The Universe in a NutshellDe (autor) Stephen Hawking
en Limba Engleză Hardback – November 2001
Stephen Hawking’s phenomenal, multimillion-copy bestseller, A Brief History of Time, introduced the ideas of this brilliant theoretical physicist to readers all over the world.
Now, in a major publishing event, Hawking returns with a lavishly illustrated sequel that unravels the mysteries of the major breakthroughs that have occurred in the years since the release of his acclaimed first book.
The Universe in a Nutshell
• Quantum mechanics
• General relativity
• 11-dimensional supergravity
• 10-dimensional membranes
• Black holes
One of the most influential thinkers of our time, Stephen Hawking is an intellectual icon, known not only for the adventurousness of his ideas but for the clarity and wit with which he expresses them. In this new book Hawking takes us to the cutting edge of theoretical physics, where truth is often stranger than fiction, to explain in laymen’s terms the principles that control our universe.
Like many in the community of theoretical physicists, Professor Hawking is seeking to uncover the grail of science — the elusive Theory of Everything that lies at the heart of the cosmos. In his accessible and often playful style, he guides us on his search to uncover the secrets of the universe — from supergravity to supersymmetry, from quantum theory to M-theory, from holography to duality.
He takes us to the wild frontiers of science, where superstring theory and p-branes may hold the final clue to the puzzle. And he lets us behind the scenes of one of his most exciting intellectual adventures as he seeks “to combine Einstein’s General Theory of Relativity and Richard Feynman’s idea of multiple histories into one complete unified theory that will describe everything that happens in the universe.”
With characteristic exuberance, Professor Hawking invites us to be fellow travelers on this extraordinary voyage through space-time. Copious four-color illustrations help clarify this journey into a surreal wonderland where particles, sheets, and strings move in eleven dimensions; where black holes evaporate and disappear, taking their secret with them; and where the original cosmic seed from which our own universe sprang was a tiny nut.
The Universe in a Nutshell is essential reading for all of us who want to understand the universe in which we live. Like its companion volume, A Brief History of Time, it conveys the excitement felt within the scientific community as the secrets of the cosmos reveal themselves.
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Ilustrații: FOUR-COLOR PHOTOS/ILLUS
Dimensiuni: 198 x 254 x 23 mm
Greutate: 0.87 kg
Editura: Bantam Books
Stephen Hawking is Lucasian Professor of Mathematics at the University of Cambridge; his other books for the general reader include the essay collection Black Holes and Baby Universes and The Universe in a Nutshell.
The Shape of Time
Einstein’s general relativity gives time a shape.
How this can be reconciled with quantum theory.
What is time? Is it an ever-rolling stream that bears all our dreams away, as the old hymn says? Or is it a railroad track? Maybe it has loops and branches, so you can keep going forward and yet return to an earlier station on the line (Fig. 2.1).
The nineteenth-century author Charles Lamb wrote: “Nothing puzzles me like time and space. And yet nothing troubles me less than time and space, because I never think of them.” Most of us don’t worry about time and space most of the time, whatever that may be; but we all do wonder sometimes what time is, how it began, and where it is leading us.
Any sound scientific theory, whether of time or of any other concept, should in my opinion be based on the most workable philosophy of science: the positivist approach put forward by Karl Popper and others. According to this way of thinking, a scientific theory is a mathematical model that describes and codifies the observations we make. A good theory will describe a large range of phenomena on the basis of a few simple postulates and will make definite predictions that can be tested. If the predictions agree with the observations, the theory survives that test, though it can never be proved to be correct. On the other hand, if the observations disagree with the predictions, one has to discard or modify the theory. (At least, that is what is supposed to happen. In practice, people often question the accuracy of the observations and the reliability and moral character of those making the observations.) If one takes the positivist position, as I do, one cannot say what time actually is. All one can do is describe what has been found to be a very good mathematical model for time and say what predictions it makes.
