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CHAPTER 2 Space and Time
Our present ideas about the motion of bodies date back toGalileo and Newton. Before them people believed Aristotle, whosaid that the natural state of a body was to be at rest andthat it moved only if driven by a force or impulse. It followedthat a heavy body should fall faster than a light one, because itwould have a greater pull toward the earth.
The Aristotelian tradition also held that one could work outall the laws that govern the universe by pure thought: it wasnot necessary to check by observation. So no one until Galileobothered to see whether bodies of different weight did in factfall at different speeds. It is said that Galileo demonstrated thatAristotle’s belief was false by dropping weights from the leaningtower of Pisa. The story is almost certainly untrue, but Galileodid do something equivalent: he rolled balls of different weightsdown a smooth slope. The situation is similar to that of heavybodies falling vertically, but it is easier to observe because theSpeeds are smaller. Galileo’s measurements indicated that eachbody increased its speed at the same rate, no matter what itsweight. For example, if you let go of a ball on a slope thatdrops by one meter for every ten meters you go along, theball will be traveling down the slope at a speed of about onemeter per second after one second, two meters per secondafter two seconds, and so on, however heavy the ball. Ofcourse a lead weight would fall faster than a feather, but thatis only because a feather is slowed down by air resistance. Ifone drops two bodies that don’t have much air resistance, suchas two different lead weights, they fall at the same rate. On themoon, where there is no air to slow things down, the astronautDavid R. Scott performed the feather and lead weightexperiment and found that indeed they did hit the ground atthe same time.
Galileo’s measurements were used by Newton as the basis ofhis laws of motion. In Galileo’s experiments, as a body rolleddown the slope it was always acted on by the same force (itsweight), and the effect was to make it constantly speed up.
This showed that the real effect of a force is always to changethe speed of a body, rather than just to set it moving, as waspreviously thought. It also meant that when-ever a body is notacted on by any force, it will keep on moving in a straight lineat the same speed. This idea was first stated explicitly inNewton’s Principia Mathematica, published in 1687, and isknown as Newton’s first law. What happens to a body when aforce does act on it is given by Newton’s second law. Thisstates that the body will accelerate, or change its speed, at arate that is proportional to the force. (For example, theacceleration is twice as great if the force is twice as great.) Theacceleration is also smaller the greater the mass (or quantity ofmatter) of the body. (The same force acting on a body oftwice the mass will produce half the acceleration.) A familiarexample is provided by a car: the more powerful the engine,the greater the acceleration, but the heavier the car, the smallerthe acceleration for the same engine. In addition to his laws ofmotion, Newton discovered a law to describe the force ofgravity, which states that every body attracts every other bodywith a force that is proportional to the mass of each body.
Thus the force between two bodies would be twice as strong ifone of the bodies (say, body A) had its mass doubled. This iswhat you might expect because one could think of the newbody A as being made of two bodies with the original mass.
Each would attract body B with the original force. Thus thetotal force between A and B would be twice the original force.
And if, say, one of the bodies had twice the mass, and theother had three times the mass, then the force would be sixtimes as strong. One can now see why all bodies fall at thesame rate: a body of twice the weight will have twice the forceof gravity pulling it down, but it will also have twice the mass.
According to Newton’s second law, these two effects will exactlycancel each other, so the acceleration will be the same in allcases.
Newton’s law of gravity also tells us that the farther apartthe bodies, the smaller the force. Newton’s law of gravity saysthat the gravitational attraction of a star is exactly one quarterthat of a similar star at half the distance. This law predicts theorbits of the earth, the moon, and the planets with greataccuracy. If the law were that the gravitational attraction of astar went down faster or increased more rapidly with distance,the orbits of the planets would not be elliptical, they wouldeither spiral in to the sun or escape from the sun.
The big difference between the ideas of Aristotle and thoseof Galileo and Newton is that Aristotle believed in a preferredstate of rest, which any body would take up if it were notdriven by some force Or impulse. In particular, he thought thatthe earth was at rest. But it follows from Newton’s laws thatthere is no unique standard of rest. One could equally well saythat body A was at rest and body B was moving at constantspeed with respect to body A, or that body B was at rest andbody A was moving. For example, if one sets aside for amoment the rotation of the earth and its orbit round the sun,one could say that the earth was at rest and that a train on itwas traveling north at ninety miles per hour or that the trainwas at rest and the earth was moving south at ninety milesper hour. If one carried out experiments with moving bodieson the train, all Newton’s laws would still hold. For instance,playing Ping-Pong on the train, one would find that the ballobeyed Newton’s laws just like a ball on a table by the track.
