A Briefer History of Time

Author: Stephen Hawking

• Most people nowadays would find the picture of our universe as an infinite tower of turtles rather ridiculous. But why should we think we know better?

• The nearest star, other than our sun, is called Proxima Centauri (also known as Alpha Centauri C), which is about four light-years away.

• Around 340 B. C., the Greek philosopher Aristotle wrote a book called On the Heavens. In that book, Aristotle made good arguments for believing that the earth was a sphere rather than flat like a plate.

• They noticed that although almost all of the thousands of lights they saw seemed to move together across the sky, five of them (not counting the moon) did not. They would sometimes wander off from a regular east-west path and then double back. These lights were named planets—the Greek word for “wanderer.”

• The Greeks observed only five planets because five are all we can see with the naked eye: Mercury, Venus, Mars, Jupiter, and Saturn.

• In 1609, Galileo started observing the night sky with a telescope, which had just been invented.

• A theory is a good theory if it satisfies two requirements. It must accurately describe a large class of observations on the basis of a model that contains only a few arbitrary elements, and it must make definite predictions about the results of future observations.

• very accurate observations of the planet Mercury revealed a small difference between its motion and the predictions of Newton’s theory of gravity. Einstein’s general theory of relativity predicted a slightly different motion than Newton’s theory did. The fact that Einstein’s predictions matched what was seen, while Newton’s did not, was one of the crucial confirmations of the new theory.

• The eventual goal of science is to provide a single theory that describes the whole universe. However, the approach most scientists actually follow is to separate the problem into two parts. First, there are the laws that tell us how the universe changes with time. (If we know what the universe is like at any one time, these physical laws tell us how it will look at any later time.) Second, there is the question of the initial state of the universe.

• Today scientists describe the universe in terms of two basic partial theories—the general theory of relativity and quantum mechanics.

• Unfortunately, however, these two theories are known to be inconsistent with each other—they cannot both be correct.

• Yet if there really were a complete unified theory, it would also presumably determine our actions—so the theory itself would determine the outcome of our search for it!

• And why should it determine that we come to the right conclusions from the evidence? Might it not equally well determine that we draw the wrong conclusion? Or no conclusion at all? The only answer that we can give to this problem is based on Darwin’s principle of natural selection.

• Because the partial theories that we already have are sufficient to make accurate predictions in all but the most extreme situations, the search for the ultimate theory of the universe seems difficult to justify on practical grounds.

• The discovery of a complete unified theory, therefore, may not aid the survival of our species.

• It is said that Galileo demonstrated that Aristotle’s belief was false by dropping weights from the Leaning Tower of Pisa in Italy. This story is almost certainly untrue,

• On the moon, where there is no air to slow things down, the astronaut David R. Scott performed the feather-and-lead-weight experiment and found that indeed they did hit the ground at the same time.

• The big difference between the ideas of Aristode and those of Galileo and Newton is that Aristotle believed in a preferred state of rest, which any body would take up if it was not driven by some force or impulse. In particular, he thought that the earth was at rest. But it follows from Newton’s laws that there is no unique standard of rest.

• For example, if you set aside for a moment the rotation of the earth and its orbit around the sun, you could say that the earth was at rest and that a train on it was traveling north at ninety miles per hour or that the train was at rest and the earth was moving south at ninety miles per hour.

• Actually, the lack of an absolute standard of rest has deep implications for physics: it means that we cannot determine whether two events that took place at different times occurred in the same position in space.

• They also discovered that time was not completely separate from and independent of space.

• They may seem counter to our experience, but although our apparently commonsense notions work well when dealing with things such as apples, or planets that travel comparatively slowly, they don’t work at all for things moving at or near the speed of light.

• THE FACT THAT LIGHT TRAVELS AT a finite but very high speed was first discovered in 1676 by the Danish astronomer Ole Christensen Roemer.

• These eclipses of Jupiter’s moons ought to occur at regular intervals, but Roemer observed that the eclipses were not evenly spaced.

• his value for the speed of light was 140,000 miles per second, compared to the modern value of 186,000 miles per second.

• A proper theory of the propagation of light didn’t come until 1865, when the British physicist James Clerk Maxwell succeeded in unifying the partial theories that up to then had been used to describe the forces of electricity and magnetism.

