,transcriber's ,notes3 ,underscores 80 before and after a word or phrase indicate italics in the original text4 ,equal signs 80 before and after a word or phrase indicate bold in the original text4 ,small capitals have been converted to ,,solid capitals4 ,illustrations have been moved so they do not break up paragraphs4 ,typographical and punctuation errors have been silently corrected4 ,,harper',s ,,modern ,,science ,,series ,,the ,a ,b ,c ,,of ,,relativity ,,by ,,bertrand ,,russell ,,author ,,of 8,,the ,,principles ,,of ,,mathematics0 8,,proposed ,,roads ,,to ,,freedom0 ,,and 8,,why ,,men ,,fight0 ,,publishers ,,harper ,,brothers ,,new ,,york ,,and ,,london ,,the ,a ,b ,c ,,of ,,relativity ,copyright1 #aibe1 by ,harper ,brothers ,printed in the ,united ,states of ,america ,contents ,,chapter ,,page ,i4 ,,touch ,,and ,,sight3 ,,the ,,earth ,,and ,,the ,,heavens #a ,,ii4 ,,what ,,happens ,,and ,,what ,,is ,,observed #ad ,,iii4 ,,the ,,velocity ,,of ,,light #bh ,,iv4 ,,clocks ,,and ,,foot ,,rules #dc ,v4 ,,space-,,time #eh ,,vi4 ,,the ,,special ,,theory ,,of ,,relativity #ga ,,vii4 ,,intervals ,,in ,,space-,,time #ia ,,viii4 ,,einstein',s ,,law ,,of ,,gravitation #aaa ,,ix4 ,,proofs ,,of ,,einstein',s ,,law ,,of ,,gravitation #aca ,x4 ,,mass1 ,,momentum1 ,,energy ,,and ,,action #add ,,xi4 ,,is ,,the ,,universe ,,finite8 #afc ,,xii4 ,,conventions ,,and ,,natural ,,laws #agg ,,xiii4 ,,the ,,abolition ,,of 8,,force0 #aib ,,xiv4 ,,what ,,is ,,matter8 #bjf ,,xv4 ,,philosophical ,,consequences #bai ,,the ,a ,b ,c ,,of ,,relativity ,,chapter ,,one3 ,,touch ,,and ,,sight3 ,,the ,,earth ,,and ,,the ,,heavens ,everybody knows that ,einstein has done something astonishing1 but very few people know exactly what it is that he has done4 ,it is generally recognized that he has revolutionized our conception of the physical world1 but his new conceptions are wrapped up in mathematical technicalities4 ,it is true that there are innumerable popular accounts of the theory of relativity1 but they generally cease to be intelligible just at the point where they begin to say something important4 ,the authors are hardly to blame for this4 ,many of the new ideas can be expressed in non-mathematical language1 but they are none the less difficult on that account4 ,what is demanded is a change in our imaginative picture of the world--a picture which has been handed down from remote1 perhaps pre-human1 ancestors1 and has been learned by each one of us in early childhood4 ,a change in our imagination is always difficult1 especially when we are no longer young4 ,the same sort of change was demanded by ,copernicus1 when he taught that the earth is not stationary and the heavens do not revolve about it once a day4 ,to us now there is no difficulty in this idea1 because we learned it before our mental habits had become fixed4 ,einstein's ideas1 similarly1 will seem easy to a generation which has grown up with them2 but for our generation a certain effort of imaginative reconstruction is unavoidable4 ,in exploring the surface of the earth1 we make use of all our senses1 more particularly of the senses of touch and sight4 ,in measuring lengths1 parts of the human body are employed in pre-scientific ages3 a 8foot10 a 8cubit10 a 8span0 are defined in this way4 ,for longer distances1 we think of the time it takes to walk from one place to another4 ,we gradually learn to judge distances roughly by the eye1 but we rely upon touch for accuracy4 ,moreover it is touch that gives us our sense of 8reality40 ,some things cannot be touched3 rainbows1 reflections in looking-glasses1 and so on4 ,these things puzzle children1 whose metaphysical speculations are arrested by the information that what is in the looking glass is not 8real40 ,macbeth's dagger was unreal because it was not 8sensible to feeling as to sight40 ,not only our geometry and physics1 but our whole conception of what exists outside us1 is based upon the sense of touch4 ,we carry this even into our metaphors3 a good speech is 8solid18 a bad speech is 8gas10 because we feel that a gas is not quite 8real40 ,in studying the heavens1 we are debarred from all senses except sight4 ,we cannot touch the sun1 or travel to it2 we cannot walk round the moon1 or apply a foot rule to the ,pleiades4 ,nevertheless1 astronomers have unhesitatingly applied the geometry and physics which they found serviceable on the surface of the earth1 and which they had based upon touch and travel4 ,in doing so1 they brought down trouble on their heads1 which it has been left for ,einstein to clear up4 ,it has turned out that much of what we learned from the sense of touch was unscientific prejudice1 which must be rejected if we are to have a true picture of the world4 ,an illustration may help us to understand how much is impossible to the astronomer as compared to the man who is interested in things on the surface of the earth4 ,let us suppose that a drug is administered to you which makes you temporarily unconscious1 and that when you wake you have lost your memory but not your reasoning powers4 ,let us suppose further that while you were unconscious you were carried into a balloon1 which1 when you come to1 is sailing with the wind in a dark night--the night of the fifth of ,november if you are in ,england1 or of the fourth of ,july if you are in ,america4 ,you can see fireworks which are being sent off from the ground1 from trains1 and from aeroplanes traveling in all directions1 but you cannot see the ground or the trains or the aeroplanes be cause of the darkness4 ,what sort of picture of the world will you form8 ,you will think that nothing is permanent3 there are only brief flashes of light1 which1 during their short existence1 travel through the void in the most various and bizarre curves4 ,you cannot touch these flashes of light1 you can only see them4 ,obviously your geometry and your physics and your metaphysics will be quite different from those of ordinary mortals4 ,if an ordinary mortal is with you in the balloon1 you will find his speech unintelligible4 ,but if ,einstein is with you1 you will understand him more easily than the ordinary mortal would1 because you will be free from a host of preconceptions which prevent most people from understanding him4 ,the theory of relativity depends1 to a considerable extent1 upon getting rid of notions which are useful in ordinary life but not to our drugged balloonist4 ,circumstances on the surface of the earth1 for various more or less accidental reasons1 suggest conceptions which turn out to be inaccurate1 although they have come to seem like necessities of thought4 ,the most important of these circumstances is that most objects on the earth's surface are fairly persistent and nearly stationary from a terrestrial point of view4 ,if this were not the case1 the idea of going a journey would not seem so definite as it does4 ,if you want to travel from ,king's ,cross to ,edinburgh1 you know that you will find ,king's ,cross where it always has been1 that the railway line will take the course that it did when you last made the journey1 and that ,waverley ,station in ,edinburgh will not have walked up to the ,castle4 ,you therefore say and think that you have traveled to ,edinburgh1 not that ,edinburgh has traveled to you1 though the latter statement would be just as accurate4 ,the success of this common sense point of view depends upon a number of things which are really of the nature of luck4 ,suppose all the houses in ,london were perpetually moving about1 like a swarm of bees2 suppose railways moved and changed their shapes like avalanches2 and finally suppose that material objects were perpetually being formed and dissolved like clouds4 ,there is nothing impossible in these suppositions3 something like them must have been verified when the earth was hotter than it is now4 ,but obviously what we call a journey to ,edinburgh would have no meaning in such a world4 ,you would begin1 no doubt1 by asking the taxi-driver3 8,where is ,king's ,cross this morning88 ,at the station you would have to ask a similar question about ,edinburgh1 but the booking-office clerk would reply3 8,what part of ,edinburgh do you mean1 ,sir8 ,prince's ,street has gone to ,glasgow1 the ,castle has moved up into the ,highlands1 and ,waverley ,station is under water in the middle of the ,firth of ,forth40 ,and on the journey the stations would not be staying quiet1 but some would be travelling north1 some south1 some east or west1 perhaps much faster than the train4 ,under these conditions you could not say where you were at any moment4 ,indeed the whole notion that one is always in some definite 8place0 is due to the fortunate immovability of most of the large objects on the earth's surface4 ,the idea of 8place0 is only a rough practical approximation3 there is nothing logically necessary about it1 and it cannot be made precise4 ,if we were not much larger than an electron1 we should not have this impression of stability1 which is only due to the grossness of our senses4 ,king's ,cross1 which to us looks solid1 would be too vast to be conceived except by a few eccentric mathematicians4 ,the bits of it that we could see would consist of little tiny points of matter1 never coming into contact with each other1 but perpetually whizzing round each other in an inconceivably rapid ballet-dance4 ,the world of our experience would be quite as mad as the one in which the different parts of ,edinburgh go for walks in different directions4 ,if--to take the opposite extreme--you were as large as the sun and lived as long1 with a corresponding slowness of perception1 you would again find a higgledy-piggledy universe without permanence--stars and planets would come and go like morning mists1 and nothing would remain in a fixed position relatively to anything else4 ,the notion of comparative stability which forms part of our ordinary outlook is thus due to the fact that we are about the size we are1 and live on a planet of which the surface is no longer very hot4 ,if this were not the case1 we should not find pre-relativity physics intellectually satisfying4 ,indeed1 we should never have invented such theories4 ,we should have had to arrive at relativity physics at one bound1 or remain ignorant of scientific laws4 ,it is fortunate for us that we were not faced with this alternative1 since it is almost inconceivable that one man could have done the work of ,euclid1 ,galileo1 ,newton1 and ,einstein4 ,yet without such an incredible genius physics could hardly have been discovered in a world where the universal flux was obvious to non-scientific observation4 ,in astronomy1 although the sun1 moon1 and stars continue to exist year after year1 yet in other respects the world we have to deal with is very different from that of everyday life4 ,as already observed1 we depend exclusively on sight3 the heavenly bodies cannot be touched1 heard1 smelt or tasted4 ,everything in the heavens is moving relatively to everything else4 ,the earth is going round the sun1 the sun is moving1 very much faster than an express train1 towards a point in the constellation 8,hercules10 the 8fixed0 stars are scurrying hither and thither like a lot of frightened hens4 ,there are no well-marked places in the sky1 like ,king's ,cross and ,edinburgh4 ,when you travel from place to place on the earth1 you say the train moves and not the stations1 because the stations preserve their topographical relations to each other and the surrounding country4 ,but in astronomy it is arbitrary which you call the train and which the station3 the question is to be decided purely by convenience and as a matter of convention4 ,in this respect1 it is interesting to contrast ,einstein and ,copernicus4 ,before ,copernicus1 people thought that the earth stood still and the heavens revolved about it once a day4 ,copernicus taught that 8really0 the earth rotates once a day1 and the daily revolution of sun and stars is only 8apparent40 ,galileo and ,newton endorsed this view1 and many things were thought to prove it--for example1 the flattening of the earth at the poles1 and the fact that bodies are heavier there than at the equator4 ,but in the modern theory the question between ,copernicus and his predecessors is merely one of convenience2 all motion is relative1 and there is no difference between the two statements3 8the earth rotates once a day0 and 8the heavens revolve about the earth once a day40 ,the two mean exactly the same thing1 just as it means the same thing if ,i say that a certain length is six feet or two yards4 ,astronomy is easier if we take the sun as fixed than if we take the earth1 just as accounts are easier in a decimal coinage4 ,but to say more for ,copernicus is to assume absolute motion1 which is a fiction4 ,all motion is relative1 and it is a mere convention to take one body as at rest4 ,all such conventions are equally legitimate1 though not all are equally convenient4 ,there is another matter of great importance1 in which astronomy differs from terrestrial physics because of its exclusive dependence upon sight4 ,both popular thought and old-fashioned physics used the notion of 8force10 which seemed intelligible because it was associated with familiar sensations4 ,when we are walking1 we have sensations connected with our muscles which we do not have when we are sitting still4 ,in the days before mechanical traction1 although people could travel by sitting in their carriages1 they could see the horses exerting themselves and evidently putting out 8force0 in the same way as human beings do4 ,everybody knew from experience what it is to push or pull1 or to be pushed or pulled4 ,these very familiar facts made 8force0 seem a natural basis for dynamics4 ,but ,newton's law of gravitation introduced a difficulty4 ,the force between two billiard balls appeared intelligible1 because we know what it feels like to bump into another person2 but the force between the earth and the sun1 which are ninety-three million miles apart1 was mysterious4 ,newton himself regarded this 8action at a distance0 as impossible1 and believed that there was some hitherto undiscovered mechanism by which the sun's influence was transmitted to the planets4 ,however1 no such mechanism was discovered1 and gravitation remained a puzzle4 ,the fact is that the whole conception of 8force0 is a mistake4 ,the sun does not exert any force on the planets2 in ,einstein's law of gravitation1 the planet only pays attention to what it finds in its own neighborhood4 ,the way in which this works will be explained in a later chapter2 for the present we are only concerned with the necessity of abandoning the notion of 8force10 which was due to misleading conceptions derived from the sense of touch4 ,as physics has advanced1 it has appeared more and more that sight is less misleading than touch as a source of fundamental notions about matter4 ,the apparent simplicity in the collision of billiard balls is quite illusory4 ,as a matter of fact1 the two billiard balls never touch at all2 what really happens is inconceivably complicated1 but is more analogous to what happens when a comet penetrates the solar system and goes away again than to what common sense supposes to happen4 ,most of what we have said hitherto was already recognized by physicists before ,einstein invented the theory of relativity4 8,force0 was known to be merely a mathematical fiction1 and it was generally held that motion is a merely relative phenomenon--that is to say1 when two bodies are changing their relative position1 we cannot say that one is moving while the other is at rest1 since the occurrence is merely a change in their relation to each other4 ,but a great labor was required in order to bring the actual procedure of physics into harmony with these new convictions4 ,newton believed in force and in absolute space and time2 he embodied these beliefs in his technical methods1 and his methods remained those of later physicists4 ,einstein invented a new technique1 free from ,newton's assumptions4 ,but in order to do so he had to change fundamentally the old ideas of space and time1 which had been unchallenged from time immemorial4 ,this is what makes both the difficulty and the interest of his theory4 ,but before explaining it there are some preliminaries which are indispensable4 ,these will occupy the next two chapters4 ,,chapter ,,ii3 ,,what ,,happens ,,and ,,what ,,is ,,observed ,a certain type of superior person is fond of asserting that 8everything is relative40 ,this is1 of course1 nonsense1 because1 if everything were relative1 there would be nothing for it to be relative to4 ,however1 without falling into metaphysical absurdities it is possible to maintain that everything in the physical world is relative to an observer4 ,this view1 true or not1 is not that adopted by the 8theory of relativity40 ,perhaps the name is unfortunate2 certainly it has led philosophers and uneducated people into confusions4 ,they imagine that the new theory proves everything in the physical world to be relative1 whereas1 on the contrary1 it is wholly concerned to exclude what is relative and arrive at a statement of physical laws that shall in no way depend upon the circumstances of the observer4 ,it is true that these circumstances have been found to have more effect upon what appears to the observer than they were formerly thought to have1 but at the same time ,einstein showed how to discount this effect completely4 ,this was the source of almost everything that is surprising in his theory4 ,when two observers perceive what is regarded as one occurrence1 there are certain similarities1 and also certain differences1 between their perceptions4 ,the differences are obscured by the requirements of daily life1 because from a business point of view they are as a rule unimportant4 ,but both psychology and physics1 from their different angles1 are compelled to emphasize the respects in which one man's perception of a given occurrence differs from another man's4 ,some of these differences are due to differences in the brains or minds of the observers1 some to differences in their sense organs1 some to differences of physical situation3 these three kinds may be called respectively psychological1 physiological1 and physical4 ,a remark made in a language we know will be heard1 whereas an equally loud remark in an unknown language may pass entirely unnoticed4 ,of two men in the ,alps1 one will perceive the beauty of the scenery while the other will notice the waterfalls with a view to obtaining power from them4 ,such differences are psychological4 ,the difference between a long-sighted and a short-sighted man1 or between a deaf man and a man who hears well1 are physiological4 ,neither of these kinds concerns us1 and ,i have mentioned them only in order to exclude them4 ,the kind that concerns us is the purely physical kind4 ,physical differences between two observers will be preserved when the observers are replaced by cameras or phonographs1 and can be reproduced on the movies or the gramophone4 ,if two men both listen to a third man speaking1 and one of them is nearer to the speaker than the other is1 the nearer one will hear louder and slightly earlier sounds than are heard by the other4 ,if two men both watch a tree falling1 they see it from different angles4 ,both these differences would be shown equally by recording instruments3 they are in no way due to idiosyncrasies in the observers1 but are part of the ordinary course of physical nature as we experience it4 ,the physicist1 like the plain man1 believes that his perceptions give him knowledge about what is really occurring in the physical world1 and not only about his private experiences4 ,professionally1 he regards the physical world as 8real10 not merely as something which human beings dream4 ,an eclipse of the sun1 for instance1 can be observed by any person who is suitably situated1 and is also observed by the photographic plates that are exposed for the purpose4 ,the physicist is persuaded that something has really happened over and above the experiences of those who have looked at the sun or at photographs of it4 ,i have emphasized this point1 which might seem a trifle obvious1 because some people imagine that ,einstein has made a difference in this respect4 ,in fact he has made none4 ,but if the physicist is justified in this belief that a number of people can observe the 8same0 physical occurrence1 then clearly the physicist must be concerned with those features which the occurrence has in common for all observers1 for the others cannot be regarded as belonging to the occurrence itself4 ,at least1 the physicist must confine himself to the features which are common to all 8equally good0 observers4 ,the observer who uses a microscope or a telescope is preferred to one who does not1 because he sees all that the latter sees and more too4 ,a sensitive photographic plate may 8see0 still more1 and is then preferred to any eye4 ,but such things as differences of perspective1 or differences of apparent size due to difference of distance1 are obviously not attributable to the object2 they belong solely to the point of view of the spectator4 ,common sense eliminates these in judging of objects2 physics has to carry the same process much further1 but the principle is the same4 ,i want to make it clear that ,i am not concerned with anything that can be called inaccuracy4 ,i am concerned with genuine physical differences between occurrences each of which is a correct record of a certain event1 from its own point of view4 ,when a man fires a gun1 people who are not quite close to him see the flash before they hear the report4 ,this is not due to any defect in their senses1 but to the fact that sound travels more slowly than light4 ,light travels so fast that1 from the point of view of phenomena on the surface of the earth1 it may be regarded as instantaneous4 ,anything that we can see on the earth happens practically at the moment when we see it4 ,in a second1 light travels #cjj1jjj kilometers 7about #ahf1jjj miles74 ,it travels from the sun to the earth in about eight minutes1 and from the stars to us in anything from three to a thousand years4 ,but of course we cannot place a clock in the sun1 and send out a flash of light from it at #ab noon1 ,greenwich ,mean ,time1 and have it received at ,greenwich at #ab4jh ,p4,m4 ,our methods of estimating the speed of light have to be more or less indirect4 ,the only direct method would be that which we apply to sound when we use an echo4 ,we could send a flash to a mirror1 and observe how long it took for the reflection to reach us2 this would give the time of the double journey to the mirror and back4 ,on the earth1 however1 the time would be so short that a great deal of theoretical physics has to be utilized if this method is to be employed--more even than is required for the employment of astronomical data4 ,the problem of allowing for the spectator's point of view1 we may be told1 is one of which physics has at all times been fully aware2 indeed it has dominated astronomy ever since the time of ,copernicus4 ,this is true4 ,but principles are often acknowledged long before their full consequences are drawn4 ,much of traditional physics is incompatible with the principle1 in spite of the fact that it was acknowledged theoretically by all