So there is no way to tell whether it is the train or the earththat is moving.
The lack of an absolute standard of rest meant that onecould not determine whether two events that took place atdifferent times occurred in the same position in space. Forexample, suppose our Ping-Pong ball on the train bouncesstraight up and down, hitting the table twice on the same spotone second apart. To someone on the track, the two bounceswould seem to take place about forty meters apart, because thetrain would have traveled that far down the track between thebounces. The nonexistence of absolute rest therefore meant thatone could not give an event an absolute position in space, asAristotle had believed. The positions of events and the distancesbetween them would be different for a person on the train andone on the track, and there would be no reason to prefer oneperson’s position to the other’s.
Newton was very worried by this lack of absolute position,or absolute space, as it was called, because it did not accordwith his idea of an absolute God. In fact, he refused to acceptlack of absolute space, even though it was implied by his laws.
He was severely criticized for this irrational belief by manypeople, most notably by Bishop Berkeley, a philosopher whobelieved that all material objects and space and time are anillusion. When the famous Dr. Johnson was told of Berkeley’sopinion, he cried, “I refute it thus!” and stubbed his toe on alarge stone.
Both Aristotle and Newton believed in absolute time. That is,they believed that one could unambiguously measure theinterval of time between two events, and that this time wouldbe the same whoever measured it, provided they used a goodclock. Time was completely separate from and independent ofspace. This is what most people would take to be thecommonsense view. However, we have had to change ourideas about space and time. Although our apparentlycommonsense notions work well when dealing with things likeapples, or planets that travel comparatively slowly, they don’twork at all for things moving at or near the speed of light.
The fact that light travels at a finite, but very high, speedwas firstdiscovered in 1676 by the Danish astronomer Ole ChristensenRoemer. He observed that the times at which the moons ofJupiter appeared to pass behind Jupiter were not evenlyspaced, as one would expect if the moons went round Jupiterat a constant rate. As the earth and Jupiter orbit around thesun, the distance between them varies. Roemer noticed thateclipses of Jupiter’s moons appeared later the farther we werefrom Jupiter. He argued that this was because the light fromthe moons took longer to reach us when we were fartheraway. His measurements of the variations in the distance of theearth from Jupiter were,? however, not very accurate, and so his value for the speedof light was 140,000 miles per second, compared to themodern value of 186,000 miles per second. Nevertheless,Roemer’s achievement, in not only proving that light travels ata finite speed, but also in measuring that speed, wasremarkable - coming as it did eleven years before Newton’spublication of Principia Mathematica. A proper theory of thepropagation of light didn’t come until 1865, when the Britishphysicist James Clerk Maxwell succeeded in unifying the partialtheories that up to then had been used to describe the forcesof electricity and magnetism. Maxwell’s equations predicted thatthere could be wavelike disturbances in the combinedelectromagnetic field, and that these would travel at a fixedspeed, like ripples on a pond. If the wavelength of these waves(the distance between one wave crest and the next) is a meteror more, they are what we now call radio waves. Shorterwavelengths are known as microwaves (a few centimeters) orinfrared (more than a ten-thousandth of a centimeter). Visiblelight has a wavelength of between only forty and eightymillionths of a centimeter. Even shorter wavelengths are knownas ultraviolet, X rays, and gamma rays.
Maxwell’s theory predicted that radio or light waves shouldtravel at a certain fixed speed. But Newton’s theory had got ridof the idea of absolute rest, so if light was supposed to travelat a fixed speed, one would have to say what that fixed speedwas to be measured relative to.
It was therefore suggested that there was a substance calledthe “ether” that was present everywhere, even in “empty”
space. Light waves should travel through the ether as soundwaves travel through air, and their speed should therefore berelative to the ether. Different observers, moving relative to theether, would see light coming toward them at different speeds,but light’s speed relative to the ether would remain fixed. Inparticular, as the earth was moving through the ether on itsorbit round the sun, the speed of light measured in thedirection of the earth’s motion through the ether (when wewere moving toward the source of the light) should be higherthan the speed of light at right angles to that motion (when wear not moving toward the source). In 1887Albert Michelson(who later became the first American to receive the Nobel Prizefor physics) and Edward Morley carried out a very carefulexperiment at the Case School of Applied Science in Cleveland.