• Maxwell showed mathematically that these electric and magnetic forces do not arise from particles acting directly on each other; rather, every electric charge and current creates a field in the surrounding space that exerts a force on every other charge and current located within that space.

• He found that a single field carries the electric and magnetic forces; thus, electricity and magnetism are inseparable aspects of the same force.

• However, in a famous paper in 1905, a hitherto unknown clerk in the Swiss patent office, Albert Einstein, pointed out that the w hole idea of an ether was unnecessary, provided one was willing to abandon the idea of absolute time (we’ll see why shortly).

• Einstein’s fundamental postulate of the theory of relativity, as it was called, stated that the laws of science should be the same for all freely moving observers, no matter what their speed. This was true for Newton’s laws of motion, but now Einstein extended the idea to include Maxwell’s theory.

• In other words, since Maxwell’s theory dictates that the speed of light has a given value, all freely moving observers must measure that same value, no matter how fast they are moving toward or away from its source. This simple idea certainly explained—without the use of the ether or any other preferred frame of reference—the meaning of the speed of light in Maxwell’s equations,

• We must accept that time is not completely separate from and independent of space but is combined with it to form an object called space-time.

• Another well-known consequence of relativity is the equivalence of mass and energy, summed up in Einstein’s famous equation E=mc2

• But according to the equivalence of energy and mass, kinetic energy adds to an object’s mass, so the faster an object moves, the harder it is to further increase the object’s speed.

• This effect is really significant only for objects moving at speeds close to the speed of light.

• According to the theory of relativity, an object can in fact never reach the speed of light, because by then its mass would have become infinite, and by the equivalence of mass and energy, it would have taken an infinite amount of energy to get it there. This is the reason that any normal object is forever confined by relativity to move at speeds slower than the speed of light.

• If, say, the sun suddenly disappeared, Maxwell’s theory tells us that the earth wouldn’t get dark for about another eight minutes (since that is how long it takes light to reach us from the sun) but, according to Newtonian gravity, the earth would immediately cease to feel the sun’s attraction and fly out of orbit.

• EINSTEIN’S THEORY OF GENERAL RELATIVITY IS based on the revolutionary suggestion that gravity is not a force like other forces but a consequence of the fact that space-time is not flat, as had been previously assumed. In general relativity, space-time is curved, or “warped,” by the distribution of mass and energy in it.

• EINSTEIN’S THEORY OF GENERAL RELATIVITY IS based on the revolutionary suggestion that gravity is not a force like other forces but a consequence of the fact that space-time is not flat, as had been previously assumed. In general relativity, space-time is curved, or “warped,” by the distribution of mass and energy in it. Bodies such as the earth are not made to move on curved orbits bv a force called gravity; instead they move in curved orbits because they follow the nearest thing to a straight path in a curved space, which is called a geodesic.

• Technically speaking, a geodesic is defined as the shortest (or longest) path between two nearby points.

• A geodesic on the earth is called a great circle. The equator is a great circle. So is any other circle on the globe whose center coincides with the center of the earth.

• In general relativity, bodies always follow geodesics in four-dimensional space-time.

• Though the phenomenon is harder to picture, the mass of the sun curves space-time in such a way that although the earth follows a straight path in four-dimensional space-time, it appears to us to move along a nearly circular orbit in three-dimensional space.

• Light rays too must follow geodesics in space-time. Again, the fact that space is curved means that light no longer appears to travel in straight lines in space, so general relativity predicts that gravitational fields should bend light.

• the theory predicts that the path of light near the sun would be slightly bent inward, on account of the mass of the sun.

• This proof of a German theory by British scientists was hailed as a great act of reconciliation between the two countries after the war.

• It is ironic, therefore, that later examination of the photographs taken on that expedition showed the errors were as great as the effect they were trying to measure. Their measurement had been sheer luck, or perhaps a case of knowing the result they wanted to get-not an uncommon occurrence in science.

• Another prediction of general relativity is that time should appear to run slower near a massive body such as the earth.

• Recall that the fundamental postulate of special relativity stated that the laws of science should be the same for all freely moving observers, no matter what speed they were moving at.

