by Brian Greene
5. For the mathematically inclined reader, Einstein’s equations of general relativity in this context reduce to . The variable a(t) is the scale factor of the universe—a number whose value, as the name indicates, sets the distance scale between objects (if the value of a(t) at two different times differs, say, by a factor of 2, then the distance between any two particular galaxies would differ between those times by a factor of 2 as well), G is Newton’s constant, is the density of matter/energy, and k is a parameter whose value can be 1, 0, or -1 according to whether the shape of space is spherical, Euclidean (“flat”), or hyperbolic. The form of this equation is usually credited to Alexander Friedmann and, as such, is called the Friedmann equation.
6. The mathematically inclined reader should note two things. First, in general relativity we typically define coordinates that are themselves dependent on the matter space contains: we use galaxies as the coordinate carriers (acting as if each galaxy has a particular set of coordinates “painted” on it—so-called co-moving coordinates). So, to even identify a specific region of space, we usually make reference to the matter that occupies it. A more precise rephrasing of the text, then, would be: The region of space containing a particular group of N galaxies at time t1 will have a larger volume at a later time t2. Second, the intuitively sensible statement regarding the density of matter and energy changing when space expands or contracts makes an implicit assumption regarding the equation of state for matter and energy. There are situations, and we will encounter one shortly, where space can expand or contract while the density of a particular energy contribution—the energy density of the so-called cosmological constant—remains unchanged. Indeed, there are even more-exotic scenarios in which space can expand while the density of energy increases. This can happen because, in certain circumstances, gravity can provide a source of energy. The important point of the paragraph is that in their original form the equations of general relativity are not compatible with a static universe.
7. Shortly we will see that Einstein abandoned his static universe when confronted by astronomical data showing that the universe is expanding. It is worth noting, though, that his misgivings about the static universe predated the data. The physicist Willem de Sitter pointed out to Einstein that his static universe was unstable: nudge it a bit bigger, and it would grow; nudge it a bit smaller, and it would shrink. Physicists shy away from solutions that require perfect, undisturbed conditions for them to persist.
8. In the big bang model, the outward expansion of space is viewed much like the upward motion of a tossed ball: attractive gravity pulls on the upward-moving ball and so slows its motion; similarly, attractive gravity pulls on the outward-moving galaxies and so slows their motion. In neither case does the ongoing motion require a repulsive force. However, you can still ask: Your arm launched the ball skyward, so what “launched” the spatial universe on its outward expansion? We will return to this question in Chapter 3, where we will see that modern theory posits a short burst of repulsive gravity, operating during the earliest moments of cosmic history. We will also see that more refined data has provided evidence that the expansion of space is not slowing over time, which has resulted in a surprising—and as later chapters will make clear—potentially profound resurrection of the cosmological constant.
The discovery of the spatial expansion was a turning point in modern cosmology. In addition to Hubble’s contributions, the achievement relied on the work and insights of many others, including Vesto Slipher, Harlow Shapley, and Milton Humason.
9. A two-dimensional torus is usually depicted as a hollow doughnut. A two-step process shows that this picture agrees with the description provided in the text. When we declare that crossing the right edge of the screen brings you back to the left edge, that’s tantamount to identifying the entire right edge with the left edge. Were the screen flexible (made of thin plastic, say) this identification could be made explicit by rolling the screen into a cylindrical shape and taping the right and left edges together. When we declare that crossing the upper edge brings you to the lower edge, that too is tantamount to identifying those edges. We can make this explicit by a second manipulation in which we bend the cylinder and tape the upper and lower circular edges together. The resulting shape has the usual doughnutlike appearance. A misleading aspect of these manipulations is that the surface of the doughnut looks curved; were it coated with reflective paint, your reflection would be distorted. This is an artifact of representing the torus as an object sitting within an ambient three-dimensional environment. Intrinsically, as a two-dimensional surface, the torus is not curved. It is flat, as is clear when it’s represented as a flat video-game screen. That’s why, in the text, I focus on the more fundamental description as a shape whose edges are identified in pairs.
10. The mathematically inclined reader will note that by “judicious slicing and paring” I am referring to taking quotients of simply connected covering spaces by various discrete isometry groups.
11. The quoted amount is for the current era. In the early universe, the critical density was higher.
12. If the universe were static, light that had been traveling for the last 13.7 billion years and has only just reached us would indeed have been emitted from a distance of 13.7 billion light-years. In an expanding universe, the object that emitted the light has continued to recede during the billions of years the light was in transit. When we receive the light, the object is thus farther away—much farther—than 13.7 billion light-years. A straightforward calculation using general relativity shows that the object (assuming it still exists and has been continually riding the swell of space) would now be about 41 billion light-years away. This means that when we look out into space we can, in principle, see light from sources that are now as far as roughly 41 billion light-years. In this sense, the observable universe has a diameter of about 82 billion light-years. The light from objects farther than this distance would not yet have had enough time to reach us and so are beyond our cosmic horizon.