They compared the speed of light in the direction of the earth’smotion with that at right angles to the earth’s motion. To theirgreat surprise, they found they were exactly the same!
Between 1887 and 1905 there were several attempts, mostnotably by the Dutch physicist Hendrik Lorentz, to explain theresult of the Michelson-Morley experiment in terms of objectscontracting and clocks slowing down when they moved throughthe ether. However, in a famous paper in 1905, a hithertounknown clerk in the Swiss patent office, Albert Einstein,pointed out that the whole idea of an ether was unnecessary,providing one was willing to abandon the idea of absolute time.
A similar point was made a few weeks later by a leadingFrench mathematician, Henri Poincare. Einstein’s argumentswere closer to physics than those of Poincare, who regardedthis problem as mathematical. Einstein is usually given the creditfor the new theory, but Poincare is remembered by having hisname attached to an important part of it.
The fundamental postulate of the theory of relativity, as itwas called, was that the laws of science should be the samefor all freely moving observers, no matter what their speed.
This was true for Newton’s laws of motion, but now the ideawas extended to include Maxwell’s theory and the speed oflight: all observers should measure the same speed of light, nomatter how fast they are moving. This simple idea has someremarkable consequences. Perhaps the best known are theequivalence of mass and energy, summed up in Einstein’sfamous equation E=mc2 (where E is energy, m is mass, and cis the speed of light), and the law that nothing may travelfaster than the speed of light. Because of the equivalence ofenergy and mass, the energy which an object has due to itsmotion will add to its mass. In other words, it will make itharder to increase its speed. This effect is only really significantfor objects moving at speeds close to the speed of light. Forexample, at 10 percent of the speed of light an object’s massis only 0.5 percent more than normal, while at 90 percent ofthe speed of light it would be more than twice its normalmass. As an object approaches the speed of light, its massrises ever more quickly, so it takes more and more energy tospeed it up further. It can in fact never reach the speed oflight, because by then its mass would have become infinite, andby the equivalence of mass and energy, it would have taken aninfinite amount of energy to get it there. For this reason, anynormal object is forever confined by relativity to move atspeeds slower than the speed of light. Only light, or otherwaves that have no intrinsic mass, can move at the speed oflight.
An equally remarkable consequence of relativity is the way ithas revolutionized our ideas of space and time. In Newton’stheory, if a pulse of light is sent from one place to another,different observers would agree on the time that the journeytook (since time is absolute), but will not always agree on howfar the light traveled (since space is not absolute). Since thespeed of the light is just the distance it has traveled divided bythe time it has taken, different observers would measuredifferent speeds for the light. In relativity, on the other hand,all observers must agree on how fast light travels. They still,however, do not agree on the distance the light has traveled,so they must therefore now also disagree over the time it hastaken. (The time taken is the distance the light has traveled -which the observers do not agree on - divided by the light’sspeed - which they do agree on.) In other words, the theoryof relativity put an end to the idea of absolute time! Itappeared that each observer must have his own measure oftime, as recorded by a clock carried with him, and thatidentical clocks carried by different observers would notnecessarily agree.
Each observer could use radar to say where and when anevent took place by sending out a pulse of light or radiowaves. Part of the pulse is reflected back at the event and theobserver measures the time at which he receives the echo. Thetime of the event is then said to be the time halfway betweenwhen the pulse was sent and the time when the reflection wasreceived back: the distance of the event is half the time takenfor this round trip, multiplied by the speed of light. (An event,in this sense, is something that takes place at a single point inspace, at a specified point in time.) This idea is shown in Fig.
2.1, which is an example of a space-time diagram. Using thisprocedure, observers who are moving relative to each other willassign different times and positions to the same event. Noparticular observer’s measurements are any more correct thanany other observer’s, but all the measurements are related. Anyobserver can work out precisely what time and position anyother observer will assign to an event, provided he knows theother observer’s relative velocity.