physicists4 ,there existed a set of rules which caused uneasiness to the philosophically minded1 but were accepted by physicists because they worked in practice4 ,locke had distinguished 8secondary0 qualities--colors1 noises1 tastes1 smells1 etc4--as subjective1 while allowing 8primary0 qualities--shapes and positions and sizes--to be genuine properties of physical objects4 ,the physicist's rules were such as would follow from this doctrine4 ,colors and noises were allowed to be subjective1 but due to waves proceeding with a definite velocity--that of light or sound as the case may be--from their source to the eye or ear of the percipient4 ,apparent shapes vary according to the laws of perspective1 but these laws are simple and make it easy to infer the 8real0 shapes from several visual apparent shapes2 moreover1 the 8real0 shapes can be ascertained by touch in the case of bodies in our neighborhood4 ,the objective time of a physical occurrence can be inferred from the time when we perceive it by allowing for the velocity of transmission--of light or sound or nerve currents according to circumstances4 ,this was the view adopted by physicists in practice1 whatever qualms they may have had in unprofessional moments4 ,this view worked well enough until physicists became concerned with much greater velocities than those that are common on the surface of the earth4 ,an express train travels about a mile in a minute2 the planets travel a few miles in a second4 ,comets1 when they are near the sun1 travel much faster1 and behave somewhat oddly2 but they were puzzling in various ways4 ,practically1 the planets were the most swiftly moving bodies to which dynamics could be adequately applied4 ,with radio-activity a new range of observations became possible4 ,individual electrons can be observed1 emanating from radium with a velocity not far short of that of light4 ,the behavior of bodies moving with these enormous speeds is not what the old theories would lead us to expect4 ,for one thing1 mass seems to increase with speed in a perfectly definite manner4 ,when an electron is moving very fast1 a bigger force is required to have a given effect upon it than when it is moving slowly4 ,then reasons were found for thinking that the size of a body is affected by its motion--for example1 if you take a cube and move it very fast1 it gets shorter in the direction of its motion1 from the point of view of a person who is not moving with it1 though from its own point of view 7i4e4 for an observer traveling with it7 it remains just as it was4 ,what was still more astonishing was the discovery that lapse of time depends on motion2 that is to say1 two perfectly accurate clocks1 one of which is moving very fast relatively to the other1 will not continue to show the same time if they come together again after a journey4 ,it follows that what we discover by means of clocks and foot rules1 which used to be regarded as the acme of impersonal science1 is really in part dependent upon our private circumstances1 i4e4 upon the way in which we are moving relatively to the bodies measured4 ,this shows that we have to draw a different line from that which is customary in distinguishing between what belongs to the observer and what belongs to the occurrence which he is observing4 ,if a man is wearing blue spectacles he knows that the blue look of everything is due to his spectacles1 and does not belong to what he is observing4 ,but if he observes two flashes of lightning1 and notes the interval of time between his observations2 if he knows where the flashes took place1 and allows1 in each case1 for the time the light took to reach him--in that case1 if his chronometer is accurate1 he naturally thinks that he has discovered the actual interval of time between the two flashes1 and not something merely personal to himself4 ,he is confirmed in this view by the fact that all other careful observers to whom he has access agree with his estimates4 ,this1 however1 is only due to the fact that all these observers are on the earth1 and share its motion4 ,even two observers in aeroplanes moving in opposite directions would have at the most a relative velocity of #djj miles an hour1 which is very little in comparison with #ahf1jjj miles a second 7the velocity of light74 ,if an electron shot out from a piece of radium with a velocity of #agj1jjj miles a second could observe the time between the two flashes1 it would arrive at a quite different estimate1 after making full allowance for the velocity of light4 ,how do you know this8 the reader may ask4 ,you are not an electron1 you cannot move at these terrific speeds1 no man of science has ever made the observations which would prove the truth of your assertion4 ,nevertheless1 as we shall see in the sequel1 there is good ground for the assertion--ground1 first of all1 in experiment1 and--what is remarkable--ground in reasonings which could have been made at any time1 but were not made until experiments had shown that the old reasonings must be wrong4 ,there is a general principle to which the theory of relativity appeals1 which turns out to be more powerful than anybody would suppose4 ,if you know that one man is twice as rich as another1 this fact must appear equally whether you estimate the wealth of both in pounds or dollars or francs or any other currency4 ,the numbers representing their fortunes will be changed1 but one number will always be double the other4 ,the same sort of thing1 in more complicated forms1 reappears in physics4 ,since all motion is relative1 you may take any body you like as your standard body of reference1 and estimate all other motions with reference to that one4 ,if you are in a train and walking to the dining-car1 you naturally1 for the moment1 treat the train as fixed and estimate your motion by relation to it4 ,but when you think of the journey you are making1 you think of the earth as fixed1 and say you are moving at the rate of sixty miles an hour4 ,an astronomer who is concerned with the solar system takes the sun as fixed1 and regards you as rotating and revolving2 in comparison with this motion1 that of the train is so slow that it hardly counts4 ,an astronomer who is interested in the stellar universe may add the motion of the sun relatively to the average of the stars4 ,you cannot say that one of these ways of estimating your motion is more correct than another2 each is perfectly correct as soon as the reference body is assigned4 ,now just as you can estimate a man's fortune in different currencies without altering its relations to the fortunes of other men1 so you can estimate a body's motion by means of different reference bodies without altering its relations to other motions4 ,and as physics is entirely concerned with relations1 it must be possible to express all the laws of physics by referring all motions to any given body as the standard4 ,we may put the matter in another way4 ,physics is intended to give information about what really occurs in the physical world1 and not only about the private perceptions of separate observers4 ,physics must1 therefore1 be concerned with those features which a physical process has in common for all observers1 since such features alone can be regarded as belonging to the physical occurrence itself4 ,this requires that the laws of phenomena should be the same whether the phenomena are described as they appear to one observer or as they appear to another4 ,this single principle is the generating motive of the whole theory of relativity4 ,now what we have hitherto regarded as the spatial and temporal properties of physical occurrences are found to be in large part dependent upon the observer2 only a residue can be attributed to the occurrences in themselves1 and only this residue can be involved in the formulation of any physical law which is to have an priori chance of being true4 ,einstein found ready to his hand an instrument of pure mathematics1 called the theory of tensors1 which enabled him to discover laws expressed in terms of the objective residue and agreeing approximately with the old laws4 ,where ,einstein's laws differed from the old ones1 they have hitherto proved more in accord with observation4 ,if there were no reality in the physical world1 but only a number of dreams dreamed by different people1 we should not expect to find any laws connecting the dreams of one man with the dreams of another4 ,it is the close connection between the perceptions of one man and the 7roughly7 simultaneous perceptions of another that makes us believe in a common external origin of the different related perceptions4 ,physics accounts both for the likenesses and for the differences between different people's perceptions of what we call the 8same0 occurrence4 ,but in order to do this it is first necessary for the physicist to find out just what are the likenesses4 ,they are not quite those traditionally assumed1 because neither space nor time separately can be taken as strictly objective4 ,what is objective is a kind of mixture of the two called 8space-time40 ,to explain this is not easy1 but the attempt must be made2 it will be begun in the next chapter4 ,,chapter ,,iii3 ,,the ,,velocity ,,of ,,light ,most of the curious things in the theory of relativity are connected with the velocity of light4 ,if the reader is to grasp the reasons for such a serious theoretical reconstruction1 he must have some idea of the facts which made the old system break down4 ,the fact that light is transmitted with a definite velocity was first established by astronomical observations4 ,jupiter's moons are sometimes eclipsed by ,jupiter1 and it is easy to calculate the times when this ought to occur4 ,it was found that when ,jupiter was unusually near the earth an eclipse of one of his moons would be observed a few minutes earlier than was expected2 and when ,jupiter was unusually remote1 a few minutes later than was expected4 ,it was found that these deviations could all be accounted for by assuming that light has a certain velocity1 so that what we observe to be happening in ,jupiter really happened a little while ago--longer ago when ,jupiter is distant than when it is near4 ,just the same velocity of light was found to account for similar facts in regard to other parts of the solar system4 ,it was therefore accepted that light in vacuo always travels at a certain constant rate1 almost exactly #cjj1jjj kilometers a second4 7,a kilometer is about five-eighths of a mile47 ,when it became established that light consists of waves1 this velocity was that of propagation of waves in the ether--at least they used to be in the ether1 but now the ether has grown somewhat shadowy1 though the waves remain4 ,this same velocity is that of the waves used in wireless telegraphy 7which are like light waves1 only longer7 and in ,x-rays 7which are like light waves1 only shorter74 ,it is generally held nowadays to be the velocity with which gravitation is propagated1 though ,eddington considers this not yet certain4 7,it used to be thought that gravitation was propagated instantaneously1 but this view is now abandoned47 ,so far1 all is plain sailing4 ,but as it became possible to make more accurate measurements1 difficulties began to accumulate4 ,the waves were supposed to be in the ether1 and therefore their velocity ought to be relative to the ether4 ,now since the ether 7if it exists7 clearly offers no resistance to the motions of the heavenly bodies1 it would seem natural to suppose that it does not share their motion4 ,if the earth had to push a lot of ether before it1 in the sort of way that a steamer pushes water before it1 one would expect a resistance on the part of the ether analogous to that offered by the water to the steamer4 ,therefore the general view was that the ether could pass through bodies without difficulty1 like air through a coarse sieve1 only more so4 ,if this were the case1 then the earth in its orbit must have a velocity relative to the ether4 ,if1 at some point of its orbit1 it happened to be moving exactly with the ether1 it must at other points be moving through it all the faster4 ,if you go for a circular walk on a windy day1 you must be walking against the wind part of the way1 whatever wind may be blowing2 the principle in this case is the same4 ,it follows that1 if you choose two days six months apart1 when the earth in its orbit is moving in exactly opposite directions1 it must be moving against an ether wind on at least one of these days4 ,now if there is an ether wind1 it is clear that1 relatively to an observer on the earth1 light signals will seem to travel faster with the wind than across it1 and faster across it than against it4 ,this is what ,michelson and ,morley set themselves to test by their famous experiment4 ,they sent out light signals in two directions at right angles2 each was reflected from a mirror1 and came back to the place from which both had been sent out4 ,now anybody can verify1 either by trial or by a little arithmetic1 that it takes longer to row a given distance on a river upstream and then back again1 than it takes to row the same distance across the stream and back again4 ,therefore1 if there were an ether wind1 one of the two light signals1 which consist of waves in the ether1 ought to have traveled to the mirror and back at a slower average rate than the other4 ,michelson and ,morley tried the experiment1 they tried it in various positions1 they tried it again later4 ,their apparatus was quite accurate enough to have detected the expected difference of speed or even a much smaller difference1 if it had existed1 but not the smallest difference could be observed4 ,the result was a surprise to them as to everybody else2 but careful repetitions made doubt impossible4 ,the experiment was first made as long ago as #ahha1 and was repeated with more elaboration in #ahhg4 ,but it was many years before it could be rightly interpreted4 ,the supposition that the earth carries the neighboring ether with it in its motion was found to be impossible1 for a number of reasons4 ,consequently a logical deadlock seemed to have arisen1 from which at first physicists sought to extricate themselves by very arbitrary hypotheses4 ,the most important of these was that of ,fitzgerald1 developed by ,lorentz1 and known as the ,fitzgerald contraction hypothesis4 ,according to this hypothesis1 when a body is in motion it becomes shortened in the direction of motion by a certain proportion depending upon its velocity4 ,the amount of the contraction was to be just enough to account for the negative result of the ,michelson-,morley experiment4 ,the journey up stream and down again was to have been really a shorter journey than the one across the stream1 and was to have been just so much shorter as would enable the slower light wave to traverse it in the same time4 ,of course the shortening could never be detected by measurement1 because our measuring rods would share it4 ,a foot rule placed in the line of the earth's motion would be shorter than the same foot rule placed at right angles to the earth's motion4 ,this point of view resembles nothing so much as the ,white ,knight's 8plan to dye my whiskers green1 and always use so large a fan that they could not be seen40 ,the odd thing was that the plan worked well enough4 ,later on1 when ,einstein propounded his special theory of relativity 7#aije71 it was found that the theory was in a certain sense correct1 but only in a certain sense4 ,that is to say1 the supposed contraction is not a physical fact1 but a result of certain conventions of measurement which1 when once the right point of view has been found1 are seen to be such as we are almost compelled to adopt4 ,but ,i do not wish yet to set forth ,einstein's solution of the puzzle4 ,for the present1 it is the nature of the puzzle itself that ,i want to make clear4 ,on the face of it1 and apart from hypotheses ad hoc1 the ,michelson-,morley experiment 7in conjunction with others7 showed that1 relatively to the earth1 the velocity of light is the same in all directions1 and that this is equally true at all times of the year1 although the direction of the earth's motion is always changing as it goes round the sun4 ,moreover1 it appeared that this is not a peculiarity of the earth1 but is true of all bodies3 if a light signal is sent out from a body1 that body will remain at the center of the waves as they travel outwards1 no matter how it may be moving--at least1 that will be the view of observers moving with the body4 ,this was the plain and natural meaning of the experiments1 and ,einstein succeeded in inventing a theory which accepted it4 ,but at first it was thought logically impossible to accept this plain and natural meaning4 ,a few illustrations will make it clear how very odd the facts are4 ,when a shell is fired1 it moves faster than sound3 the people at whom it is fired first see the flash1 then 7if they are lucky7 see the shell go by1 and last of all hear the report4 ,it is clear that if you could put a scientific observer on the shell1 he would never hear the report1 as the shell would burst and kill him before the sound had overtaken him4 ,but if sound worked on the same principles as light1 our observer would hear everything just as if he were at rest4 ,in that case1 if a screen1 suitable for producing echoes1 were attached to the shell and traveling with it1 say a hundred yards in front of it1 our observer would hear the echo of the report from the screen after just the same interval of time as if he and the shell were at rest4 ,this1 of course1 is an experiment which cannot be performed1 but others which can be performed will show the difference4 ,we might find some place on a railway where there is an echo from a place further along the railway--say a place where the railway goes into a tunnel--and when a train is traveling along the railway1 let a man on the bank fire a gun4 ,if the train is traveling towards the echo1 the passengers will hear the echo sooner than the man on the bank2 if it is traveling in the opposite direction1 they will hear it later4 ,but these are not quite the circumstances of the ,michelson-,morley experiment4 ,the mirrors in that experiment correspond to the echo1 and the mirrors are moving with the earth1 so that echo ought to move with the train4 ,let us suppose that the shot is fired from the guard's van1 and the echo comes from a screen on the engine4 ,we will suppose the distance from the guard's van to the engine to be the distance that sound can travel in a second 7about one-fifth of a mile71 and the speed of the train to be one-twelfth of the speed of sound 7about sixty miles an hour74 ,we now have an experiment which can be performed by the people in the train4 ,if the train were at rest1 the guard would hear the echo in two seconds2 as it is1 he will hear it in #b and _/ seconds4 ,from this difference1 if he knows the velocity of sound1 he can calculate the velocity of the train1 even if it is a foggy night so that he cannot see the banks4 ,but if sound behaved like light1 he would hear the echo in two seconds however fast the train might be traveling4 ,various other illustrations will help to show how extraordinary--from the point of view of tradition and common sense--are the facts about the velocity of light4 ,every one knows that if you are on an escalator you reach the top sooner if you walk up than if you stand still4 ,but if the escalator moved with the velocity of light 7which it does not do even in ,new ,york71 you would reach the top at exactly the same moment whether you walked up or stood still4 ,again3 if you are walking along a road at the rate of four miles an hour1 and a motor-car passes you going in the same direction at the rate of forty miles an hour1 if you and the motor-car both keep going the distance between you after an hour will be thirty-six miles4 ,but if the motor-car met you1 going in the opposite direction1 the distance after an hour would be forty-four miles4 ,now if the motor-car were traveling with the velocity of light1 it would make no difference whether it met or passed you3 in either case1 it would1 after a second1 be #ahf1jjj miles from you4 ,it would also be #ahf1jjj miles from any other motor-car which happened to be passing or meeting you less rapidly at the previous second4 ,this seems impossible3 how can the car be at the same distance from a number of different points along the road8 ,let us take another illustration4 ,when a fly touches the surface of a stagnant pool1 it causes ripples which move outwards in widening circles4 ,the center of the circle at any moment is the point of the pool touched by the fly4 ,if the fly moves about over the surface of the pool1 it does not remain at the center of the ripples4 ,but if the ripples were waves of light1 and the fly were a skilled physicist1 it would find that it always remained at the center of the ripples1 however it might move4 ,meanwhile a skilled physicist sitting beside the pool would judge1 as in the case of ordinary ripples1 that the center was not the fly1 but the point of the pool touched by the fly4 ,and if another fly had touched the water at the same spot at the same moment1 it also would find that it remained at the center of the ripples1 even if it separated itself widely from the first fly4 ,this is exactly analogous to the ,michelson-,morley experiment4 ,the pool corresponds to the ether2 the fly corresponds to the earth2 the contact of the fly and the pool corresponds to the light signal which ,messrs4 ,michelson and ,morley send out2 and the ripples correspond to the light waves4 ,such a state of affairs seems1 at first sight1 quite impossible4 ,it is no wonder that1 although the ,michelson-,morley experiment was made in #ahha1 it was not rightly interpreted until #aije4 ,let us see what1 exactly1 we have been saying4 ,take the man walking along a road and passed by a motor-car4 ,suppose there are a number of people at the same point of the road1 some walking1 some in motor-cars2 suppose they are going at varying rates1 some in one direction and some in another4 ,i say that if1 at this moment1 a light flash is sent out from the place where they all are1 the light waves will be #ahf1jjj miles from each one of them after a second by his watch1 although the travelers will not any longer be all in the same place4 ,at the end of a second by your watch it will be #ahf1jjj miles from you1 and it will also be #ahf1jjj miles from a person who met you when it was sent out1 but was moving in the opposite direction1 after a second by his watch--assuming both to be perfect watches4 ,how can this be8 ,there is only one way of explaining such facts1 and that is1 to assume that watches and clocks are affected by motion4 ,i do not mean that they are affected in ways that could be remedied by greater accuracy in construction2 ,i mean something much more fundamental4 ,i mean that1 if you say an hour has elapsed between two events1 and if you base this assertion upon ideally careful measurements with ideally accurate chronometers1 another equally precise person1 who has been moving rapidly relatively to you1 may judge that the time was more or less than an hour4 ,you cannot say that one is right and the other wrong1 any more than you could if one used a clock showing ,greenwich time and another a clock