• in small enough regions of space, it is impossible to tell if you are at rest in a gravitational field or uniformly accelerating in empty space.

• Einstein realized that just as you cannot tell from inside a train whether or not you are moving uniformly, you also cannot tell from inside the elevator whether you are uniformly accelerating or in a uniform gravitational field. The result was his principle of equivalence.

• The principle of equivalence, and the above example of it, is true only if inertial mass (the mass in Newton’s second law that determines how much you accelerate in response to a force) and gravitational mass (the mass in Newton’s law of gravity that determines how much gravitational force you feel) are the same thing

• because if both kinds of mass are the same, then all objects in a gravitational field will fall at the same rate, no matter what their mass.

• Now that we know the principle of equivalence, we can start to follow Einstein’s logic by doing another thought experiment that shows why time must be affected by gravity.

• just as special relativity tells us that time runs differently for observers in relative motion, general relativity tells us that time runs differently for observers at different heights in a gravitational field.

• The effect is a small one-a clock on the surface of the sun would gain only about a minute a year as compared to one on the surface of the earth. Yet with the advent of very accurate navigation systems based on signals from satellites, the difference in the speed of clocks at different heights above the earth is now of considerable practical importance. If you ignored the predictions of general relativity, the position that you calculated would be wrong by several miles!

• This is known as the twins paradox, but it is a paradox only if you have the idea of absolute time at the back of your mind. In the theory of relativity there is no unique absolute time; instead, each individual has his own personal measure of time that depends on where he is and how he is moving.

• Space and time are now dynamic quantities: when a body moves or a force acts, it affects the curvature of space and time—and in turn the structure of space-time affects the way in which bodies move and forces act.

• As the earth goes around the sun, we see the nearer stars from different positions against the background of more distant stars. The effect is the same one you see when you are driving down an open road and the relative positions of nearby trees seem to change against the background of whatever is on the horizon. The nearer the trees, the more they seem to move. This change in relative position is called parallax.

• As long ago as 1750, some astronomers were suggesting that the appearance of the Milky Way could be explained if most of the visible stars lie in a single disklike configuration, one example of what we now call a spiral galaxy. Only a few decades later, the astronomer Sir William Herschel confirmed this idea by painstakingly cataloguing the positions

• The luminosity of nearby stars can be calculated from their apparent brightness because their parallax enables us to know their distance.

• Hubble noted that these nearby stars could be classified into certain types by the kind of light they give off. The same type of stars always had the same luminosity. He then argued that if we found these types of stars in a distant galaxy, we could assume that they had the same luminosity as the similar stars nearby. With that information, we could calculate the distance to that galaxy. If we could do this for a number of stars in the same galaxy and our calculations always gave the same distance, we could be fairly confident of our estimate. In this way, Hubble worked out the distances to nine different galaxies.

• By focusing a telescope on an individual star or galaxy, one can observe the spectrum of the light from that star or galaxy. One thing this light tells us is temperature. In 1860, the German physicist Gustav Kirchhoff realized that any material body, such as a star, will give off light or other radiation when heated, just as coals glow when they are heated.

• The light such glowing objects give off is due to the thermal motion of the atoms within them. It is called blackbody radiation

• We find that certain very specific colors are missing, and these missing colors may vary from star to star. Since we know that each chemical element absorbs a characteristic set of very specific colors, by matching these to those that are missing from a star’s spectrum we can determine exactly which elements are present in that star’s atmosphere.

• To physicists, the shifting of color or frequency is known as the Doppler effect.

• The faster the car is moving, the greater the effect, so we can use the Doppler effect to measure speed.

• The behavior of light or radio waves is similar. Indeed, the police make use of the Doppler effect to measure the speed of cars by measuring the wavelength of pulses of radio waves reflected off them.

• It was quite a surprise, therefore, to find that most galaxies appeared red-shifted: nearly all were moving away from us! More surprising still was the finding that Hubble published in 1929: even the size of a galaxy’s red shift is not random but is directly proportional to the galaxy’s distance from us. In other words, the farther a galaxy is, the faster it is moving away!