Isaac Newton gave us the first mathematical model for time and space in his Principia Mathematica, published in 1687. Newton occupied the Lucasian chair at Cambridge that I now hold, though it wasn’t electrically operated in his time. In Newton’s model, time and space were a background in which events took place but which weren’t affected by them. Time was separate from space and was considered to be a single line, or railroad track, that was infinite in both directions (Fig. 2.2). Time itself was considered eternal, in the sense that it had existed, and would exist, forever. By contrast, most people thought the physical universe had been created more or less in its present state only a few thousand years ago. This worried philosophers such as the German thinker Immanuel Kant. If the universe had indeed been created, why had there been an infinite wait before the creation? On the other hand, if the universe had existed forever, why hadn’t everything that was going to happen already happened, meaning that history was over? In particular, why hadn’t the universe reached thermal equilibrium, with everything at the same temperature?
Kant called this problem an “antimony of pure reason,” because it seemed to be a logical contradiction; it didn’t have a resolution. But it was a contradiction only within the context of the Newtonian mathematical model, in which time was an infinite line, independent of what was happening in the universe. However, as we saw in Chapter 1, in 1915 a completely new mathematical model was put forward by Einstein: the general theory of relativity. In the years since Einstein’s paper, we have added a few ribbons and bows, but our model of time and space is still based on what Einstein proposed. This and the following chapters will describe how our ideas have developed in the years since Einstein’s revolutionary paper. It has been a success story of the work of a large number of people, and I’m proud to have made a small contribution.
General relativity combines the time dimension with the three dimensions of space to form what is called spacetime (see page 33, Fig. 2.3). The theory incorporates the effect of gravity by saying that the distribution of matter and energy in the universe warps and distorts spacetime, so that it is not flat. Objects in this spacetime try to move in straight lines, but because spacetime is curved, their paths appear bent. They move as if affected by a gravitational field.
As a rough analogy, not to be taken too literally, imagine a sheet of rubber. One can place a large ball on the sheet to represent the Sun. The weight of the ball will depress the sheet and cause it to be curved near the Sun. If one now rolls little ball bearings on the sheet, they won’t roll straight across to the other side but instead will go around the heavy weight, like planets orbiting the Sun (Fig. 2.4).
The analogy is incomplete because in it only a two-dimensional section of space (the surface of the rubber sheet) is curved, and time is left undisturbed, as it is in Newtonian theory. However, in the theory of relativity, which agrees with a large number of experiments, time and space are inextricably tangled up. One cannot curve space without involving time as well. Thus time has a shape. By curving space and time, general relativity changes them from being a passive background against which events take place to being active, dynamic participants in what happens. In Newtonian theory, where time existed independently of anything else, one could ask: What did God do before He created the universe? As Saint Augustine said, one should not joke about this, as did a man who said, “He was preparing Hell for those who pry too deep.” It is a serious question that people have pondered down the ages. According to Saint Augustine, before God made heaven and earth, He did not make anything at all. In fact, this is very close to modern ideas.
In general relativity, on the other hand, time and space do not exist independently of the universe or of each other. They are defined by measurements within the universe, such as the number of vibrations of a quartz crystal in a clock or the length of a ruler. It is quite conceivable that time defined in this way, within the universe, should have a minimum or maximum value–in other words, a beginning or an end. It would make no sense to ask what happened before the beginning or after the end, because such times would not be defined.
It was clearly important to decide whether the mathematical model of general relativity predicted that the universe, and time itself, should have a beginning or end. The general prejudice among theoretical physicists, including Einstein, held that time should be infinite in both directions. Otherwise, there were awkward questions about the creation of the universe, which seemed to be outside the realm of science. Solutions of the Einstein equations were known in which time had a beginning or end, but these were all very special, with a large amount of symmetry. It was thought that in a real body, collapsing under its own gravity, pressure or sideways velocities would prevent all the matter falling together to the same point, where the density would be infinite. Similarly, if one traced the expansion of the universe back in time, one would find that the matter of the universe didn’t all emerge from a point of infinite density. Such a point of infinite density was called a singularity and would be a beginning or an end of time.
In 1963, two Russian scientists, Evgenii Lifshitz and Isaac Khalatnikov, claimed to have proved that solutions of the Einstein equations with a singularity all had a special arrangement of matter and velocities. The chances that the solution representing the universe would have this special arrangement were practically zero. Almost all solutions that could represent the universe would avoid having a singularity of infinite density: Before the era during which the universe has been expanding, there must have been a previous contracting phase during which matter fell together but missed colliding with itself, moving apart again in the present expanding phase. If this were the case, time would continue on forever, from the infinite past to the infinite future.
