Book by Hawking Stephen
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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 ...
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. His phenomenal multi-million-copy bestseller A Brief History of Time introduced the fascinating world of theoretical physics to readers all over the world. Now, in a major new lavishly illustrated book, Hawking turns to the major breakthroughs that have occurred in the years since the release of his acclaimed first book. He brings to us the cutting edge of theoretical physics, where truth is often stranger than fiction, and explains in layman's terms the principles that control our universe.
Like many in the international scientific community, Professor Hawking is seeking to uncover the grail of science - the elusive Theory of Everything that lies at the heart of the cosmos. In The Universe in a Nutshell, 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. In this most exciting intellectual adventure 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'. He takes us to the wild frontiers of science where superstring theory and p-branes may hold the final clue to the puzzle.
The Universe in a Nutshell is essential reading for all those who want to understand the universe in which we live.
STEPHEN HAWKING is the Lucasian Professor of Mathematics at the University of Cambridge, and is regarded as one of the most brilliant theoretical physicists since Einstein.
Acclaim for A Brief History of Time
'This book marries a child's wonder to a genius's intellect. We journey into Hawking's universe, while marvelling at his mind.' Sunday Times
'One of the most brilliant scientific minds since Einstein.' Daily Express
'He can explain the complexities of cosmological physics with an engaging combination of clarity and wit . . . He is a brain of extraordinary power.' Observer
'It is the publishing sensation of the last decade.' Spectator
'His mind seems to soar ever more brilliantly across the vastness of space and time to unlock the secrets of the universe.' Time Magazine
'Hawking clearly possesses a natural teacher's gifts - easy, good-natured humor and an ability to illustrate highly complex propositions with analogies plucked from daily life.' New York Times
'Genius unique, tragic and triumphant . . . Hawking takes us through the evolution of modern thinking on cosmology, from Aristotle and Copernicus, through Galileo and Newton, to Einstein and, indeed, Hawking himself.' Sydney Morning Herald
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