wedge of the circle (to get the smaller circumference that you measured above), and then rejoin the paper. The paper will curve (see Figure 18-4). I walk you through the same process to illustrate the orbit of Mercury in the curved spacetime of the sun (see Chapter 12). In that case, the space is locally curved by the sun. In the present case, all of spacetime is curved. The shortening of the spinning disc is a property of spacetime and doesn’t depend on being close to the sun or to any other mass.
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Figure 18-4: Cut a wedge out of the wheel. If you rejoin the edges, you get a curved surface. |
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You can also do what the Roundworld scientists did and try to add the angles of a large triangle in space. You’ll discover that they add up to more than 180 degrees. Our spacetime is curved.
Actually, the three dimensions of space are curved. The time dimension isn’t. Spacetime is a four-dimensional cylinder, with space curved and the time dimension flat (see Figure 18-5). In what direction does space curve? Just like the Roundworlders who couldn’t visualize their curved surface, we can’t visualize our curved space. But our physicists can see it in their equations.
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Figure 18-5: Einstein’s spacetime is curved like a cylinder. |
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Sphereland
In 1960, the Dutch mathematician Dionys Burger wrote a story about a two-dimensional world, Sphereland, describing the trials and tribulations of A Hexagon, one of the two-dimensional Spherelanders. The book was a sequel to the classic Flatland, a science fiction story published in 1880 by the Shakespearean scholar Edwin Abbot.
Einstein’s Model of the Universe
Einstein’s fourth assumption was that the universe is static, existing forever in the same overall form and remaining like this forever. This assumption was very uncharacteristic of Einstein. When he first made the calculations, his field equation gave him a dynamic universe, a universe that was either expanding or collapsing.
Einstein didn’t like what his model was giving him. An expanding or collapsing universe was a strange notion. Not that it couldn’t happen. The physics of either option was fairly straightforward. A collapse would be caused by gravity pulling in all the stars from their current positions. An expansion required some sort of large explosion in the distant past that pushed all the stars away. His model could represent either possibility. But the observations of the day weren’t showing either one. The universe that the astronomers were seeing was static.
Changing his equation to fit reality
Einstein decided to modify his model so that it would represent reality. He saw that by adding a term to his equations that represented the force of repulsion and that exactly counterbalanced the pull of gravity, his universe would become static. It was a frail equilibrium, but it did the trick. He called the term the cosmological constant because, in his model, it determined the size of the universe.
It must’ve been a hard decision for Einstein to make. In his paper to the Academy he wrote: “I shall conduct the reader over the road that I have myself traveled, rather a rough and winding road . . . The field equations of gravitation . . . still need a slight modification.”
A model for an empty universe
Shortly after Einstein proposed his model of the universe, the Dutch astronomer Willem de Sitter also used Einstein’s field equation to construct a model of the universe. Like Einstein’s model, de Sitter’s universe was static, but unlike Einstein’s, it was empty. Although that may sound strange, taken as a whole, the universe is almost empty; the stars and galaxies are spread out over extremely large distances, making the density of the universe almost zero.
A few years later, scientists realized that de Sitter’s model wasn’t really static, as he’d thought. You may think that de Sitter was sloppy in developing his model or that the physicists that read the papers weren’t paying enough attention. This wasn’t the case. The problem was that Einstein’s field equation isn’t easily solved, and using it to build a model is extremely complicated. Today, physicists have developed sophisticated mathematical tools that help them use general relativity without running into these problems. In de Sitter’s model, the universe is empty and there is nothing to expand. But Arthur Eddington — the English astronomer who organized the eclipse expedition to test general relativity (see Chapter 12) — introduced particles of matter into de Sitter’s model and discovered that they moved apart.
In Einstein’s model, space is curved but time isn’t. In de Sitter’s model, both space and time are curved. If you imagine space represented by one dimension instead of three, Einstein’s model is represented by a cylinder, with time flat along the axis, while de Sitter’s universe is more like a sphere, with time also curved.
Einstein wasn’t happy with the appearance of de Sitter’s model. He’d assumed that general relativity would allow only one unique solution, which was clearly not the case. When the model was first proposed, astronomers and physicists took a great deal of interest in it. Today, we know that the mass density of the universe, although still small, isn’t small enough to model an empty universe.