Not everyone was convinced by the arguments of Lifshitz and Khalatnikov. Instead, Roger Penrose and I adopted a different approach, based not on a detailed study of solutions but on the global structure of spacetime. In general relativity, spacetime is curved not only by massive objects in it but also by the energy in it. Energy is always positive, so it gives spacetime a curvature that bends the paths of light rays toward each other.
Now consider our past light cone (Fig. 2.5), that is, the paths through spacetime of the light rays from distant galaxies that reach us at the present time. In a diagram with time plotted upward and space plotted sideways, this is a cone with its vertex, or point, at us. As we go toward the past, down the cone from the vertex, we see galaxies at earlier and earlier times. Because the universe has been expanding and everything used to be much closer together, as we look back further we are looking back through regions of higher matter density. We observe a faint background of microwave radiation that propagates to us along our past light cone from a much earlier time, when the universe was much denser and hotter than it is now. By tuning receivers to different frequencies of microwaves, we can measure the spectrum (the distribution of power arranged by frequency) of this radiation. We find a spectrum that is characteristic of radiation from a body at a temperature of 2.7 degrees above absolute zero. This microwave radiation is not much good for defrosting frozen pizza, but the fact that the spectrum agrees so exactly with that of radiation from a body at 2.7 degrees tells us that the radiation must have come from regions that are opaque to microwaves (Fig. 2.6).
Thus we can conclude that our past light cone must pass through a certain amount of matter as one follows it back. This amount of matter is enough to curve spacetime, so the light rays in our past light cone are bent back toward each other (Fig. 2.7).
As one goes back in time, the cross sections of our past light cone reach a maximum size and begin to get smaller again. Our past is pear-shaped (Fig. 2.8).
As one follows our past light cone back still further, the positive energy density of matter causes the light rays to bend toward each other more strongly. The cross section of the light cone will shrink to zero size in a finite time. This means that all the matter inside our past light cone is trapped in a region whose boundary shrinks to zero. It is therefore not very surprising that Penrose and I could prove that in the mathematical model of general relativity, time must have a beginning in what is called the big bang. Similar arguments show that time would have an end, when stars or galaxies collapse under their own gravity to form black holes. We had sidestepped Kant’s antimony of pure reason by dropping his implicit assumption that time had a meaning independent of the universe. Our paper, proving time had a beginning, won the second prize in the competition sponsored by the Gravity Research Foundation in 1968, and Roger and I shared the princely sum of $300. I don’t think the other prize essays that year have shown much enduring value.
There were various reactions to our work. It upset many physicists, but it delighted those religious leaders who believed in an act of creation, for here was scientific proof. Meanwhile, Lifshitz and Khalatnikov were in an awkward position. They couldn’t argue with the mathematical theorems that we had proved, but under the Soviet system they couldn’t admit they had been wrong and Western science had been right. However, they saved the situation by finding a more general family of solutions with a singularity, which weren’t special in the way their previous solutions had been. This enabled them to claim singularities, and the beginning or end of time, as a Soviet discovery.
Most physicists still instinctively disliked the idea of time having a beginning or end. They therefore pointed out that the mathematical model might not be expected to be a good description of spacetime near a singularity. The reason is that general relativity, which describes the gravitational force, is a classical theory, as noted in Chapter 1, and does not incorporate the uncertainty of quantum theory that governs all other forces we know. This inconsistency does not matter in most of the universe most of the time, because the scale on which spacetime is curved is very large and the scale on which quantum effects are important is very small. But near a singularity, the two scales would be comparable, and quantum gravitational effects would be important. So what the singularity theorems of Penrose and myself really established is that our classical region of spacetime is bounded to the past, and possibly to the future, by regions in which quantum gravity is important. To understand the origin and fate of the universe, we need a quantum theory of gravity, and this will be the subject of most of this book.
Quantum theories of systems such as atoms, with a finite number of particles, were formulated in the 1920s, by Heisenberg, Schrödinger, and Dirac. (Dirac was another previous holder of my chair in Cambridge, but it still wasn’t motorized.) However, people encountered difficulties when they tried to extend quantum ideas to the Maxwell field, which describes electricity, magnetism, and light.
One can think of the Maxwell field as being made up of waves of different wavelengths (the distance between one wave crest and the next). In a wave, the field will swing from one value to another like a pendulum (Fig. 2.9).
