The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos

by Brian Greene

  The Universe and the Teapot

  Einstein breathed life into spacetime. He challenged thousands of years of intuition, built up from everyday experience, that treated space and time as an unchanging backdrop. Who would ever have imagined that spacetime can writhe and flex, providing the invisible master choreographer of motion in the cosmos? That’s the revolutionary dance that Einstein envisioned and that observations have confirmed. And yet, in short order, Einstein stumbled under the weight of age-old but unfounded prejudices.

  During the year after he published the general theory of relativity, Einstein applied it on the grandest of scales: the entire cosmos. You might think this a staggering task, but the art of theoretical physics lies in simplifying the horrendously complex so as to preserve essential physical features while making the theoretical analysis tractable. It’s the art of knowing what to ignore. Through the so-called cosmological principle, Einstein established a simplifying framework that initiated the art and the science of theoretical cosmology.

  The cosmological principle asserts that if the universe is examined on the largest of scales, it will appear uniform. Think of your morning tea. On microscopic scales, there is much inhomogeneity. Some H2O molecules over here, some empty space, some polyphenol and tannin molecules over there, more empty space, and so on. But on macroscopic scales, those accessible to the naked eye, the tea is a uniform hazel. Einstein believed that the universe was like that cup of tea. The variations we observe—the earth is here, there’s some empty space, then the moon, yet more empty space, followed by Venus, Mercury, sprinkles of empty space, and then the sun—are small-scale inhomogeneities. He suggested that on cosmological scales, these variations could be ignored because, like your tea, they’d average out to something uniform.

  In Einstein’s day, evidence in support of the cosmological principle was thin at best (even the case for other galaxies was still being made), but he was guided by a strong sense that no location in the cosmos was special. He felt that, on average, every region of the universe should be on a par with every other and so should have essentially identical overall physical attributes. In the years since, astronomical observations have provided substantial support for the cosmological principle, but only if you examine space on scales at least 100 million light-years across (which is about a thousand times the end-to-end length of the Milky Way). If you take a box that’s a hundred million light-years on each side and plunk it down here, take another such box and plunk it down way over there (say, a billion light-years from here), and then measure the average overall properties inside each box—average number of galaxies, average amount of matter, average temperature, and so on—you’ll find it difficult to distinguish between the two. In short, if you’ve seen one 100-million-light-year chunk of the cosmos, you’ve pretty much seen them all.

  Such uniformity proves crucial to using the equations of general relativity to study the entire universe. To see why, think of a beautiful, uniform, smooth beach and imagine that I’ve asked you to describe its small-scale properties—the properties, that is, of each and every grain of sand. You’re stymied—the task is just too big. But if I ask you to describe only the overall features of the beach (such as the average weight of sand per cubic meter, the average reflectivity of the beach’s surface per square meter, and so on), the task becomes eminently doable. And what makes it doable is the beach’s uniformity. Measure the average sand weight, temperature, and reflectivity over here and you’re done. Doing the same measurements over there will give essentially identical answers. Likewise with a uniform universe. It would be a hopeless task to describe every planet, star, and galaxy. But describing the average properties of a uniform cosmos is monumentally easier—and, with the advent of general relativity, achievable.

  Here’s how it goes. The gross overall content of a huge volume of space is characterized by how much “stuff” it contains; more precisely, the density of matter, or, more precisely still, the density of matter and energy that the volume contains. The equations of general relativity describe how this density changes over time. But without invoking the cosmological principle, these equations are hopelessly difficult to analyze. There are ten of them, and because each equation depends intricately on the others, they form a tight mathematical Gordian knot. Happily, Einstein found that when the equations are applied to a uniform universe, the math simplifies; the ten equations become redundant and, in effect, reduce to one. The cosmological principle cuts the Gordian knot by reducing the mathematical complexity of studying matter and energy spread throughout the cosmos to a single equation (you can see it in the notes).5

  Not so happily, from Einstein’s perspective, when he studied this equation he found something unexpected and, to him, unpalatable. The prevailing scientific and philosophical stance was not only that on the largest of scales the universe was uniform, but that it was also unchanging. Much like the rapid molecular motions in your tea average out to a liquid whose appearance is static, astronomical motion such as the planets orbiting the sun and the sun moving around the galaxy would average out to an overall unchanging cosmos. Einstein, who adhered to this cosmic perspective, found to his dismay that it was at odds with general relativity. The math showed that the density of matter and energy cannot be constant through time. Either the density grows or it diminishes, but it can’t stay put.

  Although the mathematical analysis behind this conclusion is sophisticated, the underlying physics is pedestrian. Picture a baseball’s journey as it soars from home plate toward the center field fence. At first, the ball rockets upward; then it slows, reaches a high point, and finally heads back down. The ball doesn’t lazily hover like a blimp because gravity, being an attractive force, acts in one direction, pulling the baseball toward earth’s surface. A static situation, like a stalemate in a tug-of-war, requires equal and opposite forces that cancel. For a blimp, the upward push that counters downward gravity is provided by air pressure (since the blimp is filled with helium, which is lighter than air); for the ball in midair there is no counter-gravity force (air resistance does act against a ball in motion, but plays no role in a static situation), and so the ball can’t remain at a fixed height.