Rejecting a Russian model
A few years after Einstein introduced his static model of the universe, the Russian scientist Alexander Friedmann decided to remove Einstein’s cosmological constant from his model. Friedmann wanted to restore the original form of the equations and see what the consequences were. What he found was that the model predicted an expanding universe, as Einstein had first discovered. But Einstein had rejected it from the beginning and hadn’t looked into the details of the model before modifying it with the cosmological constant.
Friedmann did study it, and he found that there were two possible outcomes of this expansion, depending on the total mass density of the universe:
If the universe has a very high mass density, it will first expand, but (because in this case the gravitational field is so strong) the expansion will stop and turn into a contraction. This type of universe is called a closed universe. This universe has spherical curvature, like Einstein’s model.
If the mass density of the universe is very low, gravity is not strong enough to halt the expansion, which will continue forever at the same rate. This is an open universe. This universe is also curved, but the curvature is not closed, like the spherical curvature of the first type. It’s an open curvature that in some way curves away from itself, like the surface of a saddle (see Figure 18-6).
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Figure 18-6: The three types of Friedmann universes. |
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There is a third case that Friedmann didn’t consider at the time. If the mass density of the universe falls right in the middle, at a critical value, the universe is a flat universe which also expands forever, but the expansion gradually slows down. However, gravity is not strong enough to slow it down completely.
Friedmann published his calculations in two papers in 1924. At first, Einstein didn’t think that the model was correct. When he looked closely, he realized that Friedmann’s papers were correct, but he didn’t think that they represented the real universe. He still believed that the universe was static, not expanding. Friedmann’s model was actually his own initial model, before he introduced his correction with the cosmological constant.
If only he hadn’t. If only he’d been brave and stuck with what his equation was telling him. If only he’d been more like the rebellious Einstein who stuck with his equations even when they were telling him crazy things that hadn’t been observed — that mass warps space, that light is bent by gravity, that time is dilated. He’d been right then. This time he wasn’t.
Watching the Universe Expand
At about the time that Einstein was working on his model of the universe, a U.S. astronomer by the name of Vesto Slipher was beginning to see evidence that distant clusters of stars were moving away from us. However, because of World War I, Einstein hadn’t found out about Slipher’s research.
Slipher had been making his observations of distant stars for several years. By 1914, he’d collected data showing that the light from many of these stars was Doppler-shifted toward the red end of the spectrum (the wavelengths were stretched), meaning that they were moving away from us (see Chapter 12). Slipher was actually looking at galaxies, although at the time, astronomers didn’t know that these white smudges in the sky contained billions of stars.
Here’s a brief introduction to the structure of the universe: A typical galaxy, like our own Milky Way, contains 100 billion stars that are kept together by gravity. Galaxies are grouped in clusters (see Figure 18-7). Our cluster, the Local Group, contains some 30 galaxies. Clusters also exist in super clusters. The Local Group is part of the Virgo Super Cluster.
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Figure 18-7: Structure in the universe. |
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None of this was known when Einstein and Friedmann were working on their models. At the time, astronomers thought that the universe contained stars and nebulas (fuzzy bright patches in the sky). Some astronomers had said that these nebulas were perhaps island universes — large collections of stars. But because the nebulas were so far away, existing telescopes weren’t strong enough for astronomers to determine if they consisted of stars.
Exploring island universes
In 1924, the U.S. astronomer Edwin Hubble showed that nebulas are island universes, external to our Milky Way. With a 100-inch telescope that had been recently installed at the Mount Wilson Observatory in California, he was able to make out individual stars in the nebulas. The announcement caused a sensation among the world’s astronomers and the scientific community. Hubble had discovered galaxies, entire universes of innumerable stars very far away.
Hubble’s discovery of galaxies was one of the most important discoveries in the history of astronomy. But it wasn’t the most important discovery that Hubble would make. In 1929, he showed that the universe is expanding.
Using a yardstick for the stars
After his discovery of galaxies, Hubble began to measure their properties. The first thing he wanted to know was their distances. Astronomers had developed several methods to measure distances to stars. The simplest was the parallax method (see Chapter 4), using simple geometry to figure out the apparent shift in the position of the star as the Earth moved in its orbit around the sun. But that method was good only for nearby stars. If the star was far away, the shift was undetectable.
Eventually, astronomers discovered that some properties of stars could be used to compare their relative distances. They could calibrate a new method by comparing it to the simple parallax method.