According to quantum theory, the ground state, or lowest energy state, of a pendulum is not just sitting at the lowest energy point, pointing straight down. That would have both a definite position and a definite velocity, zero. This would be a violation of the uncertainty principle, which forbids the precise measurement of both position and velocity at the same time. The uncertainty in the position multiplied by the uncertainty in the momentum must be greater than a certain quantity, known as Planck’s constant–a number that is too long to keep writing down, so we use a symbol for it:
So the ground state, or lowest energy state, of a pendulum does not have zero energy, as one might expect. Instead, even in its ground state a pendulum or any oscillating system must have a certain minimum amount of what are called zero point fluctuations. These mean that the pendulum won’t necessarily be pointing straight down but will also have a probability of being found at a small angle to the vertical (Fig. 2.10). Similarly, even in the vacuum or lowest energy state, the waves in the Maxwell field won’t be exactly zero but can have small sizes. The higher the frequency (the number of swings per minute) of the pendulum or wave, the higher the energy of the ground state.
Calculations of the ground state fluctuations in the Maxwell and electron fields made the apparent mass and charge of the electron infinite, which is not what observations show. However, in the 1940s the physicists Richard Feynman, Julian Schwinger, and Shin‘ichiro Tomonaga developed a consistent way of removing or “subtracting out” these infinities and dealing only with the finite observed values of the mass and charge. Nevertheless, the ground state fluctuations still caused small effects that could be measured and that agreed well with experiment. Similar subtraction schemes for removing infinities worked for the Yang-Mills field in the theory put forward by Chen Ning Yang and Robert Mills. Yang-Mills theory is an extension of Maxwell theory that describes interactions in two other forces called the weak and strong nuclear forces. However, ground state fluctuations have a much more serious effect in a quantum theory of gravity. Again, each wavelength would have a ground state energy. Since there is no limit to how short the wavelengths of the Maxwell field can be, there are an infinite number of different wavelengths in any region of spacetime and an infinite amount of ground state energy. Because energy density is, like matter, a source of gravity, this infinite energy density ought to mean there is enough gravitational attraction in the universe to curl spacetime into a single point, which obviously hasn’t happened.
One might hope to solve the problem of this seeming contradiction between observation and theory by saying that the ground state fluctuations have no gravitational effect, but this would not work. One can detect the energy of ground state fluctuations by the Casimir effect. If you place a pair of metal plates parallel to each other and close together, the effect of the plates is to reduce slightly the number of wavelengths that fit between the plates relative to the number outside. This means that the energy density of ground state fluctuations between the plates, although still infinite, is less than the energy density outside by a finite amount (Fig. 2.11). This difference in energy density gives rise to a force pulling the plates together, and this force has been observed experimentally. Forces are a source of gravity in general relativity, just as matter is, so it would not be consistent to ignore the gravitational effect of this energy difference.
Another possible solution to the problem might be to suppose there was a cosmological constant such as Einstein introduced in an attempt to have a static model of the universe. If this constant had an infinite negative value, it could exactly cancel the infinite positive value of the ground state energies in free space, but this cosmological constant seems very ad hoc, and it would have to be tuned to extraordinary accuracy.
Fortunately, a totally new kind of symmetry was discovered in the 1970s that provides a natural physical mechanism to cancel the infinities arising from ground state fluctuations. Supersymmetry is a feature of our modern mathematical models that can be described in various ways. One way is to say that spacetime has extra dimensions besides the dimensions we experience. These are called Grassmann dimensions, because they are measured in numbers known as Grassmann variables rather than in ordinary real numbers. Ordinary numbers commute; that is, it does not matter in which order you multiply them: 6 times 4 is the same as 4 times 6. But Grassmann variables anticommute: x times y is the same as —y times x.
Supersymmetry was first considered for removing infinities in matter fields and Yang-Mills fields in a spacetime where both the ordinary number dimensions and the Grassmann dimensions were flat, not curved. But it was natural to extend it to ordinary numbers and Grassmann dimensions that were curved. This led to a number of theories called supergravity, with different amounts of supersymmetry. One consequence of supersymmetry is that every field or particle should have a “superpartner” with a spin that is either 1/2 greater than its own or 1/2 less (Fig 2.12).