  Einstein found that the universe is more like the baseball than the blimp. Because there’s no outward force to cancel the attractive pull of gravity, general relativity shows that the universe can’t be static. Either the fabric of space stretches or it contracts, but its size can’t remain fixed. A volume of space 100 million light-years on each side today won’t be 100 million light-years on each side tomorrow. Either it will be larger, and the density of matter within it will diminish (being spread more thinly in a larger volume), or it will be smaller, and the density of matter will increase (being packed more tightly in a smaller volume).6

  Einstein recoiled. According to the math of general relativity, the universe on the grandest of scales would be changing, because its very substrate—space itself—would be changing. The eternal and static cosmos that Einstein expected would emerge from his equations was simply not there. He had initiated the science of cosmology, but he was deeply distressed by where the math had taken him.

  Taxing Gravity

  It’s often said that Einstein blinked—that he went back to his notebooks and in desperation mangled the beautiful equations of general relativity to make them compatible with a universe that was not only uniform but also unchanging. This is only partly true. Einstein did indeed modify his equations so they would support his conviction of a static cosmos, but the change was minimal and thoroughly sensible.

  To get a feel for his mathematical move, think about filling out your tax forms. Interspersed among the lines on which you record numbers are others you leave blank. Mathematically, a blank line signifies that the entry is zero, but psychologically it connotes more. It means you’re ignoring the line because you’ve determined that it’s not relevant to your financial situation.

  If the mathematics of general relativity were arranged like
a tax form, it would have three lines. One line would describe the geometry of spacetime—its warps and curves—the embodiment of gravity. Another would describe the distribution of matter across space, the source of gravity—the cause of the warps and curves. During a decade of ardent research, Einstein had worked out the mathematical description of these two features and had thus filled in these two lines with great care. But a complete accounting of general relativity requires a third line, one that is on an absolutely equal mathematical footing with the other two but whose physical meaning is more subtle. When general relativity elevated space and time into dynamic participants in the unfolding of the cosmos, they shifted from merely providing language to delineate where and when things take place to being physical entities with their own intrinsic properties. The third line on the general relativity tax form quantifies a particular intrinsic feature of spacetime relevant for gravity: the amount of energy stitched into the very fabric of space itself. Just as every cubic meter of water contains a certain amount of energy, summarized by the water’s temperature, every cubic meter of space contains a certain amount of energy, summarized by the number on the third line. In his paper announcing the general theory of relativity, Einstein didn’t consider this line. Mathematically, this is tantamount to having set its value to zero, but much as with blank lines on your tax forms, he seems to have simply ignored it.

  When general relativity proved incompatible with a static universe, Einstein reengaged with the mathematics, and this time he took a harder look at the third line. He realized that there was no observational or experimental justification for setting it to zero. He also realized that it embodied some remarkable physics.

  If instead of zero he entered a positive number on the third line, endowing the spatial fabric with a uniform positive energy, he found (for reasons I’ll explain in the next chapter) that every region of space would push away from every other, producing something most physicists had thought impossible: repulsive gravity. Moreover, Einstein found that if he precisely adjusted the size of the number he put on the third line, the repulsive gravitational force produced across the cosmos would exactly balance the usual attractive gravitational force generated by the matter inhabiting space, giving rise to a static universe. Like a hovering blimp that neither rises nor falls, the universe would be unchanging.

  Einstein called the entry on the third line the cosmological member or the cosmological constant; with it in place, he could rest easy. Or, he could rest easier. If the universe had a cosmological constant of the right size—that is, if space were endowed with the right amount of intrinsic energy—his theory of gravity fell in line with the prevailing belief that the universe on the largest of scales was unchanging. He couldn’t explain why space would embody just the right amount of energy to ensure this balancing act, but at least he’d shown that general relativity, augmented with a cosmological constant of the right value, gave rise to the kind of cosmos he and others had expected.7

  The Primeval Atom

  It was against this backdrop that Lemaître approached Einstein at the 1927 Solvay Conference in Brussels, with his result that general relativity gave rise to a new cosmological paradigm in which space would expand. Having already wrestled with the mathematics to ensure a static universe, and having already dismissed Friedmann’s similar claims, Einstein had little patience for once again considering an expanding cosmos. He thus faulted Lemaître for blindly following the mathematics and practicing the “abominable physics” of accepting an obviously absurd conclusion.