showing ,new ,york time4 ,how this comes about1 ,i shall explain in the next chapter4 ,there are other curious things about the velocity of light4 ,one is1 that no material body can ever travel as fast as light1 however great may be the force to which it is exposed1 and however long the force may act4 ,an illustration may help to make this clear4 ,at exhibitions one sometimes sees a series of moving platforms1 going round and round in a circle4 ,the outside one goes at four miles an hour2 the next goes four miles an hour faster than the first2 and so on4 ,you can step across from each to the next2 until you find yourself going at a tremendous pace4 ,now you might think that1 if the first platform does four miles an hour1 and the second does four miles an hour relatively to the first1 then the second does eight miles an hour relatively to the ground4 ,this is an error2 it does a little less1 though so little less that not even the most careful measurements could detect the difference4 ,i want to make quite clear what it is that ,i mean4 ,i will suppose that1 in the morning1 when the apparatus is just about to start1 three men with ideally accurate chronometers stand in a row1 one on the ground1 one on the first platform1 and one on the second4 ,the first platform moves at the rate of four miles an hour with respect to the ground4 ,four miles an hour is #ceb feet in a minute4 ,the man on the ground1 after a minute by his watch1 notes the place on the ground opposite the man on the first platform1 who has been standing still while the platform carried him along4 ,the man on the ground measures the distance on the ground from himself to the point opposite the man on the first platform1 and finds it is #ceb feet4 ,the man on the first platform1 after a minute by his watch1 notes the point on his platform opposite to the man on the second platform4 ,the man on the first platform measures the distance from himself to the point opposite the man on the second platform2 it is again #ceb feet4 ,problem3 how far will the man on the ground judge that the man on the second platform has traveled in a minute8 ,that is to say1 if the man on the ground1 after a minute by his watch1 notes the place on the ground opposite the man on the second platform1 how far will this be from the man on the ground8 ,you would say1 twice #ceb feet1 that is to say1 #gjd feet4 ,but in fact it will be a little less1 though so little less as to be inappreciable4 ,the discrepancy is owing to the fact that the two watches do not keep perfect time1 in spite of the fact that each is accurate from its owner's point of view4 ,if you had a long series of such moving platforms1 each moving four miles an hour relatively to the one before it1 you would never reach a point where the last was moving with the velocity of light relatively to the ground1 not even if you had millions of them4 ,the discrepancy1 which is very small for small velocities1 becomes greater as the velocity increases1 and makes the velocity of light an unattainable limit4 ,how all this happens1 is the next topic with which we must deal4 ,note4 ,the negative result of the ,michelson-,morley experiment has recently been called in question by ,professor ,dayton ,c4 ,miller1 as a result of observations by what is said to be an improved method4 ,his claim is set forth by ,professor ,silberstein in ,nature1 ,may #bc1 #aibe1 and discussed unfavorably by ,eddington in the issue of ,june #f4 ,the matter is sub judice1 but it seems highly questionable whether the results bear out the interpretation which is put upon them4 ,,chapter ,,iv3 ,,clocks ,,and ,,foot ,,rules ,until the advent of the special theory of relativity1 no one had thought that there could be any ambiguity in the statement that two events in different places happened at the same time4 ,it might be admitted that1 if the places were very far apart1 there might be difficulty in finding out for certain whether the events were simultaneous1 but every one thought the meaning of the question perfectly definite4 ,it turned out1 however1 that this was a mistake4 ,two events in distant places may appear simultaneous to one observer who has taken all due precautions to insure accuracy 7and1 in particular1 has allowed for the velocity of light71 while another equally careful observer may judge that the first event preceded the second1 and still another may judge that the second preceded the first4 ,this would happen if the three observers were all moving rapidly relatively to each other4 ,it would not be the case that one of them would be right and the other two wrong3 they would all be equally right4 ,the time order of events is in part dependent upon the observer2 it is not always and altogether an intrinsic relation between the events themselves4 ,einstein has shown1 not only that this view accounts for the phenomena1 but also that it is the one which ought to have resulted from careful reasoning based upon the old data4 ,in actual fact1 however1 no one noticed the logical basis of the theory of relativity until the odd results of experiment had given a jog to people's reasoning powers4 ,how should we naturally decide whether two events in different places were simultaneous8 ,one would naturally say3 they are simultaneous if they are seen simultaneously by a person who is exactly half-way between them4 7,there is no difficulty about the simultaneity of two events in the same place1 such1 for example1 as seeing a light and hearing a noise47 ,suppose two flashes of lightning fall in two different places1 say ,greenwich ,observatory and ,kew ,observatory4 ,suppose that ,st4 ,paul's is half-way between them1 and that the flashes appear simultaneous to an observer on the dome of ,st4 ,paul's4 ,in that case1 a man at ,kew will see the ,kew flash first1 and a man at ,greenwich will see the ,greenwich flash first1 because of the time taken by light to travel over the intervening distance4 ,but all three1 if they are ideally accurate observers1 will judge that the two flashes were simultaneous1 because they will make the necessary allowance for the time of transmission of the light4 7,i am assuming a degree of accuracy far beyond human powers47 ,thus1 so far as observers on the earth are concerned1 the definition of simultaneity will work well enough1 so long as we are dealing with events on the surface of the earth4 ,it gives results which are consistent with each other1 and can be used for terrestrial physics in all problems in which we can ignore the fact that the earth moves4 ,but our definition is no longer so satisfactory when we have two sets of observers in rapid motion relatively to each other4 ,suppose we see what would happen if we substitute sound for light1 and defined two occurrences as simultaneous when they are heard simultaneously by a man half-way between them4 ,this alters nothing in the principle1 but makes the matter easier owing to the much slower velocity of sound4 ,let us suppose that on a foggy night two men belonging to a gang of brigands shoot the guard and engine driver of a train4 ,the guard is at the end of the train2 the brigands are on the line1 and shoot their victims at close quarters4 ,an old gentleman who is exactly in the middle of the train hears the two shots simultaneously4 ,you would say1 therefore1 that the two shots were simultaneous4 ,but a station master who is exactly half-way between the two brigands hears the shot which kills the guard first4 ,an ,australian millionaire uncle of the guard and the engine driver 7who are cousins7 has left his whole fortune to the guard1 or1 should he die first1 to the engine driver4 ,vast sums are involved in the question of which died first4 ,the case goes to the ,house of ,lords1 and the lawyers on both sides1 having been educated at ,oxford1 are agreed that either the old gentleman or the station master must have been mistaken4 ,in fact1 both may perfectly well be right4 ,the train travels away from the shot at the guard1 and towards the shot at the engine driver2 therefore the noise of the shot at the guard has farther to go before reaching the old gentleman than the shot at the engine driver has4 ,therefore if the old gentleman is right in saying that he heard the two reports simultaneously1 the station master must be right in saying that he heard the shot at the guard first4 ,we1 who live on the earth1 would naturally1 in such a case1 prefer the view of simultaneity obtained from a person at rest on the earth to the view of a person traveling in a train4 ,but in theoretical physics no such parochial prejudices are permissible4 ,a physicist on a comet1 if there were one1 would have just as good a right to his view of simultaneity as an earthly physicist has to his1 but the results would differ1 in just the same sort of way as in our illustration of the train and the shots4 ,the train is not any more 8really0 in motion than the earth2 there is no 8really0 about it4 ,you might imagine a rabbit and a hippopotamus arguing as to whether man is 8really0 a large animal2 each would think his own point of view the natural one1 and the other a pure flight of fancy4 ,there is just as little substance in an argument as to whether the earth or the train is 8really0 in motion4 ,and1 therefore1 when we are defining simultaneity between distant events1 we have no right to pick and choose among different bodies to be used in defining the point half-way between the events4 ,all bodies have an equal right to be chosen4 ,but if1 for one body1 the two events are simultaneous according to the definition1 there will be other bodies for which the first precedes the second1 and still others for which the second precedes the first4 ,we cannot therefore say unambiguously that two events in distant places are simultaneous4 ,such a statement only acquires a definite meaning in relation to a definite observer4 ,it belongs to the subjective part of our observation of physical phenomena1 not to the objective part which is to enter into physical laws4 ,this question of time in different places is perhaps1 for the imagination1 the most difficult aspect of the theory of relativity4 ,we are accustomed to the idea that everything can be dated4 ,historians make use of the fact that there was an eclipse of the sun visible in ,china on ,august #bi in the year #ggf ,b4 ,c4#a ,no doubt astronomers could tell the exact hour and minute when the eclipse began to be total at any given spot in ,north ,china4 ,and it seems obvious that we can speak of the positions of the planets at a given instant4 ,the ,newtonian theory enables us to calculate the distance between the earth and 7say7 ,jupiter at a given time by the ,greenwich clocks2 this enables us to know how long light takes at that time to travel from ,jupiter to the earth--say half an hour2 this enables us to infer that half an hour ago ,jupiter was where we see it now4 ,all this seems obvious4 ,but in fact it only works in practice because the relative velocities of the planets are very small compared with the velocity of light4 ,when we judge that an event on the earth and an event on ,jupiter have happened at the same time--for example1 that ,jupiter eclipsed one of his moons when the ,greenwich clocks showed twelve midnight--a person moving rapidly relatively to the earth would judge differently1 assuming that both he and we had made the proper allowance for the velocity of light4 ,and naturally the disagreement about simultaneity involves a disagreement about periods of time4 ,if we judged that two events on ,jupiter were separated by twenty-four hours1 another person might judge that they were separated by a longer time1 if he were moving rapidly relatively to ,jupiter and the earth4 #a ,a contemporary ,chinese ode1 after giving the day of the year correctly1 proceeds3 8,for the moon to be eclipsed ,is but an ordinary matter4 ,now that the sun has been eclipsed1 ,how bad it is40 ,the universal cosmic time which used to be taken for granted is thus no longer admissible4 ,for each body1 there is a definite time order for the events in its neighborhood2 this may be called the 8proper0 time for that body4 ,our own experience is governed by the proper time for our own body4 ,as we all remain very nearly stationary on the earth1 the proper times of different human beings agree1 and can be lumped together as terrestrial time4 ,but this is only the time appropriate to large bodies on the earth4 ,for ,beta-particles in laboratories1 quite different times would be wanted2 it is because we insist upon using our own time that these particles seem to increase in mass with rapid motion4 ,from their own point of view1 their mass remains constant1 and it is we who suddenly grow thin or corpulent4 ,the history of a physicist as observed by a ,beta-particle would resemble ,gulliver's travels4 ,the question now arises3 what really is measured by a clock8 ,when we speak of a clock in the theory of relativity1 we do not mean only clocks made by human hands3 we mean anything which goes through some regular periodic performance4 ,the earth is a clock1 because it rotates once in every twenty-three hours and fifty-six minutes4 ,an atom is a clock1 because the electrons go round the nucleus a certain number of times in a second2 its properties as a clock are exhibited to us in its spectrum1 which is due to light waves of various frequencies4 ,the world is full of periodic occurrences1 and fundamental mechanisms1 such as atoms1 show an extraordinary similarity in different parts of the universe4 ,any one of these periodic occurrences may be used for measuring time2 the only advantage of humanly manufactured clocks is that they are specially easy to observe4 ,one question is3 ,if cosmic time is abandoned1 what is really measured by a clock in the wide sense that we have just given to the term8 ,each clock gives a correct measure of its own 8proper0 time1 which1 as we shall see presently1 is an important physical quantity4 ,but it does not give an accurate measure of any physical quantity connected with events on bodies that are moving rapidly in relation to it4 ,it gives one datum towards the discovery of a physical quantity connected with such events1 but another datum is required1 and this has to be derived from measurement of distances in space4 ,distances in space1 like periods of time1 are in general not objective physical facts1 but partly dependent upon the observer4 ,how this comes about must now be explained4 ,first of all1 we have to think of the distance between two events1 not between two bodies4 ,this follows at once from what we have found as regards time4 ,if two bodies are moving relatively to each other--and this is really always the case--the distance between them will be continually changing1 so that we can only speak of the distance between them at a given time4 ,if you are in a train traveling towards ,edinburgh1 we can speak of your distance from ,edinburgh at a given time4 ,but1 as we said1 different observers will judge differently as to what is the 8same0 time for an event in the train and an event in ,edinburgh4 ,this makes the measurement of distances relative1 in just the same way as the measurement of times has been found to be relative4 ,we commonly think that there are two separate kinds of interval between two events1 an interval in space and an interval in time3 between your departure from ,london and your arrival in ,edinburgh1 there are #djj miles and ten hours4 ,we have already seen that another observer will judge the time differently2 it is even more obvious that he will judge the distance differently4 ,an observer in the sun will think the motion of the train quite trivial1 and will judge that you have traveled the distance traveled by the earth in its orbit and its diurnal rotation4 ,on the other hand1 a flea in the railway carriage will judge that you have not moved at all in space1 but have afforded him a period of pleasure which he will measure by his 8proper0 time1 not by ,greenwich ,observatory4 ,it cannot be said that you or the sun dweller or the flea are mistaken3 each is equally justified1 and is only wrong if he ascribes an objective validity to his subjective measures4 ,the distance in space between two events is1 therefore1 not in itself a physical fact4 ,but1 as we shall see1 there is a physical fact which can be inferred from the distance in time together with the distance in space4 ,this is what is called the 8interval0 in space-time4 ,taking any two events in the universe1 there are two different possibilities as to the relation between them4 ,it may be physically possible for a body to travel so as to be present at both events1 or it may not4 ,this depends upon the fact that no body can travel as fast as light4 ,suppose1 for example1 that it were possible to send out a flash of light from the earth and have it reflected back from the moon4 ,the time between the sending of the flash and the return of the reflection would be about two and a half seconds4 ,no body could travel so fast as to be present on the earth during any part of those two and a half seconds and also present on the moon at the moment of the arrival of the flash1 because in order to do so the body would have to travel faster than light4 ,but theoretically a body could be present on the earth at any time before or after those two and a half seconds and also present on the moon at the time when the flash arrived4 ,when it is physically impossible for a body to travel so as to be present at both events1 we shall say that the interval#b between the two events is 8space-like02 when it is physically possible for a body to be present at both events1 we shall say that the interval between the two events is 8time-like40 ,when the interval is 8space-like10 it is possible for a body to move in such a way that an observer on the body will judge the two events to be simultaneous4 ,in that case1 the 8interval0 between the two events is what such an observer will judge to be the distance in space between them4 ,when the interval is 8time-like10 a body can be present at both events2 in that case1 the 8interval0 between the two events is what an observer on the body will judge to be the time between them1 that is to say1 it is his 8proper0 time between the two events4 ,there is a limiting case between the two1 when the two events are parts of one light flash--or1 as we might say1 when the one event is the seeing of the other4 ,in that case1 the interval between the two events is zero4 #b ,i shall define 8interval0 in a moment4 ,there are thus three cases4 7#a7 ,it may be possible for a ray of light to be present at both events2 this happens whenever one of them is the seeing of the other4 ,in this case the interval between the two events is zero4 7#b7 ,it may happen that no body can travel from one event to the other1 because in order to do so it would have to travel faster than light4 ,in that case1 it is always physically possible for a body to travel in such a way that an observer on the body would judge the two events to be simultaneous4 ,the interval is what he would judge to be the distance in space between the two events4 ,such an interval is called 8space-like40 7#c7 ,it may be physically possible for a body to travel so as to be present at both events2 in that case1 the interval between them is what an observer on such a body will judge to be the time between them4 ,such an interval is called 8time-like40 ,the interval between two events is a physical fact about them1 not dependent upon the particular circumstances of the observer4 ,there are two forms of the theory of relativity1 the special and the general4 ,the former is in general only approximate1 but is exact at great distances from gravitating matter4 ,when the special theory can be applied1 the interval can be calculated when we know the distance in space and the distance in time between the two events1 estimated by any observer4 ,if the distance in space is greater than the distance that light would have traveled in the time1 the separation is space-like4 ,then the following construction gives the interval between the two events3 ,draw a line ,,ab as long as the distance that light would travel in the time2 round ,a describe a circle whose radius is the distance in space between the two events2 through ,b draw ,,bc perpendicular to ,,ab1 meeting the circle in ,c4 ,then ,,bc is the length of the interval between the two events4 ,illustration ,when the distance is time-like1 use the same figure1 but let ,,ac be now the distance that light would travel in the time1 while ,,ab is the distance in space between the two events4 ,the interval between them is now the time that light would take to travel the distance ,,bc4 ,although ,,ab and ,,ac are different for different observers1 ,,bc is the same length for all observers1 subject to corrections made by the general theory4 ,it represents the one interval in 8space-time0 which replaces the two intervals in space and time of the older physics4 ,so far1 this notion of interval may appear somewhat mysterious1 but as we proceed it will grow less so1 and its reason in the nature of things will gradually emerge4 ,,chapter ,v3 ,,space-,,time ,everybody who has ever heard of relativity knows the phrase 8space-time10 and knows that the correct thing is to use this phrase when formerly we should have said 8space and time40 ,but very few people who are not mathematicians have any clear idea of what is meant by this change of phraseology4 ,before dealing further with the special theory of relativity1 ,i want to try to convey to the reader what is involved in the new phrase 8space-time10 because that is1 from a philosophical and imaginative point of view1 perhaps the most important of all the novelties that ,einstein has introduced4 ,suppose you wish to say where and when some event has occurred--say an explosion on an airship--you will have to mention four quantities1 say the latitude and longitude1 the height above the ground1 and the time4 ,according to the traditional view1 the first three of these give the position in space1 while the fourth gives the position in time4 ,the three quantities that give the position in space may be assigned in all sorts of ways4 ,you might1 for instance1 take the plane of the equator1 the plane of the meridian of ,greenwich1 and the plane of the ninetieth meridian1 and say how far the airship was from each of these planes2 these three distances would be what are called 8,cartesian co-ordinates10 after ,descartes4 ,you might take any other three planes all at right angles to each other1 and you would still have ,cartesian co-ordinates4 ,or you might take the distance from ,london to a point vertically below the airship1 the direction of this distance 7northeast1 west-southwest1 or whatever it might be71 and the height of the airship above the ground4 ,there are an infinite number of such ways of fixing the position in space1 all equally legitimate2 the choice between them is merely one of convenience4 ,when people said that space had three dimensions1 they meant just this3 that three quantities were necessary in order to specify the position of a point in space1 but that the method of assigning these quantities was wholly arbitrary4 ,with regard to time1 the matter was thought to be quite different4 ,the only arbitrary elements