Nowadays we use just this method to measure distancesprecisely, because we can measure time more accurately thanlength. In effect, the meter is defined to be the distancetraveled by light in 0.000000003335640952 second, asmeasured by a cesium clock. (The reason for that particularnumber is that it corresponds to the historical definition of themeter - in terms of two marks on a particular platinum barkept in Paris.) Equally, we can use a more convenient, newunit of length called a light-second. This is simply defined asthe distance that light travels in one second. In the theory ofrelativity, we now define distance in terms of time and thespeed of light, so it follows automatically that every observer willmeasure light to have the same speed (by definition, 1 meterper 0.000000003335640952 second). There is no need tointroduce the idea of an ether, whose presence anyway cannotbe detected, as the Michelson-Morley experiment showed. Thetheory of relativity does, however, force us to changefundamentally our ideas of space and time. We must acceptthat time is not completely separate from and independent ofspace, but is combined with it to form an object calledspace-time.
It is a matter of common experience that one can describethe position of a point in space by three numbers, orcoordinates. For instance, one can say that a point in a roomis seven feet from one wall, three feet from another, and fivefeet above the floor. Or one could specify that a point was ata certain latitude and longitude and a certain height above sealevel. One is free to use any three suitable coordinates, althoughthey have only a limited range of validity. One would notspecify the position of the moon in terms of miles north andmiles west of Piccadilly Circus and feet above sea level. Instead,one might de-scribe it in terms of distance from the sun,distance from the plane of the orbits of the planets, and theangle between the line joining the moon to the sun and theline joining the sun to a nearby star such as Alpha Centauri.
Even these coordinates would not be of much use in describingthe position of the sun in our galaxy or the position of ourgalaxy in the local group of galaxies. In fact, one may describethe whole universe in terms of a collection of overlappingpatches. In each patch, one can use a different set of threecoordinates to specify the position of a point.
An event is something that happens at a particular point inspace and at a particular time. So one can specify it by fournumbers or coordinates. Again, the choice of coordinates isarbitrary; one can use any three well-defined spatial coordinatesand any measure of time. In relativity, there is no realdistinction between the space and time coordinates, just asthere is no real difference between any two space coordinates.
One could choose a new set of coordinates in which, say, thefirst space coordinate was a combination of the old first andsecond space coordinates. For instance, instead of measuringthe position of a point on the earth in miles north of Piccadillyand miles west of Piccadilly, one could use miles northeast ofPiccadilly, and miles north-west of Piccadilly. Similarly, inrelativity, one could use a new time coordinate that was the oldtime (in seconds) plus the distance (in light-seconds) north ofPiccadilly.
It is often helpful to think of the four coordinates of anevent as specifying its position in a four-dimensional spacecalled space-time. It is impossible to imagine a four-dimensionalspace. I personally find it hard enough to visualizethree-dimensional space! However, it is easy to draw diagramsof two-dimensional spaces, such as the surface of the earth.
(The surface of the earth is two-dimensional because theposition of a point can be specified by two coordinates, latitudeand longitude.) I shall generally use diagrams in which timeincreases upward and one of the spatial dimensions is shownhorizontally. The other two spatial dimensions are ignored or,sometimes, one of them is indicated by perspective. (These arecalled space-time diagrams, like Fig. 2.1.) For example, in Fig.
2.2 time is measured upward in years and the distance alongthe line from the sun to Alpha Centauri is measuredhorizontally in miles. The paths of the sun and of AlphaCentauri through space-time are shown as the vertical lines onthe left and right of the diagram. A ray of light from the sunfollows the diagonal line, and takes four years to get from thesun to Alpha Centauri.
As we have seen, Maxwell’s equations predicted that thespeed of light should be the same whatever the speed of thesource, and this has been confirmed by accurate measurements.
It follows from this that if a pulse of light is emitted at aparticular time at a particular point in space, then as time goeson it will spread out as a sphere of light whose size andposition are independent of the speed of the source. After onemillionth of a second the light will have spread out to form asphere with a radius of 300 meters; after two millionths of asecond, the radius will be 600 meters; and so on. It will belike the ripples that spread out on the surface of a pond whena stone is thrown in. The ripples spread out as a circle thatgets bigger as time goes on. If one stacks snapshots of theripples at different times one above the other, the expandingcircle of ripples will mark out a cone whose tip is at the placeand time at which the stone hit the water (Fig. 2.3). Similarly,the light spreading out from an event forms a(three-dimensional) cone in (the four-dimensional) space-time.
This cone is called the future light cone of the event. In thesame way we can draw another cone, called the past lightcone, which is the set of events from which a pulse of light isable to reach the given event (Fig. 2.4).