• Friedmann made two very simple assumptions about the universe: that the universe looks identical in whichever direction we look, and that this would also be true if we were observing the universe from anywhere else. From these two ideas alone, Friedmann showed, by solving the equations of general relativity, that we should not expect the universe to be static.

• Dicke and Peebles argued that we should still be able to see the glow of the early universe, because light from very distant parts of it would only just be reaching us now. However, the expansion of the universe meant that this light should be so greatly red-shifted that it would appear to us now as microwave radiation, rather than visible light. Dicke and Peebles were preparing to look for this radiation when Penzias and Wilson heard about their work and realized that they had already found it. For this, Penzias and Wilson were awarded the Nobel Prize in 1978 (which seems a bit hard on Dicke and Peebles, not to mention Gamow).

• Friedmann derived only one model of the universe. But if his assumptions are correct, there are actually three possible types of solutions to Einstein’s equations, that is, three different kinds of Friedmann models—and three different ways the universe can behave.

• A remarkable feature of the first kind of Friedmann model is that in it the universe is not infinite in space, but neither does space have any boundary. Gravity is so strong that space is bent round onto itself. This is rather like the surface of the earth, which is finite but has no boundary.

• If you keep traveling in a certain direction on the surface of the earth, you never come up against an impassable barrier or fall over the edge, and you eventually come back to where you started. In this model, space is just like this, but with three dimensions instead of two for the earth’s surface.

• The idea that you could go right round the universe and end up where you started makes good science fiction, but it doesn’t have much practical significance, because it can be shown that the universe would collapse to zero size before you could get around.

• It is so large, you would need to travel faster than light in order to end up where you started before the universe came to an end—and that is not allowed!

• The most basic analysis depends on two things: the present rate of expansion of the universe, and its present average density (the amount of matter in a given volume of space). The faster the current rate of expansion, the greater the gravitational force required to stop it, and thus the greater the density of matter needed. If the average density is greater than a certain critical value (determined by the rate of expansion), the gravitational attraction of the matter in the universe will succeed in halting its expansion and cause it to collapse—corresponding to the first Friedmann model. If the average density is less than the critical value, there is not enough gravitational pull to stop the expansion, and the universe will expand forever— corresponding to Friedmann’s second model.

• All our theories of cosmology are formulated on the assumption that space-time is smooth and nearly flat.

• This means that questions such as who set up the conditions for the big bang are not questions that science addresses.

• Another infinity that arises if the universe has zero size is in temperature. At the big bang itself, the universe is thought to have been infinitely hot.

• Atoms are made of smaller particles: electrons, protons, and neutrons. The protons and neutrons themselves are made of yet smaller particles called quarks.

• Any time an electron meets up with a positron, both will be annihilated, but the reverse process is not so easy: in order for two massless particles such as photons to create a particle/antiparticle pair such as an electron and a positron, the colliding massless particles must have a certain minimum energy.

• At this temperature, a force called the strong force would have played an important role. The strong force, which we will discuss in more detail in Chapter 11, is a short-range attractive force that can cause protons and neutrons to bind to each other, forming nuclei.

• It is, moreover, very difficult to explain in any other way why about one-quarter of the mass of the universe is in the form of helium.

• According to Guth, the radius of the universe increased by a million million million million million—1 with thirty zeros after it—times in only a tiny fraction of a second. Any irregularities in the universe would have been smoothed out by this expansion, just as the wrinkles in a balloon are smoothed away when you blow it up.

• As the collapsing region got smaller, it would spin faster—just as skaters spinning on ice spin faster as they draw in their arms.

• When a star runs out of fuel, it starts to cool off and gravity takes over, causing it to contract. This contraction squeezes the atoms together and causes the star to become hotter again. As the star heats up further, it would start to convert helium into heavier elements such as carbon or oxygen. This, however, would not release much more energy, so a crisis would occur. What happens next is not completely clear, but it seems likely that the central regions of the star would collapse to a very dense state, such as a black hole.

• The term “black hole” is of very recent origin. It was coined in 1969 by the American scientist John Wheeler as a graphic description of an idea that goes back at least two hundred years, to a time when there were two theories about light: one, which Newton favored, was that it was composed of particles,

• and the other was that it was made of waves.

• The outer boundary of a black hole is called the event horizon.