  A rebuke by a revered figure is no small setback, but for Lemaître it was short-lived. In 1929, using what was then the world’s largest telescope at the Mount Wilson Observatory, the American astronomer Edwin Hubble gathered convincing evidence that the distant galaxies were all rushing away from the Milky Way. The remote photons that Hubble examined had traveled to earth with a clear message: The universe is not static. It is expanding. Einstein’s reason for introducing the cosmological constant was thus unfounded. The big bang model describing a cosmos that began enormously compressed and has been expanding ever since became widely heralded as the scientific story of creation.8

  Lemaître and Friedmann were vindicated. Friedmann received credit for being the first to explore the expanding universe solutions, and Lemaître for independently developing them into robust cosmological scenarios. Their work was duly lauded as a triumph of mathematical insight into the workings of the cosmos. Einstein, by contrast, was left wishing he’d never meddled with the third line of the general relativity tax form. Had he not brought to bear his unjustified conviction that the universe is static, he wouldn’t have introduced the cosmological constant and so might have predicted cosmic expansion more than a decade before it was observed.

  Nevertheless, the cosmological constant’s story was far from over.

  The Models and the Data

  The big bang model of cosmology includes a detail that will prove essential. The model provides not one but a handful of different cosmological scenarios; all of them involve an expanding universe, but they differ with respect to the overall shape of space—and, in particular, they differ on the question of whether the full extent of space is finite or infinite. Since the finite-versus-infinite distinction will turn out to be vital in thinking about parallel worlds, I’ll lay out the possibilities.

  The cosmological principle—the assumed homogeneity of the cosmos—constrains the geometry of space because most shapes are not sufficiently uniform to qualify: they bulge here, flatten out there, or twist way over there. But the cosmological principle does not imply a unique shape for our three dimensions of space; instead, it reduces the possibilities to a sharply culled collection of candidates. To visualize them presents a challenge even for professionals, but it is a helpful fact that the situation in two dimensions provides a mathematically precise analog that we can readily picture.

  To this end, first consider a perfectly round cue ball. Its surface is two-dimensional (just as on earth’s surface, you can denote positions on the cue ball’s surface with two pieces of data—such as latitude and longitude—which is what we mean when we call a shape two-dimensional) and is completely uniform in the sense that every location looks like every other. Mathematicians call the cue ball’s surface a two-dimensional sphere and say that it has constant positive curvature. Loosely speaking, “positive” means that were you to view your reflection on a spherical mirror it would bloat outward, while “constant” means that regardless of where on the sphere your reflection is, the distortion appears the same.

  Next, picture a perfectly smooth tabletop. As with the cue ball, the tabletop’s surface is uniform. Or nearly so. Were you an ant walking on the tabletop, the view from every point would indeed look like the view from every other, but only if you stayed far from the table’s edge. Even so, complete uniformity is not hard to restore. We just need to imagine a tabletop with no edges, and there are two ways of doing so. Think of a tabletop that extends indefinitely left and right as well as back and forth. This is unusual—it’s an infinitely large surface—but it realizes the goal of having no edges since there’s now no place to fall off. Alternatively, imagine a tabletop that mimics an early video-game screen. When Ms. Pac-Man crosses the left edge, she reappears at the right; when she crosses the bottom edge she reappears at the top. No ordinary tabletop has this property, but this is a perfectly sensible geometrical space called a two-dimensional torus. I discuss this shape more fully in the notes,9 but the only features in need of emphasis here are that, like the infinite tabletop, the video-game screen shape is uniform and it has no edges. The apparent boundaries confronting Ms. Pac-Man are fictitious; she can cross through them and remain in the game.

  Mathematicians say that the infinite tabletop and the video-game screen are shapes that have constant zero curvature. “Zero” means that were you to examine your reflection on a mirrored tabletop or video-game screen, the image wouldn’t suffer any distortion, and as before, “const
ant” means that regardless of where you examine your reflection, the image looks the same. The difference between the two shapes becomes apparent only from a global perspective. If you took a journey on an infinite tabletop and maintained a constant heading, you’d never return home; on a video-game screen, you could cycle around the entire shape and find yourself back at the point of departure, even though you never turned the steering wheel.

  Finally—and this is a little more difficult to picture—a Pringles potato chip, if extended indefinitely, provides another completely uniform shape, one that mathematicians say has constant negative curvature. This means that if you view your reflection at any spot on a mirrored Pringles chip, the image will appear shrunken inward.

  Fortunately, these descriptions of two-dimensional uniform shapes extend effortlessly to our real interest in the three-dimensional space of the cosmos. Positive, negative, and zero curvatures—uniform bloating outward, shrinking inward, and no distortion at all—equally well characterize uniform three-dimensional shapes. In fact, we are doubly fortunate because although three-dimensional shapes are hard to picture (when envisioning shapes, our minds invariably place them within an environment—an airplane in space, a planet in space—but when it comes to space itself, there isn’t an outside environment to contain it); the uniform three-dimensional shapes are such tight mathematical analogs of their two-dimensional cousins that you lose little precision by doing what most physicists do: use the two-dimensional examples for your mental imagery.