Given an event P, one can divide the other events in theuniverse into three classes. Those events that can be reachedfrom the event P by a particle or wave traveling at or belowthe speed of light are said to be in the future of P. They willlie within or on the expanding sphere of light emitted from theevent P. Thus they will lie within or on the future light cone ofP in the space-time diagram. Only events in the future of Pcan be affected by what happens at P because nothing cantravel faster than light.
Similarly, the past of P can be defined as the set of allevents from which it is possible to reach the event P travelingat or below the speed of light. It is thus the set of events thatcan affect what happens at P. The events that do not lie inthe future or past of P are said to lie in the elsewhere of P(Fig. 2.5). What happens at such events can neither affect norbe affected by what happens at P. For example, if the sunwere to cease to shine at this very moment, it would not affectthings on earth at the present time because they would be inthe elsewhere of the event when the sun went out (Fig. 2.6).
We would know about it only after eight minutes, the time ittakes light to reach us from the sun. Only then would eventson earth lie in the future light cone of the event at which thesun went out. Similarly, we do not know what is happening atthe moment farther away in the universe: the light that we seefrom distant galaxies left them millions of years ago, and in thecase of the most distant object that we have seen, the light leftsome eight thousand million years ago. Thus, when we look atthe universe, we are seeing it as it was in the past.
If one neglects gravitational effects, as Einstein and Poincaredid in 1905, one has what is called the special theory ofrelativity. For every event in space-time we may construct alight cone (the set of all possible paths of light in space-timeemitted at that event), and since the speed of light is the sameat every event and in every direction, all the light cones will beidentical and will all point in the same direction. The theoryalso tells us that nothing can travel faster than light. Thismeans that the path of any object through space and timemust be represented by a line that lies within the light cone ateach event on it (Fig. 2.7). The special theory of relativity wasvery successful in explaining that the speed of light appears thesame to all observers (as shown by the Michelson-Morleyexperiment) and in describing what happens when things moveat speeds close to the speed of light. However, it wasinconsistent with the Newtonian theory of gravity, which saidthat objects attracted each other with a force that depended onthe distance between them. This meant that if one moved oneof the objects, the force on the other one would changeinstantaneously. Or in other gravitational effects should travelwith infinite velocity, instead of at or below the speed of light,as the special theory of relativity required. Einstein made anumber of unsuccessful attempts between 1908 and 1914 tofind a theory of gravity that was consistent with specialrelativity. Finally, in 1915, he proposed what we now call thegeneral theory of relativity.
Einstein made the revolutionary suggestion that gravity is nota force like other forces, but is a consequence of the fact thatspace-time is not flat, as had been previously assumed: it iscurved, or “warped,” by the distribution of mass and energy init. Bodies like the earth are not made to move on curvedorbits by a force called gravity; instead, they follow the nearestthing to a straight path in a curved space, which is called ageodesic. A geodesic is the shortest (or longest) path betweentwo nearby points. For example, the surface of the earth is atwo-dimensional curved space. A geodesic on the earth is calleda great circle, and is the shortest route between two points(Fig. 2.8). As the geodesic is the shortest path between anytwo airports, this is the route an airline navigator will tell thepilot to fly along. In general relativity, bodies always followstraight lines in four-dimensional space-time, but theynevertheless appear to us to move along curved paths in ourthree-dimensional space. (This is rather like watching anairplane flying over hilly ground. Although it follows a straightline in three-dimensional space, its shadow follows a curvedpath on the two-dimensional ground.)The mass of the sun curves space-time in such a way thatalthough the earth follows a straight path in four-dimensionalspace-time, it appears to us to move along a circular orbit inthree-dimensional space.
fact, the orbits of the planets predicted by general relativityare almost exactly the same as those predicted by theNewtonian theory of gravity. However, in the case of Mercury,which, being the nearest planet to the sun, feels the strongestgravitational effects, and has a rather elongated orbit, generalrelativity predicts that the long axis of the ellipse should rotateabout the sun at a rate of about one degree in ten thousandyears. Small though this effect is, it had been noticed before1915 and served as one of the first confirmations of Einstein’stheory. In recent years the even smaller deviations of the orbitsof the other planets from the Newtonian predictions have beenmeasured by radar and found to agree with the predictions ofgeneral relativity.