in the reckoning of time were the unit1 and the point of time from which the reckoning started4 ,one could reckon in ,greenwich time1 or in ,paris time1 or in ,new ,york time2 that made a difference as to the point of departure4 ,one could reckon in seconds1 minutes1 hours1 days1 or years2 that was a difference of unit4 ,both these were obvious and trivial matters4 ,there was nothing corresponding to the liberty of choice as to the method of fixing position in space4 ,and1 in particular1 it was thought that the method of fixing position in space and the method of fixing position in time could be made wholly independent of each other4 ,for these reasons1 people regarded time and space as quite distinct4 ,the theory of relativity has changed this4 ,there are now a number of different ways of fixing position in time1 which do not differ merely as to the unit and the starting point4 ,indeed1 as we have seen1 if one event is simultaneous with another in one reckoning1 it will precede it in another1 and follow it in a third4 ,moreover1 the space and time reckonings are no longer independent of each other4 ,if you alter the way of reckoning position in space1 you may also alter the time interval between two events4 ,if you alter the way of reckoning time1 you may also alter the distance in space between two events4 ,thus space and time are no longer independent1 any more than the three dimensions of space are4 ,we still need four quantities to determine the position of an event1 but we cannot1 as before1 divide off one of the four as quite independent of the other three4 ,it is not quite true to say that there is no longer any distinction between time and space4 ,as we have seen1 there are time-like intervals and space-like intervals4 ,but the distinction is of a different sort from that which was formerly assumed4 ,there is no longer a universal time which can be applied without ambiguity to any part of the universe2 there are only the various 8proper0 times of the various bodies in the universe1 which agree approximately for two bodies which are not in rapid relative motion1 but never agree exactly except for two bodies which are at rest relatively to each other4 ,the picture of the world which is required for this new state of affairs is as follows3 ,suppose an event ,e occurs to me1 and simultaneously a flash of light goes out from me in all directions4 ,anything that happens to any body after the light from the flash has reached it is definitely after the event ,e in any system of reckoning time4 ,any event anywhere which ,i could have seen before the event ,e occurred to me is definitely before the event ,e in any system of reckoning time4 ,but any event which happened in the intervening time is not definitely either before or after the event ,e4 ,to make the matter definite3 suppose ,i could observe a person in ,sirius1 and he could observe me4 ,anything which he does1 and which ,i see before the event ,e occurs to me1 is definitely before ,e2 anything he does after he has seen the event ,e is definitely after ,e4 ,but anything that he does before he sees the event ,e1 but so that ,i see it after the event ,e has happened1 is not definitely before or after ,e4 ,since light takes many years to travel from ,sirius to the earth1 this gives a period of twice as many years in ,sirius which may be called 8contemporary0 with ,e1 since these years are not definitely before or after ,e4 ,dr4 ,a4 ,a4 ,robb1 in his ,theory of ,time and ,space1 suggests a point of view which may or may not be philosophically fundamental1 but is at any rate a help in understanding the state of affairs we have been describing4 ,he maintains that one event can only be said to be definitely before another if it can influence that other in some way4 ,now influences spread from a center at varying rates4 ,newspapers exercise an influence emanating from ,london at an average rate of about twenty miles an hour--rather more for long distances4 ,anything a man does because of what he reads in the newspaper is clearly subsequent to the printing of the newspaper4 ,sounds travel much faster3 it would be possible to arrange a series of loud speakers along the main roads1 and have newspapers shouted from each to the next4 ,but telegraphing is quicker1 and wireless telegraphy travels with the velocity of light1 so that nothing quicker can ever be hoped for4 ,now what a man does in consequence of receiving a wireless message he does after the message was sent2 the meaning here is quite independent of conventions as to the measurement of time4 ,but anything that he does while the message is on its way cannot be influenced by the sending of the message1 and cannot influence the sender until some little time after he sent the message4 ,that is to say1 if two bodies are widely separated1 neither can influence the other except after a certain lapse of time2 what happens before that time has elapsed cannot affect the distant body4 ,suppose1 for instance1 that some notable event happens on the sun3 there is a period of sixteen minutes on the earth during which no event on the earth can have influenced or been influenced by the said notable event on the sun4 ,this gives a substantial ground for regarding that period of sixteen minutes on the earth as neither before nor after the event on the sun4 ,the paradoxes of the special theory of relativity are only paradoxes because we are unaccustomed to the point of view1 and in the habit of taking things for granted when we have no right to do so4 ,this is especially true as regards the measurement of lengths4 ,in daily life1 our way of measuring lengths is to apply a foot rule or some other measure4 ,at the moment when the foot rule is applied1 it is at rest relatively to the body which is being measured4 ,consequently the length that we arrive at by measurement is the 8proper0 length1 that is to say1 the length as estimated by an observer who shares the motion of the body4 ,we never1 in ordinary life1 have to tackle the problem of measuring a body which is in continual motion4 ,and even if we did1 the velocities of visible bodies on the earth are so small relatively to the earth that the anomalies dealt with by the theory of relativity would not appear4 ,but in astronomy1 or in the investigation of atomic structure1 we are faced with problems which cannot be tackled in this way4 ,not being ,joshua1 we cannot make the sun stand still while we measure it2 if we are to estimate its size1 we must do so while it is in motion relatively to us4 ,and similarly if you want to estimate the size of an electron1 you have to do so while it is in rapid motion1 because it never stands still for a moment4 ,this is the sort of problem with which the theory of relativity is concerned4 ,measurement with a foot rule1 when it is possible1 gives always the same result1 because it gives the 8proper0 length of a body4 ,but when this method is not possible1 we find that curious things happen1 particularly if the body to be measured is moving very fast relatively to the observer4 ,a figure like the one at the end of the previous chapter will help us to understand the state of affairs4 ,illustration ,let us suppose that the body on which we wish to measure lengths is moving relatively to ourselves1 and that in one second it moves the distance ,,om4 ,let us draw a circle round ,o whose radius is the distance that light travels in a second4 ,through ,m draw ,,mp perpendicular to ,,om1 meeting the circle in ,p4 ,thus ,,op is the distance that light travels in a second4 ,the ratio of ,,op to ,,om is the ratio of the velocity of light to the velocity of the body4 ,the ratio of ,,op to ,,mp is the ratio in which apparent lengths are altered by the motion4 ,that is to say1 if the observer judges that two points in the line of motion on the moving body are at a distance from each other represented by ,,mp1 a person moving with the body would judge that they were at a distance represented 7on the same scale7 by ,,op4 ,distances on the moving body at right angles to the line of motion are not affected by the motion4 ,the whole thing is reciprocal2 that is to say1 if an observer moving with the body were to measure lengths on the previous observer's body1 they would be altered in just the same proportion4 ,when two bodies are moving relatively to each other1 lengths on either appear shorter to the other than to themselves4 ,this is the ,fitzgerald contraction1 which was first invented to account for the result of the ,michelson-,morley experiment4 ,but it now emerges naturally from the fact that the two observers do not make the same judgment of simultaneity4 ,the way in which simultaneity comes in is this3 ,we say that two points on a body are a foot apart when we can simultaneously apply one end of a foot rule to the one and the other end to the other4 ,if1 now1 two people disagree about simultaneity1 and the body is in motion1 they will obviously get different results from their measurements4 ,thus the trouble about time is at the bottom of the trouble about distance4 ,the ratio of ,,op to ,,mp is the essential thing in all these matters4 ,times and lengths and masses are all altered in this proportion when the body concerned is in motion relatively to the observer4 ,it will be seen that1 if ,,om is very much smaller than ,,op1 that is to say1 if the body is moving very much more slowly than light1 ,,mp and ,,op are very nearly equal1 so that the alterations produced by the motion are very small4 ,but if ,,om is nearly as large as ,,op1 that is to say1 if the body is moving nearly as fast as light1 ,,mp becomes very small compared to ,,op1 and the effects become very great4 ,the apparent increase of mass in swiftly moving particles had been observed1 and the right formula had been found1 before ,einstein invented his special theory of relativity4 ,in fact1 ,lorentz had arrived at the formul called the 8,lorentz transformation10 which embody the whole mathematical essence of the special theory of relativity4 ,but it was ,einstein who showed that the whole thing was what we ought to have expected1 and not a set of makeshift devices to account for surprising experimental results4 ,nevertheless1 it must not be forgotten that experimental results were the original motive of the whole theory1 and have remained the ground for undertaking the tremendous logical reconstruction involved in ,einstein's theories4 ,we may now recapitulate the reasons which have made it necessary to substitute 8space-time0 for space and time4 ,the old separation of space and time rested upon the belief that there was no ambiguity in saying that two events in distant places happened at the same time2 consequently it was thought that we could describe the topography of the universe at a given instant in purely spatial terms4 ,but now that simultaneity has become relative to a particular observer1 this is no longer possible4 ,what is1 for one observer1 a description of the state of the world at a given instant1 is1 for another observer1 a series of events at various different times1 whose relations are not merely spatial but also temporal4 ,for the same reason1 we are concerned with events1 rather than with bodies4 ,in the old theory1 it was possible to consider a number of bodies all at the same instant1 and since the time was the same for all of them it could be ignored4 ,but now we cannot do that if we are to obtain an objective account of physical occurrences4 ,we must mention the date at which a body is to be considered1 and thus we arrive at an 8event10 that is to say1 something which happens at a given time4 ,when we know the time and place of an event in one observer's system of reckoning1 we can calculate its time and place according to another observer4 ,but we must know the time as well as the place1 because we can no longer ask what is its place for the new observer at the 8same0 time as for the old observer4 ,there is no such thing as the 8same0 time for different observers1 unless they are at rest relatively to each other4 ,we need four measurements to fix a position1 and four measurements fix the position of an event in space-time1 not merely of a body in space4 ,three measurements are not enough to fix any position4 ,that is the essence of what is meant by the substitution of space-time for space and time4 ,,chapter ,,vi3 ,,the ,,special ,,theory ,,of ,,relativity ,the special theory of relativity arose as a way of accounting for the facts of electromagnetism4 ,we have here a somewhat curious history4 ,in the eighteenth and early nineteenth centuries the theory of electricity was wholly dominated by the ,newtonian analogy4 ,two electric charges attract each other if they are of different kinds1 one positive and one negative1 but repel each other if they are of the same kind2 in each case1 the force varies as the inverse square of the distance1 as in the case of gravitation4 ,this force was conceived as an action at a distance1 until ,faraday1 by a number of remarkable experiments1 demonstrated the effect of the intervening medium4 ,faraday was no mathematician2 ,clerk ,maxwell first gave a mathematical form to the results suggested by ,faraday's experiments4 ,moreover ,clerk ,maxwell gave grounds for thinking that light is an electromagnetic phenomenon1 consisting of electromagnetic waves4 ,the medium for the transmission of electromagnetic effects could therefore be taken to be the ether1 which had long been assumed for the transmission of light4 ,the correctness of ,maxwell's theory of light was proved by the experiments of ,hertz in manufacturing electromagnetic waves2 these experiments afforded the basis for wireless telegraphy4 ,so far1 we have a record of triumphant progress1 in which theory and experiment alternately assume the leading role4 ,at the time of ,hertz's experiments1 the ether seemed securely established1 and in just as strong a position as any other scientific hypothesis not capable of direct verification4 ,but a new set of facts began to be discovered1 and gradually the whole picture was changed4 ,the movement which culminated with ,hertz was a movement for making everything continuous4 ,the ether was continuous1 the waves in it were continuous1 and it was hoped that matter would be found to consist of some continuous structure in the ether4 ,then came the discovery of the electron1 a small finite unit of negative electricity1 and the proton1 a small finite unit of positive electricity4 ,the most modern view is that electricity is never found except in the form of electrons and protons2 all electrons have the same amount of negative electricity1 and all protons have an exactly equal and opposite amount of positive electricity4 ,it appeared that an electric current1 which had been thought of as a continuous phenomenon1 consists of electrons traveling one way and positive ions traveling the other way2 it is no more strictly continuous than the stream of people going up and down an escalator4 ,then came the discovery of quanta1 which seems to show a fundamental discontinuity in all such natural processes as can be measured with sufficient precision4 ,thus physics has had to digest new facts and face new problems4 ,but the problems raised by the electron and the quantum are not those that the theory of relativity can solve1 at any rate at present2 as yet1 it throws no light upon the discontinuities which exist in nature4 ,the problems solved by the special theory of relativity are typified by the ,michelson-,morley experiment4 ,assuming the correctness of ,maxwell's theory of electromagnetism1 there should have been certain discoverable effects of motion through the ether2 in fact1 there were none4 ,then there was the observed fact that a body in very rapid motion appears to increase its mass2 the increase is in the ratio of ,,op to ,,mp in the figure in the preceding chapter4 ,facts of this sort gradually accumulated1 until it became imperative to find some theory which would account for them all4 ,maxwell's theory reduced itself to certain equations1 known as 8,maxwell's equations40 ,through all the revolutions which physics has undergone in the last fifty years1 these equations have remained standing2 indeed they have continually grown in importance as well as in certainty--for ,maxwell's arguments in their favor were so shaky that the correctness of his results must almost be ascribed to intuition4 ,now these equations were1 of course1 obtained from experiments in terrestrial laboratories1 but there was a tacit assumption that the motion of the earth through the ether could be ignored4 ,in certain cases1 such as the ,michelson-,morley experiment1 this ought not to have been possible without measurable error2 but it turned out to be always possible4 ,physicists were faced with the odd difficulty that ,maxwell's equations were more accurate than they should be4 ,a very similar difficulty was explained by ,galileo at the very beginning of modern physics4 ,most people think that if you let a weight drop it will fall vertically4 ,but if you try the experiment in the cabin of a moving ship1 the weight falls1 in relation to the cabin1 just as if the ship were at rest2 for instance1 if it starts from the middle of the ceiling it will drop onto the middle of the floor4 ,that is to say1 from the point of view of an observer on the shore it does not fall vertically1 since it shares the motion of the ship4 ,so long as the ship's motion is steady1 everything goes on inside the ship as if the ship were not moving4 ,galileo explained how this happens1 to the great indignation of the disciples of ,aristotle4 ,in orthodox physics1 which is derived from ,galileo1 a uniform motion in a straight line has no discoverable effects4 ,this was1 in its day1 as astonishing a form of relativity as that of ,einstein is to us4 ,einstein1 in the special theory of relativity1 set to work to show how electromagnetic phenomena could be unaffected by uniform motion through the ether if there be an ether4 ,this was a more difficult problem1 which could not be solved by merely adhering to the principles of ,galileo4 ,the really difficult effort required for solving this problem was in regard to time4 ,it was necessary to introduce the notion of 8proper0 time which we have already considered1 and to abandon the old belief in one universal time4 ,the quantitative laws of electromagnetic phenomena are expressed in ,maxwell's equations1 and these equations are found to be true for any observer1 however he may be moving4#c ,it is a straight-forward mathematical problem to find out what differences there must be between the measures applied by one observer and the measures applied by another1 if1 in spite of their relative motion1 they are to find the same equations verified4 ,the answer is contained in the 8,lorentz transformation10 found as a formula by ,lorentz1 but interpreted and made intelligible by ,einstein4 ,the ,lorentz transformation tells us what estimate of distances and periods of time will be made by an observer whose relative motion is known1 when we are given those of another observer4 ,we may suppose that you are in a train on a railway which travels due east4 ,you have been traveling for a time which1 by the clocks at the station from which you started1 is t4 ,at a distance x from your starting point1 as measured by the people on the line1 an event occurs at this moment--say the line is struck by lightning4 ,you have been traveling all the time with a uniform velocity v4 ,the question is3 ,how far from you will you judge that this event has taken place1 and how long after you started will it be by your watch1 assuming that your watch is correct from the point of view of an observer on the train8 #c ,so long as he has no considerable acceleration4 ,the treatment of acceleration belongs to the general theory of relativity4 ,our solution of this problem has to satisfy certain conditions4 ,it has to bring out the result that the velocity of light is the same for all observers1 however they may be moving4 ,and it has to make physical phenomena--in particular1 those of electromagnetism--obey the same laws for different observers1 however they may find their measures of distances and times affected by their motion4 ,and it has to make all such effects on measurement reciprocal4 ,that is to say1 if you are in a train and your motion affects your estimate of distances outside the train1 there must be an exactly similar change in the estimate which people outside the train make of distances inside it4 ,these conditions are sufficient to determine the solution of the problem1 but the method of obtaining the solution cannot be explained without more mathematics than is possible in the present work4 ,before dealing with the matter in general terms1 let us take an example4 ,let us suppose that you are in a train on a long straight railway1 and that you are traveling at three-fifths of the velocity of light4 ,suppose that you measure the length of your train1 and find that it is a hundred yards4 ,suppose that the people who catch a glimpse of you as you pass succeed1 by skilful scientific methods1 in taking observations which enable them to calculate the length of your train4 ,if they do their work correctly1 they will find that it is eighty yards long4 ,everything in the train will seem to them shorter in the direction of the train than it does to you4 ,dinner plates1 which you see as ordinary circular plates1 will look to the outsider as if they were oval3 they will seem only four-fifths as broad in the direction in which the train is moving as in the direction of the breadth of the train4 ,and all this is reciprocal4 ,suppose you see out of the window a man carrying a fishing rod which1 by his measurement1 is fifteen feet long4 ,if he is holding it upright1 you will see it as he does2 so you will if he is holding it horizontally at right angles to the railway4 ,but if he is pointing it along the railway1 it will seem to you to be only twelve feet long4 ,all lengths in the direction of motion are diminished by twenty per cent1 both for those who look into the train from outside and for those who look out of the train from inside4 ,but the effects in regard to time are even more strange4 ,this matter has been explained with almost ideal lucidity by ,eddington in ,space1 ,time and ,gravitation4 ,he supposes an aviator traveling1 relatively to the earth1 at a speed of #afa1jjj miles a second1 and he says3 8,if we observed the aviator carefully we should infer that he was unusually slow in his movements2 and events in the conveyance moving with him would be similarly retarded--as though time had forgotten to go on4 ,his cigar lasts twice as long as one of ours4 ,i said #8infer' deliberately2 we should see a still more extravagant slowing down of time2 but that is easily explained1 because the aviator is rapidly increasing his distance from us and the light impressions take longer and longer to reach us4 ,the more moderate retardation referred to remains after we have allowed for the time of transmission of light4 ,but here again reciprocity comes in1 because in the aviator's opinion it is we who are traveling at #afa1jjj miles a second past him2 and when he has made all allowances1 he finds that it is we who are sluggish4 ,our cigar lasts twice as long as his40 ,what a situation