• In order to understand what you would see if you were watching a massive star collapse to form a black hole, it is necessary to remember that in the theory of relativity there is no absolute time. In other words, each observer has his own measure of time. The passage of time for someone on a star’s surface will be different from that for someone at a distance, because the gravitational field is stronger on the star’s surface.

• We learned from our earlier thought experiment aboard the rocket ship that gravity slows time, and the stronger the gravity, the greater the effect. The astronaut on the star is in a stronger gravitational field than his companions in orbit, so what to him is one second will be more than one second on their clocks.

• Sometimes, when a very massive star collapses, the outer regions of the star may get blown off in a tremendous explosion called a supernova.

• In fact, it was recently proposed that a die-off of marine creatures that occurred at the interface of the Pleistocene and Pliocene epochs about two million years ago was caused by cosmic ray radiation from a supernova in a nearby cluster of stars called the Scorpius-Centaurus association.

• Some scientists believe that advanced life is likely to evolve only in regions of galaxies in which there are not too many stars—“zones of life”— because in denser regions phenomena such as supernovas would be common enough to regularly snuff out any evolutionary beginnings.

• The leading candidate for the next supernova explosion in our galaxy is a star called Rho Cassiopeiae. Fortunately, it is a safe and comfortable ten thousand light-years from us.

• In a supernova, some of the heavier elements produced near the end of the star’s life are flung back into the galaxy and provide some of the raw material for the next generation of stars.

• Our own sun contains about 2 percent of these heavier elements. It is a second- or third-generation star, formed some five billion years ago out of a cloud of rotating gas containing the debris of earlier supernovas.

• few of the errors would have produced new macromolecules that were even better at reproducing themselves. They would have therefore had an advantage and would have tended to replace the original macromolecules. In this way a process of evolution was started that led to the development of more and more complicated, self-reproducing organisms.

• The first primitive forms of life consumed various materials, including hydrogen sulfide, and released oxygen. This gradually changed the atmosphere to the composition that it has today, and allowed the development of higher forms of life such as fish, reptiles, mammals, and ultimately the human race.

• because mathematics cannot really handle infinite numbers, by predicting that the universe began with the big bang, a time when the density of the universe and the curvature of space-time would have been infinite, the theory of general relativity predicts that there is a point in the universe where the theory itself breaks down, or fails. Such a point is an example of what mathematicians call a singularity.

• When a theory predicts singularities such as infinite density and curvature, it is a sign that the theory must somehow be modified.

• The only input these laws would need is the complete state of the universe at any one time. This is called an initial condition or a boundary condition.

• The equations on which physical theories are based can generally have very different solutions, and you must rely on the initial or boundary conditions to decide which solutions apply. It’s a little like saying that your bank account has large amounts going in and out of it. Whether you end up bankrupt or rich depends not only on the sums paid in and out but also on the boundary or initial condition of how much was in the account to start with.

• The quantum hypothesis explained the observed rate of emission of radiation from hot bodies very well, but its implications for determinism were not realized until 1926, when another German scientist, Werner Heisenberg, formulated his famous uncertainty principle.

• light of a given wavelength has only limited sensitivity: you will not be able to determine the position of the particle more accurately than the distance between the wave crests of the light. Thus, in order to measure the position of the particle precisely, it is necessary to use light of a short wavelength, that is, of a high frequency.

• Because Planck’s constant is so tiny, the effects of the trade-off, and of quantum theory in general, are, like the effects of relativity, not directly noticeable in our everyday lives.

• The limit dictated by the uncertainty principle does not depend on the way in which you try to measure the position or velocity of the particle, or on the type of particle. Heisenberg’s uncertainty principle is a fundamental, inescapable property of the world, and it has had profound implications for the way in which we view the world.

• The uncertainty principle signaled an end to Laplace’s dream of a theory of science, a model of the universe that would be completely deterministic.

• It seems better to employ the principle of economy known as Occam’s razor and cut out all the features of the theory that cannot be observed.

• This approach led Heisenberg, Erwin Schrödinger, and Paul Dirac in the 1920s to reformulate Newton’s mechanics into a new theory called quantum mechanics, based on the uncertainty principle.