Light rays too must follow geodesics in space-time. Again, thefact that space is curved means that light no longer appears totravel in straight lines in space. So general relativity predictsthat light should be bent by gravitational fields. For example,the theory predicts that the light cones of points near the sunwould be slightly bent inward, on account of the mass of thesun. This means that light from a distant star that happened topass near the sun would be deflected through a small angle,causing the star to appear in a different position to anobserver on the earth (Fig. 2.9). Of course, if the light fromthe star always passed close to the sun, we would not be ableto tell whether the light was being deflected or if instead thestar was really where we see it. However, as the earth orbitsaround the sun, different stars appear to pass behind the sunand have their light deflected. They therefore change theirapparent position relative to other stars. It is normally verydifficult to see this effect, because the light from the sun makesit impossible to observe stars that appear near to the sun thesky. However, it is possible to do so during an eclipse of thesun, when the sun’s light is blocked out by the moon.
Einstein’s prediction of light deflection could not be testedimmediately in 1915, because the First World War was inprogress, and it was not until 1919 that a British expedition,observing an eclipse from West Africa, showed that light wasindeed deflected by the sun, just as predicted by the theory.
This proof of a German theory by British scientists was hailedas a great act of reconciliation between the two countries afterthe war. It is ionic, therefore, that later examination of thephotographs taken on that expedition showed the errors wereas great as the effect they were trying to measure. Theirmeasurement had been sheer luck, or a case of knowing theresult they wanted to get, not an uncommon occurrence inscience. The light deflection has, however, been accuratelyconfirmed by a number of later observations.
Another prediction of general relativity is that time shouldappear to slower near a massive body like the earth. This isbecause there is a relation between the energy of light and itsfrequency (that is, the number of waves of light per second):
the greater the energy, the higher frequency. As light travelsupward in the earth’s gravitational field, it loses energy, and soits frequency goes down. (This means that the length of timebetween one wave crest and the next goes up.) To someonehigh up, it would appear that everything down below wasmaking longer to happen. This prediction was tested in 1962,using a pair of very accurate clocks mounted at the top andbottom of a water tower. The clock at the bottom, which wasnearer the earth, was found to run slower, in exact agreementwith general relativity. The difference in the speed of clocks atdifferent heights above the earth is now of considerablepractical importance, with the advent of very accurate navigationsystems based on signals from satellites. If one ignored thepredictions of general relativity, the position that one calculatedwould be wrong by several miles!
Newton’s laws of motion put an end to the idea of absoluteposition in space. The theory of relativity gets rid of absolutetime. Consider a pair of twins. Suppose that one twin goes tolive on the top of a mountain while the other stays at sealevel. The first twin would age faster than the second. Thus, ifthey met again, one would be older than the other. In thiscase, the difference in ages would be very small, but it wouldbe much larger if one of the twins went for a long trip in aspaceship at nearly the speed of light. When he returned, hewould be much younger than the one who stayed on earth.
This is known as the twins paradox, but it is a paradox only ifone has the idea of absolute time at the back of one’s mind.
In the theory of relativity there is no unique absolute time, butinstead each individual has his own personal measure of timethat depends on where he is and how he is moving.
Before 1915, space and time were thought of as a fixedarena in which events took place, but which was not affectedby what happened in it. This was true even of the specialtheory of relativity. Bodies moved, forces attracted and repelled,but time and space simply continued, unaffected. It was naturalto think that space and time went on forever.
The situation, however, is quite different in the general theoryof relativity. Space and time are now dynamic quantities: whena body moves, or a force acts, it affects the curvature of spaceand time - and in turn the structure of space-time affects theway in which bodies move and forces act. Space and time notonly affect but also are affected by everything that happens inthe universe. Just as one cannot talk about events in theuniverse without the notions of space and time, so in generalrelativity it became meaningless to talk about space and timeoutside the limits of the universe.
In the following decades this new understanding of spaceand time was to revolutionize our view of the universe. The oldidea of an essentially unchanging universe that could haveexisted, and could continue to exist, forever was replaced bythe notion of a dynamic, expanding universe that seemed tohave begun a finite time ago, and that might end at a finitetime in the future. That revolution forms the subject of thenext chapter. And years later, it was also to be the startingpoint for my work in theoretical physics. Roger Penrose and Ishowed that Einstein’s general theory of relativity implied thatthe universe must have a beginning and, possibly, an end.

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