In 1912, Henrietta Leavitt at the Harvard College Observatory discovered one such method. She was a research assistant at the observatory and was observing and studying a type of stars known as variable stars, which change brightness periodically. She was concentrating on one specific group of these stars in the Small Magellanic Cloud, a cloud of stars visible from the southern hemisphere.
Leavitt measured the times it takes these stars to go from dim to maximum brightness and found that it varies from one day to several months. These stars had been seen before in other places and were known as Cepheid stars, because the first one to be discovered was in the constellation Cepheus. Cepheids are very bright giant stars that can be seen at great distances.
After many measurements over four years, Leavitt realized that there is a direct relationship between a Cepheid’s brightness and the time it takes the star to go from dim to bright. If a star that is 1,000 times brighter than the sun goes from dim to bright in 3 days, a star that is 10,000 times as bright as the sun does it in 30 days. If you measure these times, you know the brightness of the star. And this is the actual brightness of the star, not the apparent brightness that we see from Earth, because these particular Cepheids are all in the Small Magellanic Cloud, meaning that they are all roughly at the same distance from Earth.
Leavitt figured out that she could use the period-luminosity relationship, as her method was called, to measure the distances to the stars. But first she needed to calibrate her method. To calibrate, all she had to do was to find a Cepheid close enough to the Earth, so she could apply one of the other distance methods to find the distance. When she knew the distance based on the other method, she reversed the method and used her relationship to find the distance to other Cepheids in other locations.
This is how Leavitt’s method works. Suppose you are observing a distant galaxy and you want to know its distance from Earth. You look for Cepheids and measure how long one of them takes to change from dim to bright. The time you measure tells you the actual brightness of the star. When you know the brightness, you can figure out how far it is, in the same way that if you know a light bulb you see shining at a distance is a 100-watt bulb, you can figure out how far it is.
Henrietta Leivitt had discovered the first yardstick to measure the distances to distant galaxies. Her method was perfected over the years and is one of the main distance measurement tools in use today.
Discovering that galaxies are moving away
With the telescopes at Mount Wilson and Henrietta Leavitt’s method, Hubble began to measure the distances to his new galaxies. By 1929, he had measured the distances of about two dozen galaxies. He then compared the distances that he was measuring with the spectra that Vesto Slipher had been taking. What he obtained was astonishing. He found that all the galaxies in the universe are rushing away from each other, and the farther they are, the faster they move. The universe is expanding.
Hubble had made the most important discovery in the study of the nature of the universe. Expansion means some sort of explosion. The universe was born out of an explosion, and we are living in it.
Hubble published his results later that year. In his paper, he mentioned that the models of the universe based on Einstein’s general relativity had predicted this expansion. But not Einstein’s own model. Einstein introduced the cosmological constant to keep his universe from expanding. If he had listened to his equations, he would’ve predicted Hubble’s discovery.
In the face of Hubble’s finding, Einstein realized that his static model was wrong and gave in. He hadn’t accepted Friedmann’s model of the universe, which was essentially his own without the cosmological constant. He now recognized that he’d been wrong. In his book The Meaning of Relativity, Einstein later wrote that “Friedmann found a way out of his possible dilemma. He showed that it is possible, according to the field equations, to have a finite [universe] without enlarging these field equations.” And without a need to add his cosmological constant.
He told the physicist and cosmologist George Gamow that the cosmological constant idea was the biggest blunder he had made in his entire life.
Was it really? In Chapter 19, I show you why there may be doubts about that claim.
Edwin Hubble
Edwin Hubble became one of the towering figures in astronomy. He was also a bit of a ham. He spoke with an English accent, which he acquired when he went to England as a Rhodes scholar. He smoked a pipe and enjoyed blowing smoke rings across the table. He was a showman. But he didn’t need to be. Knowing how to strike a pose is the ritual of lesser men. His discoveries and papers were of great historical importance and needed no help.
He had started out his professional life as a lawyer. After studying physics at the University of Chicago, he attended Oxford on his Rhodes Scholarship and studied law there, passing the bar exam when he returned to the United States in 1913. After practicing for a year, he decided to go back to the University of Chicago to study astronomy, obtaining his PhD in 1917.
Soon after graduation, Hubble received an invitation to join the staff at the Mount Wilson Observatory, at the time the best in the world. Hubble sent a telegram to the director of the observatory: “Regret cannot accept your invitation. Am off to war.”
After his return from World War I, Hubble did go to Mount Wilson to start his long and successful career there.