• One of the revolutionary properties of quantum mechanics is that it does not predict a single definite result for an observation. Instead, it predicts a number of different possible outcomes and tells us how likely each of these is.

• according to quantum theory there is a certain probability that the dart will hit the bull’s-eye, and also a nonzero probability that it will land in any other given area of the board.

• A dart made of a single atom might have a 90 percent probability of hitting the bull’s-eye, with a 5 percent chance of hitting elsewhere on the board, and another 5 percent chance of missing it completely.

• All you can say is that if you repeat the experiment many times, you can expect that, on average, ninety times out of each hundred times you repeat the experiment, the dart will hit the bull’s-eye.

• he never accepted that the universe was governed by chance; his feelings were summed up in his famous statement “God does not play dice.”

• However, in general, the distance the light has to travel from the light source to the point via one of the slits will be different than for the light traveling via the other slit.

• But if you open the second slit, the number of electrons hitting the screen increases at some points and decreases at others, just as if the electrons were interfering as waves do, rather than acting as particles.

• Now imagine sending the electrons through the slits one at a time. Is there still interference? One might expect each electron to pass through one slit or the other, doing away with the interference pattern. In reality, however, even when the electrons are sent through one at a time, the interference pattern still appears. Each electron, therefore, must be passing through both slits at the same time and interfering with itself!

• The phenomenon of interference between particles has been crucial to our understanding of the structure of atoms, the basic units out of which we, and everything around us, are made.

• It revealed that an electron orbiting around the nucleus could be thought of as a wave, with a wavelength that depended on its velocity. Imagine the wave circling the nucleus at specified distances, as Bohr had postulated. For certain orbits, the circumference of the orbit would correspond to a whole number (as opposed to a fractional number) of wavelengths of the electron. For these orbits the wave crest would be in the same position each time round, so the waves would reinforce each other. These orbits would correspond to Bohr’s allowed orbits. However, for orbits whose lengths were not a whole number of wavelengths, each wave crest would eventually be canceled out by a trough as the electrons went round. These orbits would not be allowed. Bohr’s law of allowed and forbidden orbits now had an explanation.

• Since the structure of molecules and their reactions with each other underlie all of chemistry and biology, quantum mechanics allows us in principle to predict nearly everything we see around us, within the limits set by the uncertainty principle.

• By predicting points of infinite density—singularities—classical (that is, nonquantum) general relativity predicts its own downfall, just as classical mechanics predicted its downfall by suggesting that blackbodies should radiate infinite energy or that atoms should collapse to infinite density. And as with classical mechanics, we hope to eliminate these unacceptable singularities by making classical general relativity into a quantum theory—that is, by creating a quantum theory of gravity.

• When we apply Feynman’s sum over histories to Einstein’s view of gravity, the analogue of the history of a particle is now a complete curved space-time that represents the history of the whole universe.

• it is possible for space-time to be finite in extent and yet to have no singularities that formed a boundary or edge. Space-time would be like the surface of the earth, only with two more dimensions.

• the quantum theory of gravity has opened up a new possibility in w hich there would be no singularities at w hich the laws of science broke down.

• If there is no boundary to space-time, there is no need to specify the behavior at the boundary—no need to know the initial state of the universe. There is no edge of space-time at which we would have to appeal to God or some new law to set the boundary conditions for space-time. We could say: “The boundary condition of the universe is that it has no boundary.” The universe would be completely self-contained and not affected by anything outside itself. It would neither be created nor destroyed. It would just BE. As long as we believed the universe had a beginning, the role of a creator seemed clear. But if the universe is really completely self-contained, having no boundary or edge, having neither beginning nor end, then the answer is not so obvious: what is the role of a creator?

• Thus time became a more personal concept, relative to the observer who measured it.

• Gödel was a mathematician who was famous for proving that it is impossible to prove all true statements, even if you limit yourself to trying to prove all the true statements in a subject as apparently cut-and-dried as arithmetic.

• There seem to be two possible resolutions to the paradoxes posed by time travel. The first may be called the consistent histories approach. It says that even if space-time is warped so that it would be possible to travel into the past, what happens in space-time must be a consistent solution of the laws of physics.