The ground state energies of bosons, fields whose spin is a whole number (0, 1, 2 , etc.), are positive. On the other hand, the ground state energies of fermions, fields whose spin is a half number (1/2, 3/2 , etc.), are negative. Because there are equal numbers of bosons and fermions, the biggest infinities cancel in supergravity theories (see Fig 2.13, page 50).
There remained the possibility that there might be smaller but still infinite quantities left over. No one had the patience needed to calculate whether these theories were actually completely finite. It was reckoned it would take a good student two hundred years, and how would you know he hadn’t made a mistake on the second page? Still, up to 1985, most people believed that most supersymmetric supergravity theories would be free of infinities.
Then suddenly the fashion changed. People declared there was no reason not to expect infinities in supergravity theories, and this was taken to mean they were fatally flawed as theories. Instead, it was claimed that a theory named supersymmetric string theory was the only way to combine gravity with quantum theory. Strings, like their namesakes in everyday experience, are one-dimensional extended objects. They have only length. Strings in string theory move through a background spacetime. Ripples on the string are interpreted as particles (Fig. 2.14).
If the strings have Grassmann dimensions as well as their ordinary number dimensions, the ripples will correspond to bosons and fermions. In this case, the positive and negative ground state energies will cancel so exactly that there will be no infinities even of the smaller sort. Superstrings, it was claimed, were the TOE, the Theory of Everything.
Historians of science in the future will find it interesting to chart the changing tide of opinion among theoretical physicists. For a few years, strings reigned supreme and supergravity was dismissed as just an approximate theory, valid at low energy. The qualification “low energy” was considered particularly damning, even though in this context low energies meant particles with energies of less than a billion billion times those of particles in a TNT explosion. If supergravity was only a low energy approximation, it could not claim to be the fundamental theory of the universe. Instead, the underlying theory was supposed to be one of five possible superstring theories. But which of the five string theories described our universe? And how could string theory be formulated, beyond the approximation in which strings were pictured as surfaces with one space dimension and one time dimension moving through a flat background spacetime? Wouldn’t the strings curve the background spacetime?
In the years after 1985, it gradually became apparent that string theory wasn’t the complete picture. To start with, it was realized that strings are just one member of a wide class of objects that can be extended in more than one dimension. Paul Townsend, who, like me, is a member of the Department of Applied Mathematics and Theoretical Physics at Cambridge, and who did much of the fundamental work on these objects, gave them the name “p-branes.” A p-brane has length in p directions. Thus a p=1 brane is a string, a p=2 brane is a surface or membrane, and so on (Fig. 2.15). There seems no reason to favor the p=1 string case over other possible values of p. Instead, we should adopt the principle of p-brane democracy: all p-branes are created equal.
All the p-branes could be found as solutions of the equations of supergravity theories in 10 or 11 dimensions. While 10 or 11 dimensions doesn’t sound much like the spacetime we experience, the idea was that the other 6 or 7 dimensions are curled up so small that we don’t notice them; we are only aware of the remaining 4 large and nearly flat dimensions.
I must say that personally, I have been reluctant to believe in extra dimensions. But as I am a positivist, the question “Do extra dimensions really exist?” has no meaning. All one can ask is whether mathematical models with extra dimensions provide a good description of the universe. We do not yet have any observations that require extra dimensions for their explanation. However, there is a possibility we may observe them in the Large Hadron Collider in Geneva. But what has convinced many people, including myself, that one should take models with extra dimensions seriously is that there is a web of unexpected relationships, called dualities, between the models. These dualities show that the models are all essentially equivalent; that is, they are just different aspects of the same underlying theory, which has been given the name M-theory. Not to take this web of dualities as a sign we are on the right track would be a bit like believing that God put fossils into the rocks in order to mislead Darwin about the evolution of life.
These dualities show that the five superstring theories all describe the same physics and that they are also physically equivalent to supergravity (Fig. 2.16). One cannot say that superstrings are more fundamental than supergravity, or vice versa. Rather, they are different expressions of the same underlying theory, each useful for calculations in different kinds of situations. Because string theories don’t have any infinities, they are good for calculating what happens when a few high energy particles collide and scatter off each other. However, they are not of much use for describing how the energy of a very large number of particles curves the universe or forms a bound state, like a black hole. For these situations, one needs supergravity, which is basically Einstein’s theory of curved spacetime with some extra kinds of matter. It is this picture that I shall mainly use in what follows.