for envy6 ,each man thinks that the other's cigar lasts twice as long as his own4 ,it may1 however1 be some consolation to reflect that the other man's visits to the dentist also last twice as long4 ,this question of time is rather intricate1 owing to the fact that events which one man judges to be simultaneous another considers to be separated by a lapse of time4 ,in order to try to make clear how time is affected1 ,i shall revert to our railway train traveling due east at a rate three-fifths of that of light4 ,for the sake of illustration1 ,i assume that the earth is large and flat1 instead of small and round4 ,if we take events which happen at a fixed point on the earth1 and ask ourselves how long after the beginning of the journey they will seem to be to the traveler1 the answer is that there will be that retardation that ,eddington speaks of1 which means in this case that what seems an hour in the life of the stationary person is judged to be an hour and a quarter by the man who observes him from the train4 ,reciprocally1 what seems an hour in the life of the person in the train is judged by the man observing him from outside to be an hour and a quarter4 ,each makes periods of time observed in the life of the other a quarter as long again as they are to the person who lives through them4 ,the proportion is the same in regard to times as in regard to lengths4 ,but when1 instead of comparing events at the same place on the earth1 we compare events at widely separated places1 the results are still more odd4 ,let us now take all the events along the railway which1 from the point of view of a person who is stationary on the earth1 happen at a given instant1 say the instant when the observer in the train passes the stationary person4 ,of these events1 those which occur at points towards which the train is moving will seem to the traveler to have already happened1 while those which occur at points behind the train will1 for him1 be still in the future4 ,when ,i say that events in the forward direction will seem to have already happened1 ,i am saying something not strictly accurate1 because he will not yet have seen them2 but when he does see them1 he will1 after allowing for the velocity of light1 come to the conclusion that they must have happened before the moment in question4 ,an event which happens in the forward direction along the railway1 and which the stationary observer judges to be now 7or rather1 will judge to have been now when he comes to know of it71 if it occurs at a distance along the line which light could travel in a second1 will be judged by the traveler to have occurred three-quarters of a second ago4 ,if it occurs at a distance from the two observers which the man on the earth judges that light could travel in a year1 the traveler will judge 7when he comes to know of it7 that it occurred nine months earlier than the moment when he passed the earth dweller4 ,and generally1 he will ante-date events in the forward direction along the railway by three-quarters of the time that it would take light to travel from them to the man on the earth whom he is just passing1 and who holds that these events are happening now--or rather1 will hold that they happened now when the light from them reaches him4 ,events happening on the railway behind the train will be post-dated by an exactly equal amount4 ,we have thus a two-fold correction to make in the date of an event when we pass from the terrestrial observer to the traveler4 ,we must first take five-fourths of the time as estimated by the earth dweller1 and then subtract three-fourths of the time that it would take light to travel from the event in question to the earth dweller4 ,take some event in a distant part of the universe1 which becomes visible to the earth dweller and the traveler just as they pass each other4 ,the earth dweller1 if he knows how far off the event occurred1 can judge how long ago it occurred1 since he knows the speed of light4 ,if the event occurred in the direction towards which the traveler is moving1 the traveler will infer that it happened twice as long ago as the earth dweller thinks4 ,but if it occurred in the direction from which he has come1 he will argue that it happened only half as long ago as the earth dweller thinks4 ,if the traveler moves at a different speed1 these proportions will be different4 ,suppose now that 7as sometimes occurs7 two new stars have suddenly flared up1 and have just become visible to the traveler and to the earth dweller whom he is passing4 ,let one of them be in the direction towards which the train is traveling1 the other in the direction from which it has come4 ,suppose that the earth dweller is able1 in some way1 to estimate the distance of the two stars1 and to infer that light takes fifty years to reach him from the one in the direction towards which the traveler is moving1 and one hundred years to reach him from the other4 ,he will then argue that the explosion which produced the new star in the forward direction occurred fifty years ago1 while the explosion which produced the other new star occurred a hundred years ago4 ,the traveler will exactly reverse these figures3 he will infer that the forward explosion occurred a hundred years ago1 and the backward one fifty years ago4 ,i assume that both argue correctly on correct physical data4 ,in fact1 both are right1 unless they imagine that the other must be wrong4 ,it should be noted that both will have the same estimate of the velocity of light1 because their estimates of the distances of the two new stars will vary in exactly the same proportion as their estimates of the times since the explosions4 ,indeed1 one of the main motives of this whole theory is to secure that the velocity of light shall be the same for all observers1 however they may be moving4 ,this fact1 established by experiment1 was incompatible with the old theories1 and made it absolutely necessary to admit something startling4 ,the theory of relativity is just as little startling as is compatible with the facts4 ,indeed1 after a time1 it ceases to seem startling at all4 ,there is another feature of very great importance in the theory we have been considering1 and that is that1 although distances and times vary for different observers1 we can derive from them the quantity called 8interval10 which is the same for all observers4 ,the 8interval10 in the special theory of relativity1 is obtained as follows3 ,take the square of the distance between two events1 and the square of the distance traveled by light in the time between the two events2 subtract the lesser of these from the greater1 and the result is defined as the square of the interval between the events4 ,the interval is the same for all observers1 and represents a genuine physical relation between the two events1 which the time and the distance do not4 ,we have already given a geometrical construction for the interval at the end of ,chapter ,,iv2 this gives the same result as the above rule4 ,the interval is 8time-like0 when the time between the events is longer than light would take to travel from the place of the one to the place of the other2 in the contrary case it is 8space-like40 ,when the time between the two events is exactly equal to the time taken by light to travel from one to the other1 the interval is zero2 the two events are then situated on parts of one light ray1 unless no light happens to be passing that way4 ,when we come to the general theory of relativity1 we shall have to generalize the notion of interval4 ,the more deeply we penetrate into the structure of the world1 the more important this concept becomes2 we are tempted to say that it is the reality of which distances and periods of time are confused representations4 ,the theory of relativity has altered our view of the fundamental structure of the world2 that is the source both of its difficulty and of its importance4 ,the remainder of this chapter may be omitted by readers who have not even the most elementary acquaintance with geometry or algebra4 ,but for the benefit of those whose education has not been entirely neglected1 ,i will add a few explanations of the general formula of which ,i have hitherto given only particular examples4 ,the general formula in question is the 8,lorentz transformation10 which tells1 when one body is moving in a given manner relatively to another1 how to infer the measures of lengths and times appropriate to the one body from those appropriate to the other4 ,before giving the algebraical formul1 ,i will give a geometrical construction4 ,as before1 we will suppose that there are two observers1 whom we will call ,o and ,o1 one of whom is stationary on the earth while the other is traveling at a uniform speed along a straight railway4 ,at the beginning of the time considered1 the two observers were at the same point of the railway1 but now they are separated by a certain distance4 ,a flash of lightning strikes a point ,x on the railway1 and ,o judges that at the moment when the flash takes place the observer in the train has reached the point ,o4 ,the problem is3 how far will ,o judge that he is from the flash1 and how long after the beginning of the journey 7when he was at ,o7 will he judge that the flash took place8 ,we are supposed to know ,os estimates1 and we want to calculate those of ,o4 ,illustration ,in the time that1 according to ,o1 has elapsed since the beginning of the journey1 let ,,oc be the distance that light would have traveled along the railway4 ,describe a circle about ,o1 with ,,oc as radius1 and through ,o draw a perpendicular to the railway1 meeting the circle in ,d4 ,on ,,od take a point ,y such that ,,oy is equal to ,,ox 7,x is the point of the railway where the lightning strikes74 ,draw ,,ym perpendicular to the railway1 and ,,os perpendicular to ,,od4 ,let ,,ym and ,,os meet in ,s4 ,also let ,,do produced and ,,os produced meet in ,r4 ,through ,x and ,c draw perpendiculars to the railway meeting ,,os produced in ,q and ,z respectively4 ,then ,,rq 7as measured by ,o7 is the distance at which ,o will believe himself to be from the flash1 not ,o,x as it would be according to the old view4 ,and whereas ,o thinks that1 in the time from the beginning of the journey to the flash1 light would travel a distance ,,oc1 ,o thinks that the time elapsed is that required for light to travel the distance ,,sz 7as measured by ,o74 ,the interval as measured by ,o is got by subtracting the square on ,,ox from the square on ,,oc2 the interval as measured by ,o is got by subtracting the square on ,,rq from the square on ,,sz4 ,a little very elementary geometry shows that these are equal4 ,the algebraical formul embodied in the above construction are as follows3 ,from the point of view of ,o1 let an event occur at a distance x along the railway1 and at a time t after the beginning of the journey 7when ,o was at ,o74 ,from the point of view of ,o1 let the same event occur at a distance x along the railway1 and at a time t after the beginning of the journey4 ,let c be the velocity of light1 and v the velocity of ,o relative to ,o4 ,put c ------------------------ 7c - v7 ,then x 7x - vt7 7 vx 7 t 7t - ----------7 7 c 7 ,this is the ,lorentz transformation1 from which everything in this chapter can be deduced4 ,,chapter ,,vii3 ,,intervals ,,in ,,space-,,time ,the special theory of relativity1 which we have been considering hitherto1 solved completely a certain definite problem3 to account for the experimental fact that1 when two bodies are in uniform relative motion1 all the laws of physics1 both those of ordinary dynamics and those connected with electricity and magnetism1 are exactly the same for the two bodies4 8,uniform0 motion1 here1 means motion in a straight line with constant velocity4 ,but although one problem was solved by the special theory1 another was immediately suggested3 what if the motion of the two bodies is not uniform8 ,suppose1 for instance1 that one is the earth while the other is a falling stone4 ,the stone has an accelerated motion3 it is continually falling faster and faster4 ,nothing in the special theory enables us to say that the laws of physical phenomena will be the same for an observer on the stone as for one on the earth4 ,this is particularly awkward1 as the earth itself is1 in an extended sense1 a falling body3 ,it has at every moment an acceleration#d towards the sun1 which makes it go round the sun instead of moving in a straight line4 ,as our knowledge of physics is derived from experiments on the earth1 we cannot rest satisfied with a theory in which the observer is supposed to have no acceleration4 ,the general theory of relativity removes this restriction1 and allows the observer to be moving in any way1 straight or crooked1 uniformly or with an acceleration4 ,in the course of removing the restriction1 ,einstein was led to his new law of gravitation1 which we shall consider presently4 ,the work was extraordinarily difficult1 and occupied him for ten years4 ,the special theory dates from #aije1 the general theory from #aiae4 #d ,this does not mean that its velocity is increasing1 but that it is changing its direction4 ,the only sort of motion which is called 8unaccelerated0 is motion with uniform velocity in a straight line4 ,it is obvious from experiences with which we are all familiar that an accelerated motion is much more difficult to deal with than a uniform one4 ,when you are in a train which is traveling steadily1 the motion is not noticeable so long as you do not look out of the window2 but when the brakes are applied suddenly you are precipitated forwards1 and you become aware that something is happening without having to notice anything outside the train4 ,similarly in a lift everything seems ordinary while it is moving steadily1 but at starting and stopping1 when its motion is accelerated1 you have odd sensations in the pit of the stomach4 7,we call a motion 8accelerated0 when it is getting slower as well as when it is getting quicker2 when it is getting slower the acceleration is negative47 ,the same thing applies to dropping a weight in the cabin of a ship4 ,so long as the ship is moving uniformly1 the weight will behave1 relatively to the cabin1 just as if the ship were at rest3 if it starts from the middle of the ceiling1 it will hit the middle of the floor4 ,but if there is an acceleration everything is changed4 ,if the boat is increasing its speed very rapidly1 the weight will seem to an observer in the cabin to fall in a curve directed towards the stern2 if the speed is being rapidly diminished1 the curve will be directed towards the bow4 ,all these facts are familiar1 and they led ,galileo and ,newton to regard an accelerated motion as something radically different1 in its own nature1 from a uniform motion4 ,but this distinction could only be maintained by regarding motion as absolute1 not relative4 ,if all motion is relative1 the earth is accelerated relatively to the lift just as truly as the lift relatively to the earth4 ,yet the people on the ground have no sensations in the pits of their stomachs when the lift starts to go up4 ,this illustrates the difficulty of our problem4 ,in fact1 though few physicists in modern times have believed in absolute motion1 the technique of mathematical physics still embodied ,newton's belief in it1 and a revolution in method was required to obtain a technique free from this assumption4 ,this revolution was accomplished in ,einstein's general theory of relativity4 ,it is somewhat optional where we begin in explaining the new ideas which ,einstein introduced1 but perhaps we shall do best by taking the conception of 8interval40 ,this conception1 as it appears in the special theory of relativity1 is already a generalization of the traditional notion of distance in space and time2 but it is necessary to generalize it still further4 ,however1 it is necessary first to explain a certain amount of history1 and for this purpose we must go back as far as ,pythagoras4 ,pythagoras1 like many of the greatest characters in history1 perhaps never existed3 he is a semi-mythical character1 who combined mathematics and priestcraft in uncertain proportions4 ,i shall1 however1 assume that he existed1 and that he discovered the theorem attributed to him4 ,he was roughly a contemporary of ,confucius and ,buddha2 he founded a religious sect1 which thought it wicked to eat beans1 and a school of mathematicians1 who took a particular interest in right-angled triangles4 ,the theorem of ,pythagoras 7the forty-seventh proposition of ,euclid7 states that the sum of the squares on the two shorter sides of a right-angled triangle is equal to the square on the side opposite the right angle4 ,no proposition in the whole of mathematics has had such a distinguished history4 ,we all learned to 8prove0 it in youth4 ,it is true that the 8proof0 proved nothing1 and that the only way to prove it is by experiment4 ,it is also the case that the proposition is not quite true--it is only approximately true4 ,but everything in geometry1 and subsequently in physics1 has been derived from it by successive generalizations4 ,the latest of these generalizations is the general theory of relativity4 ,the theorem of ,pythagoras was itself1 in all probability1 a generalization of an ,egyptian rule of thumb4 ,in ,egypt1 it had been known for ages that a triangle whose sides are #c1 #d1 and #e units of length is a right-angled triangle2 the ,egyptians used this knowledge practically in measuring their fields4 ,now if the sides of a triangle are #c1 #d1 and #e inches1 the squares on these sides will contain respectively #i1 #af1 and #be square inches2 and #i and #af added together make #be4 ,three times three is written 8#c02 four times four1 8#d02 five times five1 8#e40 ,so that we have #c #d #e4 ,it is supposed that ,pythagoras noticed this fact1 after he had learned from the ,egyptians that a triangle whose sides are #c1 #d and #e has a right angle4 ,he found that this could be generalized1 and so arrived at his famous theorem3 ,in a right-angled triangle1 the square on the side opposite the right angle is equal to the sum of the squares on the other two sides4 ,illustration ,similarly in three dimensions3 if you take a right-angled solid block1 the square on the diagonal 7the dotted line in the figure7 is equal to the sum of the squares on the three sides4 ,this is as far as the ancients got in this matter4 ,illustration ,the next step of importance is due to ,descartes1 who made the theorem of ,pythagoras the basis of his method of analytical geometry4 ,suppose you wish to map out systematically all the places on a plain--we will suppose it small enough to make it possible to ignore the fact that the earth is round4 ,we will suppose that you live in the middle of the plain4 ,one of the simplest ways of describing the position of a place is to say3 starting from my house1 go first such and such a distance east1 then such and such a distance north 7or it may be west in the first case1 and south in the second74 ,this tells you exactly where the place is4 ,in the rectangular cities of ,america1 it is the natural method to adopt3 in ,new ,york you will be told to go so many blocks east 7or west7 and then so many blocks north 7or south74 ,the distance you have to go east is called x1 and the distance you have to go north is called y4 7,if you have to go west1 x is negative2 if you have to go south1 y is negative47 ,let ,o be your starting point 7the 8origin072 let ,,om be the distance you go east1 and ,,mp the distance you go north4 ,how far are you from home in a direct line when you reach ,p8 ,the theorem of ,pythagoras gives the answer4 ,the square on ,,op is the sum of the squares on ,,om and ,,mp4 ,if ,,om is four miles1 and ,,mp is three miles1 ,,op is #e miles4 ,if ,,om is #ab miles and ,,mp is #e miles1 ,,op is #ac miles1 because #ab #e #ac4 ,so that if you adopt ,descartes' method of mapping1 the theorem of ,pythagoras is essential in giving you the distance from place to place4 ,in three dimensions the thing is exactly analogous4 ,suppose that1 instead of wanting merely to fix positions on the plain1 you want to fix stations for captive balloons above it1 you will then have to add a third quantity1 the height at which the balloon is to be4 ,if you call the height z1 and if r is the direct distance from ,o to the balloon1 you will have r x y z1 and from this you can calculate r when you know x1 y1 and z4 ,for example1 if you can get to the balloon by going #ab miles east1 #d miles north1 and then #c miles up1 your distance from the balloon in a straight line is #ac miles1 because #ab #ab #add1 #d #d #af1 #c #c #i1 #add #af #i #afi #ac #ac4 ,but now suppose that1 instead of taking a small piece of the earth's surface which can be regarded as flat1 you consider making a map of the world4 ,an accurate map of the world on flat paper is impossible4 ,a globe can be accurate1 in the sense that everything is produced to scale1 but a flat map cannot be4 ,i am not talking of practical difficulties1 ,i am talking of a theoretical impossibility4 ,for example3 the northern halves of the meridian of ,greenwich and the ninetieth meridian of west longitude1 together with the piece of the equator between them1 make a triangle whose sides are all equal and whose angles are all right angles4 ,on a flat surface1 a triangle of that sort would be impossible4 ,on the other hand1 it is possible to make a square on a flat surface1 but on a sphere it is impossible4 ,suppose you try on the earth3 walk #ajj miles west1 then #ajj miles north1 then #ajj miles east1 then #ajj miles south4 ,you might think this would make a square1 but it wouldn't1 because you would not at the end have come back to your starting point4 ,if you have time1 you may convince yourself of this by experiment4 ,if not1 you can easily see that it must be so4 ,when you are nearer the pole1 #ajj miles takes you through more longitude than when you are nearer the equator1 so that in doing your #ajj miles east 7if you are in the northern hemisphere7 you get to a point further east than that from which you started4 ,as you walk due south after this1 you remain further east than your starting point1 and end up at a different place from that in which you began4 ,suppose1 to take another illustration1 that you start on the equator #d1jjj miles east of the ,greenwich meridian2 you travel till you reach the meridian1 then you travel northwards along it for #d1jjj miles1 through ,greenwich and up to the neighborhood of the ,shetland ,islands2 then you travel eastward for #d1jjj miles1 and then #d1jjj miles south4 ,this will take you to the equator at a point #d1jjj miles further east than the point from which you started4 ,in a sense1 what we have just been saying is not quite fair1 because1 except on the equator1 traveling due east is not the shortest route from a place to another place due east of it4 ,a ship traveling 7say7 from ,new ,york to ,lisbon1 which is nearly due east1 will start by going a certain distance northward4 ,it will sail on a 8great circle10 that is to say1 a circle whose centre is the centre of the earth4 ,this is the nearest approach to a straight line that can be drawn on the surface of the earth4 ,meridians of longitude are great circles1 and so is the equator1 but the other parallels of latitude