• The other possible way to resolve the paradoxes of time travel might be called the alternative histories hypothesis. The idea here is that when time travelers go back to the past, they enter alternative histories that differ from recorded history.

• the chronology protection conjecture. This says that the laws of physics conspire to prevent macroscopic bodies from carrying information into the past.

• the possibility of time travel remains open. But don’t bet on it. Your opponent might have the unfair advantage of knowing the future.

• Some call these numbers fundamental constants; others call them fudge factors.

• Einstein refused to believe in the reality of quantum mechanics. Yet it seems that the uncertainty principle is a fundamental feature of the universe we live in.

• in 1928 physicist and Nobel Prize winner Max Born told a group of visitors to Göttingen University, “Physics, as we know it, will be over in six months.”

• Force-carrying particles can be grouped into four categories. It should be emphasized that this division into four classes is man-made; it is convenient for the construction of partial theories, but it may not correspond to anything deeper.

• Gravitational attraction is pictured as being caused by the exchange of virtual particles called gravitons.

• The next category is the electromagnetic force, which interacts with electrically charged particles such as electrons and quarks, but not with uncharged particles such as neutrinos.

• The electromagnetic attraction between negatively charged electrons and positively charged protons in the nucleus causes the electrons to orbit the nucleus of the atom, just as gravitational attraction causes the earth to orbit the sun.

• The electromagnetic attraction is pictured as being caused by the exchange of large numbers of virtual particles called photons. Again, the photons that are exchanged are virtual particles.

• It is believed that this force is carried by a particle, called the gluon, which interacts only with itself and with the quarks.

• The main difficulty in finding a theory that unifies gravity with the other forces is that the theory of gravity—general relativity—is the only one that is not a quantum theory: it does not take into account the uncertainty principle.

• the value of a field and its rate of change with time are like the position and velocity (i.e., change of position) of a particle: the uncertainty principle implies that the more accurately one knows one of these quantities, the less accurately one can know the other.

• In the case of fluctuations of the electromagnetic field, these particles are virtual photons, and in the case of fluctuations of the gravitational field, they are virtual gravitons.

• In the case of fluctuations of the weak and strong force fields, however, the virtual pairs are pairs of matter particles, such as electrons or quarks, and their antiparticles.

• Renormalization involves introducing new infinities that have the effect of canceling the infinities that arise in the theory.

• Before string theory, each of the fundamental particles was thought to occupy a single point of space. In string theories, the basic objects are not point particles but things that have a length but no other dimension, like an infinitely thin piece of string.

• If the fundamental objects in the universe are strings, what are the point particles we seem to observe in our experiments? In string theories, what were previously thought of as different point particles are now pictured as various waves on the string, like waves on a vibrating kite string.

• String theories, however, have a bigger problem: they seem to be consistent only if space-time has either ten or twenty-six dimensions, instead of the usual four!

• Indeed, they provide an ideal way of overcoming the normal restriction of general relativity that one cannot travel faster than light or back in time (see Chapter 10). The idea is to take a shortcut through the extra dimensions.

• anthropic principle, which can be paraphrased as “We see the universe the way it is because we exist.”

• only in the few universes that are like ours would intelligent beings develop and ask the question, “Why is the universe the way we see it?” The answer is then simple: if it had been different, we would not be here!

• It seems clear then that life, at least as we know it, can exist only in regions of space-time in which one time dimension and exactly three space dimensions are not curled up small.

• It’s a bit like the old paradox: can God make a stone so heavy that He can’t lift it?

• we had a particle with an energy above what is called the Planck energy, its mass would be so concentrated that it would cut itself off from the rest of the universe and form a little black hole.

• On the other hand, seventy years ago, if Eddington is to be believed, only two people understood the general theory of relativity.

• To try to answer these questions, we adopt some picture of the world. Just as an infinite tower of tortoises supporting the flat earth is such a picture, so is the theory of superstrings.

• In effect, God was confined to the areas that nineteenth-century science did not understand.

• The question remains, however: how or why were the laws and the initial state of the universe chosen?

• the fact that gravity is always attractive implies that the universe must be either expanding or contracting.

• Wittgenstein, the most famous philosopher of the twentieth century, said, “The sole remaining task for philosophy is the analysis of language.”