To describe how quantum theory shapes time and space, it is helpful to introduce the idea of imaginary time. Imaginary time sounds like something from science fiction, but it is a well-defined mathematical concept: time measured in what are called imaginary numbers. One can think of ordinary real numbers such as 1, 2, -3.5, and so on as corresponding to positions on a line stretching from left to right: zero in the middle, positive real numbers on the right, and negative real numbers on the left (Fig. 2.17).
Imaginary numbers can then be represented as corresponding to positions on a vertical line: zero is again in the middle, positive imaginary numbers plotted upward, and negative imaginary numbers plotted downward. Thus imaginary numbers can be thought of as a new kind of number at right angles to ordinary real numbers. Because they are a mathematical construct, they don’t need a physical realization; one can’t have an imaginary number of oranges or an imaginary credit card bill (Fig. 2.18).
One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?
Einstein’s classical (i.e., nonquantum) general theory of relativity combined real time and the three dimensions of space into a four-dimensional spacetime. But the real time direction was distinguished from the three spatial directions; the world line or history of an observer always increased in the real time direction (that is, time always moved from past to future), but it could increase or decrease in any of the three spatial directions. In other words, one could reverse direction in space, but not in time (Fig. 2.19).
On the other hand, because imaginary time is at right angles to real time, it behaves like a fourth spatial direction. It can therefore have a much richer range of possibilities than the railroad track of ordinary real time, which can only have a beginning or an end or go around in circles. It is in this imaginary sense that time has a shape.
To see some of the possibilities, consider an imaginary time spacetime that is a sphere, like the surface of the Earth. Suppose that imaginary time was degrees of latitude (Fig. 2.20, see page 61). Then the history of the universe in imaginary time would begin at the South Pole. It would make no sense to ask, “What happened before the beginning?” Such times are simply not defined, any more than there are points south of the South Pole. The South Pole is a perfectly regular point of the Earth’s surface, and the same laws hold there as at other points. This suggests that the beginning of the universe in imaginary time can be a regular point of spacetime, and that the same laws can hold at the beginning as in the rest of the universe. (The quantum origin and evolution of the universe will be discussed in the next chapter.)
Another possible behavior is illustrated by taking imaginary time to be degrees of longitude on the Earth. All the lines of longitude meet at the North and South Poles (Fig. 2.21, see page 61) Thus time stands still there, in the sense that an increase of imaginary time, or of degrees of longitude, leaves one in the same spot. This is very similar to the way that ordinary time appears to stand still on the horizon of a black hole. We have come to recognize that this standing still of real and imaginary time (either both stand still or neither does) means that the spacetime has a temperature, as I discovered for black holes. Not only does a black hole have a temperature, it also behaves as if it has a quantity called entropy. The entropy is a measure of the number of internal states (ways it could be configured on the inside) that the black hole could have without looking any different to an outside observer, who can only observe its mass, rotation, and charge. This black hole entropy is given by a very simple formula I discovered in 1974. It equals the area of the horizon of the black hole: there is one bit of information about the internal state of the black hole for each fundamental unit of area of the horizon. This shows that there is a deep connection between quantum gravity and thermodynamics, the science of heat (which includes the study of entropy). It also suggests that quantum gravity may exhibit what is called holography (Fig. 2.22).
Information about the quantum states in a region of spacetime may be somehow coded on the boundary of the region, which has two dimensions less. This is like the way that a hologram carries a three-dimensional image on a two-dimensional surface. If quantum gravity incorporates the holographic principle, it may mean that we can keep track of what is inside black holes. This is essential if we are to be able to predict the radiation that comes out of black holes. If we can’t do that, we won’t be able to predict the future as fully as we thought. This is discussed in Chapter 4. Holography is discussed again in Chapter 7. It seems we may live on a 3-brane–a four-dimensional (three space plus one time) surface that is the boundary of a five-dimensional region, with the remaining dimensions curled up very small. The state of the world on a brane encodes what is happening in the five-dimensional region.
“Provocative and informed ... plenty of comprehensible analogies and no small amount of humor, often self-deprecating.... Best of all, the book is liberally sprinkled with well-conceived, gorgeously rendered and frequently whimsical illustrations.”—Time