are not4 ,we ought1 therefore1 to have supposed that1 when you reach the ,shetland ,islands1 you travel #d1jjj miles1 not due east1 but along a great circle which lands you at a point due east of the ,shetland ,islands4 ,this1 however1 only reinforces our conclusion3 you will end at a point even further east of your starting point than before4 ,what are the differences between the geometry on a sphere and the geometry on a plane8 ,if you make a triangle on the earth1 whose sides are great circles1 you will not find that the angles of the triangle add up to two right angles3 they will add up to rather more4 ,the amount by which they exceed two right angles is proportional to the size of the triangle4 ,on a small triangle such as you could make with strings on your lawn1 or even on a triangle formed by three ships which can just see each other1 the angles will add up to so little more than two right angles that you will not be able to detect the difference4 ,but if you take the triangle made by the equator1 the ,greenwich meridian1 and the ninetieth meridian1 the angles add up to three right angles4 ,and you can get triangles in which the angles add up to anything up to six right angles4 ,all this you could discover by measurements on the surface of the earth1 without having to take account of anything in the rest of space4 ,the theorem of ,pythagoras also will fail for distances on a sphere4 ,from the point of view of a traveler bound to the earth1 the distance between two places is their great circle distance1 that is to say1 the shortest journey that a man can make without leaving the surface of the earth4 ,now suppose you take three bits of great circles which make a triangle1 and suppose one of them is at right angles to another--to be definite1 let one be the equator and one a bit of the meridian of ,greenwich going northward from the equator4 ,suppose you go #c1jjj miles along the equator1 and then #d1jjj miles due north2 how far will you be from your starting point1 estimating the distance along a great circle8 ,if you were on a plane1 your distance would be #e1jjj miles1 as we saw before4 ,in fact1 however1 your great circle distance will be considerably less than this4 ,in a right-angled triangle on a sphere1 the square on the side opposite the right angle is less than the sum of the squares on the other two sides4 ,these differences between the geometry on a sphere and the geometry on a plane are intrinsic differences2 that is to say1 they enable you to find out whether the surface on which you live is like a plane or like a sphere1 without requiring that you should take account of anything outside the surface4 ,such considerations led to the next step of importance in our subject1 which was taken by ,gauss1 who flourished a hundred years ago4 ,he studied the theory of surfaces1 and showed how to develop it by means of measurements on the surfaces themselves1 without going outside them4 ,in order to fix the position of a point in space1 we need three measurements2 but in order to fix the position of a point on a surface we need only two--for example1 a point on the earth's surface is fixed when we know its latitude and longitude4 ,now ,gauss found that1 whatever system of measurement you adopt1 and whatever the nature of the surface1 there is always a way of calculating the distance between two not very distant points of the surface1 when you know the quantities which fix their positions4 ,the formula for the distance is a generalization of the formula of ,pythagoras3 it tells you the square of the distance in terms of the squares of the differences between the measure quantities which fix the points1 and also the product of these two quantities4 ,when you know this formula1 you can discover all the intrinsic properties of the surface1 that is to say1 all those which do not depend upon its relations to points outside the surface4 ,you can discover1 for example1 whether the angles of a triangle add up to two right angles1 or more1 or less1 or more in some cases and less in others4 ,but when we speak of a 8triangle10 we must explain what we mean1 because on most surfaces there are no straight lines4 ,on a sphere1 we shall replace straight lines by great circles1 which are the nearest possible approach to straight lines4 ,in general1 we shall take1 instead of straight lines1 the lines that give the shortest route on the surface from place to place4 ,such lines are called 8geodesics40 ,on the earth1 the geodesics are great circles4 ,in general1 they are the shortest way of traveling from point to point if you are unable to leave the surface4 ,they take the place of straight lines in the intrinsic geometry of a surface4 ,when we inquire whether the angles of a triangle add up to two right angles or not1 we mean to speak of a triangle whose sides are geodesics4 ,and when we speak of the distance between two points1 we mean the distance along a geodesic4 ,the next step in our generalizing process is rather difficult3 it is the transition to non-,euclidean geometry4 ,we live in a world in which space has three dimensions1 and our empirical knowledge of space is based upon measurement of small distances and of angles4 7,when ,i speak of small distances1 ,i mean distances that are small compared to those in astronomy2 all distances on the earth are small in this sense47 ,it was formerly thought that we could be sure priori that space is ,euclidean--for instance1 that the angles of a triangle add up to two right angles4 ,but it came to be recognized that we could not prove this by reasoning2 if it was to be known1 it must be known as the result of measurements4 ,before ,einstein1 it was thought that measurements confirm ,euclidean geometry within the limits of exactitude attainable2 now this is no longer thought4 ,it is still true that we can1 by what may be called a natural artifice1 cause ,euclidean geometry to seem true throughout a small region1 such as the earth2 but in explaining gravitation ,einstein is led to the view that over large regions where there is matter we cannot regard space as ,euclidean4 ,the reasons for this will concern us later4 ,what concerns us now is the way in which non-,euclidean geometry results from a generalization of the work of ,gauss4 ,there is no reason why we should not have the same circumstances in three-dimensional space as we have1 for example1 on the surface of a sphere4 ,it might happen that the angles of a triangle would always add up to more than two right angles1 and that the excess would be proportional to the size of the triangle4 ,it might happen that the distance between two points would be given by a formula analogous to what we have on the surface of a sphere1 but involving three quantities instead of two4 ,whether this does happen or not1 can only be discovered by actual measurements4 ,there are an infinite number of such possibilities4 ,this line of argument was developed by ,riemann1 in his dissertation 8,on the hypotheses which underlie geometry0 7#ahed71 which applied ,gauss's work on surfaces to different kinds of three-dimensional spaces4 ,he showed that all the essential characteristics of a kind of space could be deduced from the formula for small distances4 ,he assumed that1 from the small distances in three given directions which would together carry you from one point to another not far from it1 the distances between the two points could be calculated4 ,for instance1 if you know that you can get from one point to another by first moving a certain distance east1 then a certain distance north1 and finally a certain distance straight up in the air1 you are to be able to calculate the distance from the one point to the other4 ,and the rule for the calculation is to be an extension of the theorem of ,pythagoras1 in the sense that you arrive at the square of the required distance by adding together multiples of the squares of the component distances1 together possibly with multiples of their products4 ,from certain characteristics in the formula1 you can tell what sort of space you have to deal with4 ,these characteristics do not depend upon the particular method you have adopted for determining the positions of points4 ,in order to arrive at what we want for the theory of relativity1 we now have one more generalization to make3 we have to substitute the 8interval0 between events for the distance between points4 ,this takes us to space-time4 ,we have already seen that1 in the special theory of relativity1 the square of the interval is found by subtracting the square of the distance between the events from the square of the distance that light would travel in the time between them4 ,in the general theory1 we do not assume this special form of interval1 except at a great distance from matter4 ,elsewhere1 we assume to begin with a general form1 like that which ,riemann used for distances4 ,moreover1 like ,riemann1 ,einstein only assumes his formula for neighboring events1 that is to say1 events which have only a small interval between them4 ,what goes beyond these initial assumptions depends upon observation of the actual motion of bodies1 in ways which we shall explain in later chapters4 ,we may now sum up and re-state the process we have been describing4 ,in three dimensions1 the position of a point relatively to a fixed point 7the 8origin07 can be determined by assigning three quantities 78co-ordinates074 ,for example1 the position of a balloon relatively to your house is fixed if you know that you will reach it by going first a given distance due east1 then another given distance due north1 then a third given distance straight up4 ,when1 as in this case1 the three co-ordinates are three distances all at right angles to each other1 which1 taken successively1 transport you from the origin to the point in question1 the square of the direct distance to the point in question is got by adding up the squares of the three co-ordinates4 ,in all cases1 whether in ,euclidean or in non-,euclidean spaces1 it is got by adding multiples of the squares and products of the co-ordinates according to an assignable rule4 ,the co-ordinates may be any quantities which fix the position of a point1 provided that neighboring points must have neighboring quantities for their co-ordinates4 ,in the general theory of relativity1 we add a fourth co-ordinate to give the time1 and our formula gives 8interval0 instead of spatial distance2 moreover we assume the accuracy of our formula for small distances only4 ,we assume further that1 at great distances from matter1 the formula approximates more and more closely to the formula for interval which is used in the special theory4 ,we are now at last in a position to tackle ,einstein's theory of gravitation4 ,,chapter ,,viii3 ,,einstein',s ,,law ,,of ,,gravitation ,before tackling ,einstein's new law1 it is as well to convince ourselves1 on logical grounds1 that ,newton's law of gravitation cannot be quite right4 ,newton said that between any two particles of matter there is a force which is proportional to the product of their masses and inversely proportional to the square of their distance4 ,that is to say1 ignoring for the present the question of mass1 if there is a certain attraction when the particles are a mile apart1 there will be a quarter as much attraction when they are two miles apart1 a ninth as much when they are three miles apart1 and so on3 the attraction diminishes much faster than the distance increases4 ,now1 of course1 ,newton1 when he spoke of the distance1 meant the distance at a given time3 ,he thought there could be no ambiguity about time4 ,but we have seen that this was a mistake4 ,what one observer judges to be the same moment on the earth and the sun1 another will judge to be two different moments4 8,distance at a given moment0 is therefore a subjective conception1 which can hardly enter into a cosmic law4 ,of course1 we could make our law unambiguous by saying that we are going to estimate times as they are estimated by ,greenwich ,observatory4 ,but we can hardly believe that the accidental circumstances of the earth deserve to be taken so seriously4 ,and the estimate of distance1 also1 will vary for different observers4 ,we cannot1 therefore1 allow that ,newton's form of the law of gravitation can be quite correct1 since it will give different results according to which of many equally legitimate conventions we adopt4 ,this is as absurd as it would be if the question whether one man had murdered another were to depend upon whether they were described by their ,christian names or their surnames4 ,it is obvious that physical laws must be the same whether distances are measured in miles or in kilometers1 and we are concerned with what is essentially only an extension of the same principle4 ,our measurements are conventional to an even greater extent than is admitted by the special theory of relativity4 ,moreover1 every measurement is a physical process carried out with physical material2 the result is certainly an experimental datum1 but may not be susceptible of the simple interpretation which we ordinarily assign to it4 ,we are1 therefore1 not going to assume to begin with that we know how to measure anything4 ,we assume that there is a certain physical quantity1 called 8interval10 which is a relation between two events that are not widely separated2 but we do not assume in advance that we know how to measure it1 beyond taking it for granted that it is given by some generalization of the theorem of ,pythagoras such as we spoke of in the preceding chapter4 ,we do assume1 however1 that events have an order1 and that this order is four-dimensional4 ,we assume1 that is to say1 that we know what we mean by saying that a certain event is nearer to another than to a third1 so that before making accurate measurements we can speak of the 8neighborhood0 of an event2 and we assume that1 in order to assign the position of an event in space-time1 four quantities 7co-ordinates7 are necessary--e4g4 in our former case of an explosion on an airship1 latitude1 longitude1 altitude and time4 ,but we assume nothing about the way in which these co-ordinates are assigned1 except that neighboring co-ordinates are assigned to neighboring events4 ,the way in which these numbers1 called co-ordinates1 are to be assigned is neither wholly arbitrary nor a result of careful measurement--it lies in an intermediate region4 ,while you are making any continuous journey1 your co-ordinates must never alter by sudden jumps4 ,in ,america one finds that the houses between 7say7 ,fourteenth ,street and ,fifteenth ,street are likely to have numbers between #adjj and #aejj1 while those between ,fifteenth ,street and ,sixteenth ,street have numbers between #aejj and #afjj1 even if the #adjj's were not used up4 ,this would not do for our purposes1 because there is a sudden jump when we pass from one block to the next4 ,or again we might assign the time co-ordinate in the following way3 take the time that elapses between two successive births of people called ,smith2 an event occurring between the births of the #cjjjth and the #cjjast ,smith known to history shall have a co-ordinate lying between #cjjj and #cjja2 the fractional part of its co-ordinate shall be the fraction of a year that has elapsed since the birth of the #cjjjth ,smith4 7,obviously there could never be as much as a year between two successive additions to the ,smith family47 ,this way of assigning the time co-ordinate is perfectly definite1 but it is not admissible for our purposes1 because there will be sudden jumps between events just before the birth of a ,smith and events just after1 so that in a continuous journey your time co-ordinate will not change continuously4 ,it is assumed that1 independently of measurement1 we know what a continuous journey is4 ,and when your position in space-time changes continuously1 each of your four co-ordinates must change continuously4 ,one1 two1 or three of them may not change at all2 but whatever change does occur must be smooth1 without sudden jumps4 ,this explains what is not allowable in assigning co-ordinates4 ,to explain all the changes that are legitimate in your co-ordinates1 suppose you take a large piece of soft india-rubber4 ,while it is in an unstretched condition1 measure little squares on it1 each one-tenth of an inch each way4 ,put in little tiny pins at the corners of the squares4 ,we can take as two of the co-ordinates of one of these pins the number of pins passed in going to the right from a given pin until we come just below the pin in question1 and then the number of pins we pass on the way up to this pin4 ,in the figure1 let ,o be the pin we start from and ,p the pin to which we are going to assign co-ordinates4 ,p is in the fifth column and the third row1 so its co-ordinates in the plane of the india-rubber are to be #e and #c4 ,illustration3 ,fig4 #a4 ,illustration3 ,fig4 #b4 ,now take the india-rubber and stretch it and twist it as much as you like4 ,let the pins now be in the shape they have in ,fig4 #b4 ,the divisions now no longer represent distances according to our usual notions1 but they will still do just as well as co-ordinates4 ,we may still take ,p as having the co-ordinates #e and #c in the plane of the india-rubber2 and we may still regard the india-rubber as being in a plane1 even if we have twisted it out of what we should ordinarily call a plane4 ,such continuous distortions do not matter4 ,to take another illustration3 instead of using a steel measuring rod to fix our co-ordinates1 let us use a live eel1 which is wriggling all the time4 ,the distance from the tail to the head of the eel is to count as one from the point of view of co-ordinates1 whatever shape the creature may be assuming at the moment4 ,the eel is continuous1 and its wriggles are continuous1 so it may be taken as our unit of distance in assigning co-ordinates4 ,beyond the requirement of continuity1 the method of assigning co-ordinates is purely conventional1 and therefore a live eel is just as good as a steel rod4 ,we are apt to think that1 for really careful measurements1 it is better to use a steel rod than a live eel4 ,this is a mistake3 not because the eel tells us what the steel rod was thought to tell1 but because the steel rod really tells no more than the eel obviously does4 ,the point is1 not that eels are really rigid1 but that steel rods really wriggle4 ,to an observer in just one possible state of motion1 the eel would appear rigid1 while the steel rod would seem to wriggle just as the eel does to us4 ,for everybody moving differently both from this observer and ourselves1 both the eel and the rod would seem to wriggle4 ,and there is no saying that one observer is right and another wrong4 ,in such matters1 what is seen does not belong solely to the physical process observed1 but also to the standpoint of the observer4 ,measurements of distances and times do not directly reveal properties of the things measured1 but relations of the things to the measurer4 ,what observation can tell us about the physical world is therefore more abstract than we have hitherto believed4 ,it is important to realize that geometry1 as taught in schools since ,greek times1 ceases to exist as a separate science1 and becomes merged in physics4 ,the two fundamental notions in elementary geometry were the straight line and the circle4 ,what appears to you as a straight road1 whose parts all exist now1 may appear to another observer to be like the flight of a rocket1 some kind of curve whose parts come into existence successively4 ,the circle depends upon measurement of distances1 since it consists of all the points at a given distance from its center4 ,and measurement of distances1 as we have seen1 is a subjective affair1 depending upon the way in which the observer is moving4 ,the failure of the circle to have objective validity was demonstrated by the ,michelson-,morley experiment1 and is thus1 in a sense1 the starting point of the whole theory of relativity4 ,rigid bodies1 which we need for measurement1 are only rigid for certain observers2 for others1 they will be constantly changing all their dimensions4 ,it is only our obstinately earth-bound imagination that makes us suppose a geometry separate from physics to be possible4 ,that is why we do not trouble to give physical significance to our co-ordinates from the start4 ,formerly1 the co-ordinates used in physics were supposed to be carefully measured distances2 now we realize that this care at the start is thrown away4 ,it is at a later stage that care is required4 ,our co-ordinates now are hardly more than a systematic way of cataloguing events4 ,but mathematics provides1 in the method of tensors1 such an immensely powerful technique that we can use co-ordinates assigned in this apparently careless way just as effectively as if we had applied the whole apparatus of minutely accurate measurement in arriving at them4 ,the advantage of being haphazard at the start is that we avoid making surreptitious physical assumptions1 which we can hardly help making1 if we suppose that our co-ordinates have initially some particular physical significance4 ,we assume that1 if two events are close together 7but not necessarily otherwise71 there is an interval between them which can be calculated from the differences between their co-ordinates by some such formula as we considered in the preceding chapter4 ,that is to say1 we take the squares and products of the differences of co-ordinates1 we multiply them by suitable amounts 7which in general will vary from place to place71 and we add the results together4 ,the sum obtained is the square of the interval4 ,we do not assume in advance that we know the amounts by which the squares and products must be multiplied2 this is going to be discovered by observing physical phenomena4 ,we know1 however1 certain things4 ,we know that the old ,newtonian physics is very nearly accurate when our co-ordinates have been chosen in a certain way4 ,we know that the special theory of relativity is still more nearly accurate for suitable co-ordinates4 ,from such facts we can infer certain things about our new co-ordinates1 which1 in a logical deduction1 appear as postulates of the new theory4 ,as such postulates we take3 #a4 ,that every body travels in a geodesic in space-time1 except in so far as electromagnetic forces act upon it4 #b4 ,that a light ray travels so that the interval between two parts of it is zero4 #c4 ,that at a great distance from gravitating matter1 we can transform our co-ordinates by mathematical manipulation so that the interval shall be what it is in the special theory of relativity2 and that this is approximately true wherever gravitation is not very powerful4 ,each of these postulates requires some explanation4 ,we saw that a geodesic on a surface is the shortest line that can be drawn on the surface from one point to another2 for example1 on the earth the geodesics are great circles4 ,when we come to space-time1 the mathematics is the same1 but the verbal explanations have to be rather different4 ,in the general theory of relativity1 it is only neighboring events that have a definite interval1 independently of the route by which we travel from one to the other4 ,the interval between distant events depends upon the route pursued1 and has to be calculated