13. In loose language, you can envision that because of quantum mechanics, particles always experience what I like to call “quantum jitter”: a kind of inescapable random quantum vibration that renders the very notion of the particle having a definite position and speed (momentum) approximate. In this sense, changes to position/speed that are so small that they’re on par with the quantum jitters are within the “noise” of quantum mechanics and hence are not meaningful.
In more precise language, if you multiply the imprecision in the measurement of position by the imprecision in the measurement of momentum, the result—the uncertainty—is always larger than a number called Planck’s constant, named after Max Planck, one of the pioneers of quantum physics. In particular, this implies that fine resolutions in measuring the position of a particle (small imprecision in position measurement) necessarily entail large uncertainty in the measurement of its momentum and, by association, its energy. Since energy is always limited, the resolution in position measurements is thus limited too.
Also note that we will always apply these concepts in a finite spatial domain—generally in regions the size of today’s cosmic horizon (as in the next section). A finite-sized region, however large, implies a maximum uncertainty in position measurements. If a particle is assumed to be in a given region, the uncertainty of its position is surely no larger than the size of the region. Such a maximum uncertainty in position then entails, from the uncertainty principle, a minimum amount of uncertainty in momentum measurements—that is, limited resolution in momentum measurements. Together with the limited resolution in position measurements, we see the reduction from an infinite to a finite number of possible distinct configurations of a particle’s position and speed.
You might still wonder about the barrier to building a device capable of measuring a particle’s position with ever greater precision. It too is a matter of energy. As in the text, if you want to measure a particle’s position with ever greater precision, you need to use an ever mor
e refined probe. To determine whether a fly is in a room, you can turn on an ordinary, diffuse overhead light. To determine if an electron is in a cavity, you need to illuminate it with the sharp beam of a powerful laser. And to determine the electron’s position with ever greater accuracy you need to make that laser ever more powerful. Now, when an ever more powerful laser zaps an electron, it imparts an ever greater disturbance to its velocity. Thus, the bottom line is that precision in determining particles’ positions comes at the cost of huge changes in the particles’ velocities—and hence huge changes in particle energies. If there’s a limit to how much energy particles can have, as there always will be, there’s a limit to how finely their positions can be resolved.
Limited energy in a limited spatial domain thus gives finite resolution on both position and velocity measurements.
14. The most direct way to make this calculation is by invoking a result I will describe in nontechnical terms in Chapter 9: the entropy of a black hole—the logarithm of the number of distinct quantum states—is proportional to its surface area measured in square Planck units. A black hole that fills our cosmic horizon would have a radius of about 1028 centimeters, or roughly 1061 Planck lengths. Its entropy would therefore be about 10122 in square Planck units. Hence the total number of distinct states is roughly 10 raised to the power 10122, or 1010122.
15. You might be wondering why I’m not also incorporating fields. As we will see, particles and fields are complementary languages—a field can be described in terms of the particles of which it’s composed, much like an ocean wave can be described in terms of its constituent water molecules. The choice of using a particle or field language is largely one of convenience.
16. The distance that light can travel in a given time interval depends sensitively on the rate at which space expands. In later chapters we will encounter evidence that the rate of spatial expansion is accelerating. If so, there is a limit to how far light can travel through space, even if we wait an arbitrarily long time. Distant regions of space would be receding from us so quickly that light we emit could not reach them; similarly, light they emit could not reach us. This would mean that cosmic horizons—the portion of space with which we can exchange light signals—would not grow in size indefinitely. (For the mathematically inclined reader, the essential formulae are in Chapter 6, note 7.)
17. G. Ellis and G. Bundrit studied duplicate realms in an infinite classical universe; J. Garriga and A. the quantum context.
Chapter 3: Eternity and Infinity
1. One point of departure from the earlier work was Dicke’s perspective, which focused on the possibility of an oscillating universe that would repeatedly go through a series of cycles—big bang, expansion, contraction, big crunch, big bang again. In any given cycle there would be remnant radiation suffusing space.
2. It is worth noting that even though they don’t have jet engines, galaxies generally do exhibit some motion above and beyond that arising from the expansion of space—typically the result of large-scale intergalactic gravitational forces as well as the intrinsic motion of the swirling gas cloud from which stars in the galaxies formed. Such motion is called peculiar velocity and is generally small enough that it can be safely ignored for cosmological purposes.
3. The horizon problem is subtle, and my description of inflationary cosmology’s solution slightly nonstandard, so for the interested reader let me elaborate here in a little more detail. First the problem, again: Consider two regions in the night sky that are so distant from one another that they have never communicated. And to be concrete, let’s say each region has an observer who controls a thermostat that sets his or her region’s temperature. The observers want the two regions to have the same temperature, but because the observers have been unable to communicate, they don’t know how to set their respective thermostats. The natural thought is that since billions of years ago the observers were much closer, it would have been easy for them, way back then, to have communicated and thus to have ensured the two regions had equal temperatures. However, as noted in the main text, in the standard big bang theory this reasoning fails. Here’s more detail on why. In the standard big bang theory, the universe is expanding, but because of gravity’s attractive pull, the rate of expansion slows over time. It’s much like what happens when you toss a ball in the air. During its ascent it first moves away from you quickly, but because of the tug of earth’s gravity, it steadily slows. The slowing down of spatial expansion has a profound effect. I’ll use the tossed ball analogy to explain the essential idea. Imagine a ball that undergoes, say, a six second ascent. Since it initially travels quickly (as it leaves your hand), it might cover the first half of the journey in only two seconds, but due to its diminishing speed it takes four more seconds to cover the second half of the journey. At the halfway point in time, three seconds, it was thus beyond the halfway mark in distance. Similarly, with spatial expansion that slows over time: at the halfway point in cosmic history, our two observers would be separated by more than half their current distance. Think about what this means. The two observers would be closer together, but they would find it harder—not easier—to have communicated. Signals one observer sends would have half the time to reach the other, but the distance the signals would need to traverse is more than half of what it is today. Being allotted half the time to communicate across more than half their current separation only makes communication more difficult.
The distance between objects is thus only one consideration when analyzing their ability to influence each other. The other essential consideration is the amount of time that’s elapsed since the big bang, as this constrains how far any purported influence could have traveled. In the standard big bang, although everything was indeed closer in the past, the universe was also expanding more quickly, resulting in less time, proportionally speaking, for influences to be exerted.
The resolution offered by inflationary cosmology is to insert a phase in the earliest moments of cosmic history in which the expansion rate of space doesn’t decrease like the speed of the ball tossed upwards; instead, the spatial expansion starts out slow and then continually picks up speed: the expansion accelerates. By the same reasoning we just followed, at the halfway point of such an inflationary phase our two observers will be separated by less than half their distance at the end of that phase. And being allotted half the time to communicate across less than half the distance means it is easier at earlier times for them to communicate. More generally, at ever earlier times, accelerated expansion means there is more time, proportionally speaking—not less—for influences to be exerted. This would have allowed today’s distant regions to have easily communicated in the early universe, explaining the common temperature they now have.
Because the accelerated expansion results in a much greater total spatial expansion of space than in the standard big bang theory, the two regions would have been much closer together at the onset of inflation than at a comparable moment in the standard big bang theory. This size disparity in the very early universe is an equivalent way of understanding why communication between the regions, which would have proved impossible in the standard big bang, can be easily accomplished in the inflationary theory. If at a given moment after the beginning, the distance between two regions is less, it is easier for them to exchange signals.
Taking the expansion equations seriously to arbitrarily early times (and for definiteness, imagine that space is spherically shaped), we also see that the two regions would have initially separated more quickly in the standard big bang than in the inflationary model: that’s how they became so much farther apart in the standard big bang compared with their separation in the inflationary theory. In this sense, the inflationary framework involves a period of time during which the rate of separation between these regions is slower than in the usual big bang framework.
Often, in describing inflationary cosmology, the focus is solely on the fantastic increase in expansion speed over the conventional framework, not on a dec
rease in speed. The difference in description derives from which physical features between the two frameworks one compares. If one compares the trajectories of two regions of a given distance apart in the very early universe, then in the inflationary theory those regions separate much faster than in the standard big bang theory; by today they are also much farther apart in the inflationary theory than in the conventional big bang. But if one considers two regions of a given distance apart today (like the two regions on opposite sides of the night sky upon which we’ve been focused), the description I’ve given is relevant. Namely, at a given moment in time in the very early universe, those regions were much closer together, and had been moving apart much more slowly, in a theory that invokes inflationary expansion as compared with one that doesn’t. The role of inflationary expansion is to make up for the slower start by then propelling those regions apart ever more quickly, ensuring that they arrive at the same location in the sky that they would have in the standard big bang theory.
A fuller treatment of the horizon problem would include a more detailed specification of the conditions from which the inflationary expansion emerges as well as the subsequent processes by which, for example, the cosmic microwave background radiation is produced. But this discussion highlights the essential distinction between accelerated and decelerated expansion.
4. Note that by squeezing the bag, you inject energy into it, and since both mass and energy give rise to the resulting gravitational warpage, the increase in weight will be partially due to the increase in energy. The point, however, is that the increase in pressure itself also contributes to the increase in weight. (Also note that to be precise, we should imagine doing this “experiment” in a vacuum chamber, so we don’t need to consider the buoyant forces due to the air surrounding the bag.) For everyday examples the increase is tiny. However, in astrophysical settings the increase can be significant. In fact, it plays a role in understanding why, in certain situations, stars necessarily collapse to form black holes. Stars generally maintain their equilibrium through a balance between outward-pushing pressure, generated by nuclear processes in the star’s core, and inward-pulling gravity, generated by the star’s mass. As the star exhausts its nuclear fuel, the positive pressure decreases, causing the star to contract. This brings all its constituents closer together and so increases their gravitational attraction. To avoid further contraction, additional outward pressure (what is labeled positive pressure, as in the next paragraph in the text) is needed. But the additional positive pressure itself generates additional attractive gravity and thus makes the need for additional positive pressure all the more urgent. In certain situations, this leads to a spiraling instability and the very thing that the star usually relies upon to counteract the inward pull of gravity—positive pressure—contributes so strongly to that very inward pull that a complete gravitational collapse becomes unavoidable. The star will implode and form a black hole.