by dividing the route into a number of little bits and adding up the intervals for the various little bits4 ,if the interval is space-like1 a body cannot travel from one event to the other2 therefore when we are considering the way bodies move1 we are confined to time-like intervals4 ,the interval between neighboring events1 when it is time-like1 will appear as the time between them for an observer who travels from the one event to the other4 ,and so the whole interval between two events will be judged by a person who travels from one to the other to be what his clocks show to be the time that he has taken on the journey4 ,for some routes this time will be longer1 for others shorter2 the more slowly the man travels1 the longer he will think he has been on the journey4 ,this must not be taken as a platitude4 ,i am not saying that if you travel from ,london to ,edinburgh you will take longer if you travel more slowly4 ,i am saying something much more odd4 ,i am saying that if you leave ,london at #aj ,a4,m4 and arrive in ,edinburgh at #f4cj ,p4,m4 ,greenwich time1 the more slowly you travel the longer you will take--if the time is judged by your watch4 ,this is a very different statement4 ,from the point of view of a person on the earth1 your journey takes eight and a half hours4 ,but if you had been a ray of light traveling round the solar system1 starting from ,london at #aj ,a4,m41 reflected from ,jupiter to ,saturn1 and so on1 until at last you were reflected back to ,edinburgh and arrived there at #f4cj ,p4,m41 you would judge that the journey had taken you exactly no time4 ,and if you had gone by any circuitous route1 which enabled you to arrive in time by traveling fast1 the longer your route the less time you would judge that you had taken2 the diminution of time would be continual as your speed approached that of light4 ,now ,i say that when a body travels1 if it is left to itself1 it chooses the route which makes the time between two stages of the journey as long as possible2 if it had traveled from one event to another by any other route1 the time1 as measured by its own clocks1 would have been shorter4 ,this is a way of saying that bodies left to themselves do their journeys as slowly as they can2 it is a sort of law of cosmic laziness4 ,its mathematical expression is that they travel in geodesics1 in which the total interval between any two events on the journey is greater than by any alternative route4 7,the fact that it is greater1 not less1 is due to the fact that the sort of interval we are considering is more analogous to time than to distance47 ,for example1 if a person could leave the earth and travel about for a time and then return1 the time between his departure and return would be less by his clocks than by those on the earth3 the earth1 in its journey round the sun1 chooses the route which makes the time of any bit of its course by its clocks longer than the time as judged by clocks which move by a different route4 ,this is what is meant by saying that bodies left to themselves move in geodesics in space-time4 ,we assume that the body considered is not acted upon by electromagnetic forces4 ,we are concerned at present with the law of gravitation1 not with the effects of electromagnetism4 ,these effects have been brought into the framework of the general theory of relativity by ,weyl1#e but for the present we will ignore his work4 ,the planets1 in any case1 are not subject1 as wholes1 to appreciable electromagnetic forces2 it is only gravitation that has to be considered in accounting for their motions1 with which we are concerned in this chapter4 #e ,see his ,space1 ,time1 ,matter1 ,methuen1 #aibb4 ,our second postulate1 that a light ray travels so that the interval between two parts of it is zero1 has the advantage that it does not have to be stated only for small distances4 ,if each little bit of interval is zero1 the sum of them all is zero1 and so even distant parts of the same light ray have a zero interval4 ,the course of a light ray is also a geodesic according to the definition4 ,thus we now have two empirical ways of discovering what are the geodesics in space-time1 namely light rays and bodies moving freely4 ,among freely-moving bodies are included all which are not subject to constraints or to electromagnetic forces1 that is to say1 the sun1 stars1 planets and satellites1 and also falling bodies on the earth1 at least when they are falling in a vacuum4 ,when you are standing on the earth1 you are subject to electromagnetic forces3 the electrons and protons in the neighborhood of your feet exert a repulsion on your feet which is just enough to overcome the earth's gravitation4 ,this is what prevents you from falling through the earth1 which1 solid as it looks1 is mostly empty space4 ,the third postulate1 which relates the general to the special theory1 is very useful4 ,it is not necessary for the application of the special theory to a limited region that there should be no gravitation in the region2 it is enough if the intensity of gravitation is practically the same throughout the region4 ,this enables us to apply the special theory within any small region4 ,how small it will have to be1 depends upon the neighborhood4 ,on the surface of the earth1 it would have to be small enough for the curvature of the earth to be negligible4 ,in the spaces between the planets1 it need only be small enough for the attraction of the sun and the planets to be sensibly constant throughout the region4 ,in the spaces between the stars it might be enormous--say half the distance from one star to the next--without introducing measurable inaccuracies4 ,at a great distance from gravitating matter1 we can so choose our co-ordinates as to obtain a ,euclidean space2 this is really only another way of saying that the special theory of relativity applies4 ,in the neighborhood of matter1 although we can make our space ,euclidean in any small region1 we cannot do so throughout any region within which gravitation varies sensibly--at least1 if we do1 we shall have to abandon the view that bodies move in geodesics4 ,in the neighborhood of a piece of matter1 there is1 as it were1 a hill in space-time2 this hill grows steeper and steeper as it gets nearer the top1 like the neck of a champagne bottle4 ,it ends in a sheer precipice4 ,now by the law of cosmic laziness which we mentioned earlier1 a body coming into the neighborhood of the hill will not attempt to go straight over the top1 but will go round4 ,this is the essence of ,einstein's view of gravitation4 ,what a body does1 it does because of the nature of space-time in its own neighborhood1 not because of some mysterious force emanating from a distant body4 ,an analogy will serve to make the point clear4 ,suppose that on a dark night a number of men with lanterns were walking in various directions across a huge plain1 and suppose that in one part of the plain there was a hill with a flaring beacon on the top4 ,our hill is to be such as we have described1 growing steeper as it goes up1 and ending in a precipice4 ,i shall suppose that there are villages dotted about the plain1 and the men with lanterns are walking to and from these various villages4 ,paths have been made showing the easiest way from any one village to any other4 ,these paths will all be more or less curved1 to avoid going too far up the hill2 they will be more sharply curved when they pass near the top of the hill than when they keep some way off from it4 ,now suppose that you are observing all this1 as best you can1 from a place high up in a balloon1 so that you cannot see the ground1 but only the lanterns and the beacon4 ,you will not know that there is a hill1 or that the beacon is at the top of it4 ,you will see that people turn out of the straight course when they approach the beacon1 and that the nearer they come the more they turn aside4 ,you will naturally attribute this to an effect of the beacon2 you may think that it is very hot and people are afraid of getting burnt4 ,but if you wait for daylight you will see the hill1 and you will find that the beacon merely marks the top of the hill and does not influence the people with lanterns in any way4 ,now in this analogy the beacon corresponds to the sun1 the people with lanterns correspond to the planets and comets1 the paths correspond to their orbits1 and the coming of daylight corresponds to the coming of ,einstein4 ,einstein says that the sun is at the top of a hill1 only the hill is in space-time1 not in space4 7,i advise the reader not to try to picture this1 because it is impossible47 ,each body1 at each moment1 adopts the easiest course open to it1 but owing to the hill the easiest course is not a straight line4 ,each little bit of matter is at the top of its own little hill1 like the cock on his own dung-heap4 ,what we call a big bit of matter is a bit which is at the top of a big hill4 ,the hill is what we know about2 the bit of matter at the top is assumed for convenience4 ,perhaps there is really no need to assume it1 and we could do with the hill alone1 for we can never get to the top of any one else's hill1 any more than the pugnacious cock can fight the peculiarly irritating bird that he sees in the looking glass4 ,i have given only a qualitative description of ,einstein's law of gravitation2 to give its exact quantitative formulation is impossible without more mathematics than ,i am permitting myself4 ,the most interesting point about it is that it makes the law no longer the result of action at a distance3 the sun exerts no force on the planets whatever4 ,just as geometry has become physics1 so1 in a sense1 physics has become geometry4 ,the law of gravitation has become the geometrical law that every body pursues the easiest course from place to place1 but this course is affected by the hills and valleys that are encountered on the road4 ,,chapter ,,ix3 ,,proofs ,,of ,,einstein',s ,,law ,,of ,,gravitation ,the reasons for accepting ,einstein's law of gravitation rather than ,newton's are partly empirical1 partly logical4 ,we will begin with the former4 ,einstein's law of gravitation gives very nearly the same results as ,newton's1 when applied to the calculation of the orbits of the planets and their satellites4 ,if it did not1 it could not be true1 since the consequences deduced from ,newton's law have been found to be almost exactly verified by observation4 ,when1 in #aiae1 ,einstein first published his new law1 there was only one empirical fact to which he could point to show that his theory was better than ,newton's4 ,this was what is called the 8motion of the perihelion of ,mercury40 ,the planet ,mercury1 like the other planets1 moves round the sun in an ellipse1 with the sun in one of the foci4 ,at some points of its orbit it is nearer to the sun than at other points4 ,the point where it is nearest to the sun is called its 8perihelion40 ,now it was found by observation that1 from one occasion when ,mercury is nearest to the sun until the next1 ,mercury does not go exactly once round the sun1 but a little bit more4 ,the discrepancy is very small2 it amounts to an angle of forty-two seconds in a century4 ,that is to say1 in each year the planet has to move rather less than half a second of angle after it has finished a complete revolution from the last perihelion before it reaches the next perihelion4 ,this very minute discrepancy from ,newtonian theory had puzzled astronomers4 ,there was a calculated effect due to perturbations caused by the other planets1 but this small discrepancy was the residue after allowing for these perturbations4 ,einstein's theory accounted for this residue1 as well as for its absence in the case of the other planets4 7,in them it exists1 but is too small to be observed47 ,this was1 at first1 his only empirical advantage over ,newton4 ,his second success was more sensational4 ,according to orthodox opinion1 light in a vacuum ought always to travel in straight lines4 ,not being composed of material particles1 it ought to be unaffected by gravitation4 ,however1 it was possible1 without any serious breach with old ideas1 to admit that1 in passing near the sun1 light might be deflected out of the straight path as much as if it were composed of material particles4 ,einstein1 however1 maintained1 as a deduction from his law of gravitation1 that light would be deflected twice as much as this4 ,that is to say1 if the light of a star passed very near the sun1 ,einstein maintained that the ray from the star would be turned through an angle of just under one and three-quarters seconds4 ,his opponents were willing to concede half of this amount4 ,now it is not every day that a star almost in line with the sun can be seen4 ,this is only possible during a total eclipse1 and not always then1 because there may be no bright stars in the right position4 ,eddington points out that1 from this point of view1 the best day of the year is ,may #bi1 because then there are a number of bright stars close to the sun4 ,it happened by incredible good fortune that there was a total eclipse of the sun on ,may #bi1 #aiai--the first year after the armistice4 ,two ,british expeditions photographed the stars near the sun during the eclipse1 and the results confirmed ,einstein's prediction4 ,some astronomers who remained doubtful whether sufficient precautions had been taken to insure accuracy were convinced when their own observations in a subsequent eclipse gave exactly the same result4 ,einstein's estimate of the amount of the deflection of light by gravitation is therefore now universally accepted4 ,the third experimental test is on the whole favorable to ,einstein1 though the quantities concerned are so small that it is only just possible to measure them1 and the result is therefore not decisive4 ,but successive investigations have made it more and more probable that the small effect predicted by ,einstein really occurs4 ,before explaining the effect in question1 a few preliminary explanations are necessary4 ,the spectrum of an element consists of certain lines of various shades of light1 separated by a prism1 and emitted by the element when it glows4 ,they are the same 7to a very close approximation7 whether the element is in the earth or the sun or a star4 ,each line is of some definite shade of color1 with some definite wave length4 ,longer wave lengths are towards the red end of the spectrum1 shorter ones towards the violet end4 ,when the source of light is moving towards you1 the apparent wave lengths grow shorter1 just as waves at sea come quicker when you are traveling against the wind4 ,when the source of light is moving away from you1 the apparent wave lengths grow longer1 for the same reason4 ,this enables us to know whether the stars are moving towards us or away from us4 ,if they are moving towards us1 all the lines in the spectrum of an element are moved a little toward violet2 if away from us1 toward red4 ,you may notice the analogous effect in sound any day4 ,if you are in a station and an express comes through whistling1 the note of the whistle seems much more shrill while the train is approaching you than when it has passed4 ,probably many people think the note has 8really0 changed1 but in fact the change in what you hear is only due to the fact that the train was first approaching and then receding4 ,to people in the train1 there was no change of note4 ,this is not the effect with which ,einstein is concerned4 ,the distance of the sun from the earth does not change much2 for our present purposes1 we may regard it as constant4 ,einstein deduces from his law of gravitation that any periodic process which takes place in an atom in the sun 7whose gravitation is very intense7 must1 as measured by our clocks1 take place at a slightly slower rate than it would in a similar atom on the earth4 ,the 8interval0 involved will be the same in the sun and on the earth1 but the same interval in different regions does not correspond to exactly the same time2 this is due to the 8hilly0 character of space-time which constitutes gravitation4 ,consequently any given line in the spectrum ought1 when the light comes from the sun1 to seem to us a little nearer the red end of the spectrum than if the light came from a source on the earth4 ,the effect to be expected is very small--so small that there is still some slight uncertainty as to whether it exists or not4 ,but it now seems highly probable that it exists4 ,no other measurable differences between the consequences of ,einstein's law and those of ,newton's have hitherto been discovered4 ,but the above experimental tests are quite sufficient to convince astronomers that1 where ,newton and ,einstein differ as to the motions of the heavenly bodies1 it is ,einstein's law that gives the right results4 ,even if the empirical grounds in favor of ,einstein stood alone1 they would be conclusive4 ,whether his law represents the exact truth or not1 it is certainly more nearly exact than ,newton's1 though the inaccuracies in ,newton's were all exceedingly minute4 ,but the considerations which originally led ,einstein to his law were not of this detailed kind4 ,even the consequence about the perihelion of ,mercury1 which could be verified at once from previous observations1 could only be deduced after the theory was complete1 and could not form any part of the original grounds for inventing such a theory4 ,these grounds were of a more abstract logical character4 ,i do not mean that they were not based upon observed facts1 and ,i do not mean that they were priori fantasies such as philosophers indulged in formerly4 ,what ,i mean is that they were derived from certain general characteristics of physical experience1 which showed that ,newton must be wrong and that something like ,einstein's law must be substituted4 ,the arguments in favor of the relativity of motion are1 as we saw in earlier chapters1 quite conclusive4 ,in daily life1 when we say that something moves1 we mean that it moves relatively to the earth4 ,in dealing with the motions of the planets1 we consider them as moving relatively to the sun1 or to the center of mass of the solar system4 ,when we say that the solar system itself is moving1 we mean that it is moving relatively to the stars4 ,there is no physical occurrence which can be called 8absolute motion40 ,consequently the laws of physics must be concerned with relative motions1 since these are the only kind that occur4 ,we now take the relativity of motion in conjunction with the experimental fact that the velocity of light is the same relatively to one body as relatively to another1 however the two may be moving4 ,this leads us to the relativity of distances and times4 ,this in turn shows that there is no objective physical fact which can be called 8the distance between two bodies at a given time10 since the time and the distance will both depend on the observer4 ,therefore ,newton's law of gravitation is logically untenable1 since it makes use of 8distance at a given time40 ,this shows that we cannot rest content with ,newton1 but it does not show what we are to put in his place4 ,here several considerations enter in4 ,we have in the first place what is called 8the equality of gravitational and inertial mass40 ,what this means is as follows3 ,when you apply a given force#f to a heavy body1 you do not give it as much acceleration as you would to a light body4 ,what is called the 8inertial0 mass of a body is measured by the amount of force required to produce a given acceleration4 ,at a given point of the earth's surface1 the 8mass0 is proportional to the 8weight40 ,what is measured by scales is rather the mass than the weight3 the weight is defined as the force with which the earth attracts the body4 ,now this force is greater at the poles than at the equator1 because at the equator the rotation of the earth produces a 8centrifugal force0 which partially counteracts gravitation4 ,the force of the earth's attraction is also greater on the surface of the earth than it is at a great height or at the bottom of a very deep mine4 ,none of these variations are shown by scales1 because they affect the weights used just as much as the body weighed2 but they are shown if we use a spring balance4 ,the mass does not vary in the course of these changes of weight4 #f ,although 8force0 is no longer to be regarded as one of the fundamental concepts of dynamics1 but only as a convenient way of speaking1 it can still be employed1 like 8sunrise0 and 8sunset10 provided we realize what we mean4 ,often it would require very roundabout expressions to avoid the term 8force40 ,the 8gravitational0 mass is differently defined4 ,it is capable of two meanings4 ,we may mean 7#a71 the way a body responds in a situation where gravitation has a known intensity1 for example1 on the surface of the earth1 or on the surface of the sun2 or 7#b71 the intensity of the gravitational force produced by the body1 as1 for example1 the sun produces stronger gravitational forces than the earth does4 ,newton says that the force of gravitation between two bodies is proportional to the product of their masses4 ,now let us consider the attraction of different bodies to one and the same body1 say the sun4 ,then different bodies are attracted by forces which are proportional to their masses1 and which1 therefore1 produce exactly the same acceleration in all of them4 ,thus if we mean 8gravitational mass0 in sense 7#a71 that is to say1 the way a body responds to gravitation1 we find that 8the equality of inertial and gravitational mass10 which sounds formidable1 reduces to this3 that in a given gravitational situation1 all bodies behave exactly alike4 ,as regards the surface of the earth1 this was one of the first discoveries of ,galileo4 ,aristotle thought that heavy bodies fall faster than light ones2 ,galileo showed that this is not the case1 when the resistance of the air is eliminated4 ,in a vacuum1 a feather falls as fast as a lump of lead4 ,as regards the planets1 it was ,newton who established the corresponding facts4 ,at a given distance from the sun1 a comet1 which has a very small mass1 experiences exactly the same acceleration towards the sun as a planet experiences at the same distance4 ,thus the way in which gravitation affects a body depends only upon where the body is1 and in no degree upon the nature of the body4 ,this suggests that the gravitational effect is a characteristic of the locality1 which is what ,einstein makes it4 ,as for the gravitational mass in sense 7#b71 i4e41 the intensity of the force produced by a body1 this is no longer exactly proportional to its inertial mass4 ,the question involves some rather complicated mathematics1 and ,i shall not go into it4#g #g ,see ,eddington1 ,the ,mathematical ,theory of ,relativity1 ,cambridge ,university ,press1 #b;d edition1 p4 #abh4 ,we have another indication as to what sort of thing the law of gravitation must be1 if it is to be a characteristic of a neighborhood1 as we have seen reason to suppose that it is4 ,it must be expressed in some law which is unchanged when we adopt a different kind of co-ordinates4 ,we saw that we must not1 to begin with1 regard our co-ordinates as having any physical significance3 they are merely systematic ways of naming different parts of space-time4 ,being conventional1 they cannot enter into physical laws4 ,that means to say that1 if we have expressed a law correctly in terms of one set of co-ordinates1 it must be expressed by the same formula in terms of another set of co-ordinates4 ,or1 more exactly1 it must be possible to find a formula which expresses the law1 and which is unchanged however we change the co-ordinates4 ,it is the business of the theory of tensors to deal with such formul4 ,and the theory of tensors shows that there is one formula which obviously suggests itself as being possibly the law of gravitation4 ,when this possibility is examined1 it is found to give the right results2 it is here that the empirical confirmations come in4 ,but if ,einstein's law had not been found to agree with experience1 we could not have gone back to ,newton's law4 ,we should have been compelled by logic to seek some law expressed in terms of 8tensors10 and therefore independent of our choice of co-ordinates4 ,it is impossible without mathematics to explain the theory of tensors2 the non-mathematician must be content to know that it is the technical method by which we eliminate the conventional element from our measurements and laws1 and thus arrive at physical laws which are independent of the observer's point of view4 ,of this method1 ,einstein's law of gravitation is the most splendid example4 ,,chapter ,x3 ,,mass1 ,,momentum1 ,,energy ,,and ,,action ,the pursuit of quantitative precision is as arduous as it is important4 ,physical measurements are made with extraordinary exactitude2 if they were made less carefully1 such minute discrepancies as form the experimental data for the theory of relativity could never be revealed4 ,mathematical physics1 before the coming of relativity1 used a set of conceptions which were supposed to be as precise as physical measurements1 but it has turned out that they were logically defective1 and that this defectiveness showed itself in very small deviations from expectations based upon calculation4 ,in this chapter ,i want to show how the fundamental ideas of pre-relativity physics are affected1 and what modifications they have had to undergo4 ,we have already had occasion to speak of mass4 ,for purposes of daily life1 mass is much the same as weight2 the usual measures of weight--ounces1 grams1 etc4--are really measures of mass4 ,but as soon as we begin to make accurate measurements1 we are compelled to distinguish between mass and weight4 ,two different methods of weighing are in common use1 one1 that of scales1 the other that of the spring balance4 ,when you go a journey and your luggage is weighed1 it is not put on scales1 but on a spring2 the weight depresses the spring a certain amount1 and the result is indicated by a needle on a dial4 ,the same principle is used in automatic machines for finding your weight4 ,the spring balance shows weight1 but scales show mass4 ,so long as you stay in one part of the world1 the difference does not matter2 but if you test two weighing machines of different kinds in a number of different places1 you will find1 if they are accurate1 that their results do not always agree4 ,scales will give the same result anywhere1 but a spring balance will not4 ,that is to say1 if you have a lump of lead weighing ten pounds by the scales1 it will also weigh ten pounds by scales in any other part of the world4 ,but if it weighs ten pounds by a spring balance in ,london1 it will weigh more at the ,north ,pole1 less at the equator1 less high up in an aeroplane1 and less at the bottom of a coal mine1 if it is weighed in all those places on the same spring balance4 ,the fact is that the two instruments measure quite different quantities4 ,the scales measure what may be called 7apart from refinements which will concern us presently7 8quantity of matter40 ,there is the same 8quantity of matter0 in a pound of feathers as in a pound of lead4 ,standard 8weights10 which are really standard 8masses10 will measure the amount of mass in any substance put into the opposite scales4 ,but 8weight0 is a properly due to the earth's gravitation3 ,it is the amount of the force by which the earth attracts a body4 ,this force varies from place to place4 ,in the first place1 anywhere outside the earth the attraction varies inversely as the square of the distance from the center of the earth2 it is therefore less at great heights4 ,in the second place1 when you go down a coal mine1 part of the earth is above you1 and attracts matter upwards instead of downwards1 so that the net attraction downwards is less than on the surface of the earth4 ,in the third place1 owing to the rotation of the earth1 there is what is called a 8centrifugal force10 which acts against gravitation4 ,this is greatest at the equator1 because there the rotation of the earth involves the fastest motion2 at the poles it does not exist1 because they are on the axis of rotation4 ,for all these reasons1 the force with which a given body is attracted to the earth is measureably different at different places4 ,it is this force that is measured by a spring balance2 that is why a spring balance gives different results in different places4 ,in the case of scales1 the standard 8weights0 are altered just as much as the body to be weighed1 so that the result is the same everywhere2 but the result is the 8mass10 not the 8weight40 ,a standard 8weight0 has the same mass everywhere1 but not the same 8weight02 it is in fact a unit of mass1 not of weight4 ,for theoretical purposes1 mass1 which is almost invariable for a given body1 is much more important than weight1 which varies according to circumstances4 ,mass may be regarded1 to begin with1 as 8quantity of matter02 we shall see that this view is not strictly correct1 but it will serve as a starting point for subsequent refinements4 ,for theoretical purposes1 a mass is defined as being determined by the amount of force required to produce a given acceleration3 ,the more massive a body is1 the greater will be the force required to alter its velocity by a given amount in a given time4 ,it takes a more powerful engine to make a long train attain a speed of ten miles an hour at the end of the first half-minute1 than it does to make a short train do so4 ,or we may have circumstances where the force is the same for a number of different bodies2 in that case1 if we can measure the accelerations produced in them1 we can tell the ratios of their masses3 the greater the mass1 the smaller the acceleration4 ,we may take1 in illustration of this method1 an example which is important in connection with relativity4 ,radio-active bodies emit beta-particles 7electrons7 with enormous velocities4 ,we can observe their path by making them travel through water vapor and form a cloud as they go4 ,we can at the same time subject them to known electric and magnetic forces1 and observe how much they are bent out of a straight line by these forces4 ,this makes it possible to compare their masses4 ,it is found that the faster they travel1 the greater is their mass1 as measured by the stationary observer2 the increase is greatest as applied to their mass as measured by the effect of a force in the line of motion4 ,in regard to forces at right angles to the line of motion1 there is a change of mass with velocity in the same proportion as the changes of length and time4 ,it is known otherwise that1 apart from the effect of motion1 all electrons have the same mass4 ,all this was known before the theory of relativity was invented1 but it showed that the traditional conception of mass had not quite the definiteness that had been ascribed to it4 ,mass used to be regarded as 8quantity of matter10 and supposed to be quite invariable4 ,now mass was found to be relative to the observer1 like length and time1 and to be altered by motion in exactly the same proportion4 ,however1 this could be remedied4 ,we could take the 8proper mass10 the mass as measured by an observer who shares the motion of the body4 ,this was easily inferred from the measured mass1 by taking the same proportion as in the case of lengths and times4 ,but there is a more curious fact1 and that is1 that after we have made this correction we still have not obtained a quantity which is at all times exactly the same for the same body4 ,when a body absorbs energy--for example1 by growing hotter--its 8proper mass0 increases slightly4 ,the increase is very slight1 since it is measured by dividing the increase of energy by the square of the velocity of light4 ,on the other hand1 when a body parts with energy it loses mass4 ,the most notable case of this is that four hydrogen atoms can come together to make one helium atom1 but a helium atom has rather less than four times the mass of one hydrogen atom4 ,we have thus two kinds of mass1 neither of which quite fulfils the old ideal4 ,the mass as measured by an observer who is in motion relative to the body in question is a relative quantity1 and has no physical significance as a property of the body4 ,the 8proper mass0 is a genuine property of the body1 not dependent upon the observer2 but it1 also1 is not strictly constant4 ,as we shall see shortly1 the notion of mass becomes absorbed into the notion of energy2 it represents1 so to speak1 the energy which the body expends internally1 as opposed to that which it displays to the outer world4 ,conservation of mass1 conservation of momentum1 and conservation of energy were the great principles of classical mechanics4 ,let us next consider conservation of momentum4 ,the momentum of a body in a given direction is its velocity in that direction multiplied by its mass4 ,thus a heavy body moving slowly may have the same momentum as a light body moving fast4 ,when a number of bodies interact in any way1 for instance by collisions1 or by mutual gravitation1 so long as no outside influences come in1 the total momentum of all the bodies in any direction remains unchanged4 ,this law remains true in the theory of relativity4 ,for different observers1 the mass will be different1 but so will the velocity2 these two differences neutralize each other1 and it turns out that the principle still remains true4 ,the momentum of a body is different in different directions4 ,the ordinary way of measuring it is to take the velocity in a given direction 7as measured by the observer7 and multiply it by the mass 7as measured by the observer74 ,now the velocity in a given direction is the distance traveled in that direction in unit time4 ,suppose we take instead the distance traveled in that direction while the body moves through unit 8interval40 7,in ordinary cases1 this is only a very slight change1 because1 for velocities considerably less than that of light1 interval is very nearly equal to lapse of time47 ,and suppose that instead of the mass as measured by the observer we take the proper mass4 ,these two changes increase the velocity and diminish the mass1 both in1 the same proportion4 ,thus the momentum remains the same1 but the quantities that vary according to the observer have been replaced by quantities which are fixed independently of the observer--with the exception of the distance traveled by the body in the given direction4 ,when we substitute space-time for time1 we find that the measured mass 7as opposed to the proper mass7 is a quantity of the same kind as the momentum in a given direction2 it might be called the momentum in the time direction4 ,the measured mass is obtained by multiplying the invariant mass by the time traversed in traveling through unit interval2 the momentum is obtained by multiplying the same invariant mass by the distance traversed 7in the given direction7 in traveling through unit interval4 ,from a space-time point of view1 these naturally belong together4 ,although the measured mass of a body depends upon the way the observer is moving relatively to the body1 it is none the less a very important quantity4 ,for any given observer1 the measured mass of the whole physical universe is constant4#h ,the proper mass of all the bodies in the world is not necessarily the same at one time as at another1 so that in this respect the measured mass has an advantage4 ,the conservation of measured mass is the same thing as the conservation of energy4 ,this may seem surprising1 since at first sight mass and energy are very different things4 ,but it has turned out that energy is the same thing as measured mass4 ,to explain how this comes about is not easy2 nevertheless we will make the attempt4 #h ,this is subject to the explanations given below as regards conservation of energy4 ,in popular talk1 8mass0 and 8energy0 do not mean at all the same thing4 ,we associate 8mass0 with the idea of a fat man in a chair1 very slow to move1 while 8energy0 suggests a thin person full of hustle and 8pep40 ,popular talk associates 8mass0 and 8inertia10 but its view of inertia is one-sided3 it includes slowness in beginning to move1 but not slowness in stopping1 which is equally involved4 ,all these terms have technical meanings in physics1 which are only more or less analogous to the meanings of the terms in popular talk4 ,for the present1 we are concerned with the technical meaning of 8energy40 ,throughout the latter half of the nineteenth century1 a great deal was made of the 8conservation of energy10 or the 8persistence of force10 as ,herbert ,spencer preferred to call it4 ,this principle was not easy to state in a simple way1 because of the different forms of energy2 but the essential point was that energy is never created or destroyed1 though it can be transformed from one kind into another4 ,the principle acquired its position through ,joule's discovery of 8the mechanical equivalent of heat10 which showed that there was a constant proportion between the work required to produce a given amount of heat and the work required to raise a given weight through a given height3 in fact1 the same sort of work could be utilized for either purpose according to the mechanism4 ,when heat was found to consist in motion of molecules1 it was seen to be natural that it should be analogous to other forms of energy4 ,broadly speaking1 by the help of a certain amount of theory1 all forms of energy were reduced to two1 which were called respectively 8kinetic0 and 8potential40 ,these were defined as follows3 ,the kinetic energy of a particle is half the mass multiplied by the square of the velocity4 ,the kinetic energy of a number of particles is the sum of the kinetic energies of the separate particles4 ,the potential energy is more difficult to define4 ,it represents any state of strain1 which can only be preserved by the application of force4 ,to take the easiest case3 ,if a weight is lifted to a height and kept suspended1 it has potential energy1 because1 if left to itself1 it will fall4 ,its potential energy is equal to the kinetic energy which it would acquire in falling through the same distance through which it was lifted4 ,similarly when a comet goes round the sun in a very eccentric orbit1 it moves much faster when it is near the sun than when it is far from it1 so that its kinetic energy is much greater when it is near the sun4 ,on the other hand1 its potential energy is greatest when it is farthest from the sun1 because it is then like the stone which has been lifted to a height4 ,the sum of the kinetic and potential energies of the comet is constant1 unless it suffers collisions or loses matter by forming a tail4 ,we can determine accurately the change of potential energy in passing from one position to another1 but the total amount of it is to a certain extent arbitrary1 since we can fix the zero level where we like4 ,for example1 the potential energy of our stone may be taken to be the kinetic energy it would acquire in falling to the surface of the earth1 or what it would acquire in falling down a well to the center of the earth1 or any assigned lesser distance4 ,it does not matter which we take1 so long as we stick to our decision4 ,we are concerned with a profit-and-loss account1 which is unaffected by the amount of the assets with which we start4 ,both the kinetic and the potential energies of a given set of bodies will be different for different observers4 ,in classical dynamics1 the kinetic energy differed according to the state of motion of the observer1 but only by a constant amount2 the potential energy did not differ at all4 ,consequently1 for each observer1 the total energy was constant--assuming always that the observers concerned were moving in straight lines with uniform velocities1 or1 if not1 were able to refer their motions to bodies which were so moving4 ,but in relativity dynamics the matter becomes more complicated4 ,we cannot profitably adapt the idea of potential energy to the theory of relativity1 and therefore the conservation of energy1 in a strict sense1 cannot be maintained4 ,but we obtain a property1 closely analogous to conservation1 which applies to kinetic energy alone4 ,as ,eddington puts it3 the kinetic energy is not always strictly conserved1 and the classical theory therefore introduces a supplementary quantity1 the potential energy1 so that the sum of the two is strictly conserved4 ,the relativity treatment1 on the other hand1 discovers another formula1 analogous to the one expressing conservation1 which holds always for the kinetic energy4 8,the relativity treatment adheres to the physical quantity and modifies the law2 the classical treatment adheres to the law and modifies the physical quantity40 ,the new formula1 he continues1 may be spoken of 8as the law of conservation of energy and momentum1 because1 though it is not formally a law of conservation1 it expresses exactly the phenomena which classical mechanics attributes to conservation40#i ,it is only in this modified and less rigorous sense that the conservation of energy remains true4 #i ,mathematical ,theory of ,relativity1 p4 #ace4 ,what is meant by 8conservation0 in practice is not exactly what it means in theory4 ,in theory we say that a quantity is conserved when the amount of it in the world is the same at any one time as at any other4 ,but in practice we cannot survey the whole world1 so we have to mean something more manageable4 ,we mean that1 taking any given region1 if the amount of the quantity in the region has changed1 it is because some of the quantity has passed across the boundary of the region4 ,if there were no births and deaths1 population would be conserved2 in that case the population of a country could only change by emigration or immigration1 that is to say1 by passing across the boundaries4 ,we might be unable to take an accurate census of ,china or ,central ,africa1 and1 therefore1 we might not be able to ascertain the total population of the world4 ,but we should be justified in assuming it to be constant if1 wherever statistics were possible1 the population never changed except through people crossing the frontiers4 ,in fact1 of course1 population is not conserved4 ,a physiologist of my acquaintance once put four mice into a thermos4 ,some hours later1 when he went to take them out1 there were eleven of them4 ,but mass is not subject to these fluctuations3 the mass of the eleven mice at the end of the time was no greater than the mass of the four at the beginning4 ,this brings us back to the problem for the sake of which we have been discussing energy4 ,we stated that1 in relativity theory1 measured mass and energy are regarded as the same thing1 and we undertook to explain why4 ,it is now time to embark upon this explanation4 ,but here1 as at the end of ,chapter ,,vi1 the totally unmathematical reader will do well to skip1 and begin again at the following paragraph4 ,let us take the velocity of light as the unit of velocity2 this is always convenient in relativity theory4 ,let m be the proper mass of a particle1 v its velocity relative to the observer4 ,then its measured mass will be m -------------------- 7#a - v7 while its kinetic energy1 according to the usual formula1 will be mv ,as we saw before1 energy only occurs in a profit-and-loss account1 so that we can add any constant quantity to it that we like4 ,we may therefore take the energy to be m 7mv74 ,now if v is a small fraction of the velocity of light1 m mv is almost exactly equal to m ------------------ 7#a - v74 ,consequently1 for velocities such as large bodies have1 the energy and the measured mass turn out to be indistinguishable within the limits of accuracy attainable4 ,in fact1 it is better to alter our definition of energy1 and take it to be m -------------------- 7#a - v71 because this is the quantity for which the law analogous to conservation holds4 ,and when the velocity is very great1 it gives a better measure of energy than the traditional formula4 ,the traditional formula must therefore be regarded as an approximation1 of which the new formula gives the exact version4 ,in this way1 energy and measured mass become identified4 ,i come now to the notion of 8action10 which is less familiar to the general public than energy1 but has become more important in relativity physics1 as well as in the theory of quanta4#aj 7,the quantum is a small amount of action47 ,the word 8action0 is used to denote energy multiplied by time4 ,that is to say1 if there is one unit of energy in a system1 it will exert one unit of action in a second1 #ajj units of action in #ajj seconds1 and so on2 a system which has #ajj units of energy will exert #ajj units of action in a second1 and #aj1jjj in #ajj seconds1 and so on4 8,action0 is thus1 in a loose sense1 a measure of how much has been accomplished3 it is increased both by displaying more energy and by working for a longer time4 ,since energy is the same thing as measured mass1 we may also take action to be measured mass multiplied by time4 ,in classical mechanics1 the 8density0 of matter in any region is the mass divided by the volume2 that is to say1 if you know the density in a small region1 you discover the total amount of matter by multiplying the density by the volume of the small region4 ,in relativity mechanics1 we always want to substitute space-time for space2 therefore a 8region0 must no longer be taken to be merely a volume1 but a volume lasting for a time2 a small region will be a small volume lasting for a small time4 ,it follows that1 given the density1 a small region in the new sense contains1 not a small mass merely1 but a small mass multiplied by a small time1 that is to say1 a small amount of 8action40 ,this explains why it is to be expected that 8action0 will prove of fundamental importance in relativity mechanics4 ,and so in fact it is4 #aj ,on this subject1 see the present author's ,a4,b4,c4 of ,atoms1 chaps4 ,,vi and ,,xiii4 ,all the laws of dynamics have been put together into one principle1 called 8,the ,principle of ,least ,action40 ,this states that1 in passing from one state to another1 a body chooses a route involving less action than any slightly different route--again a law of cosmic laziness4 ,the principle is subject to certain limitations1 which have been pointed out by ,eddington1#aa but it remains one of the most comprehensive ways of stating the purely formal part of mechanics4 ,the fact that the quantum is a unit of action seems to show that action is also fundamental in the empirical structure of the world4 ,but at present there is no bridge connecting the quantum with the theory of relativity4 #aa ,op4 cit4 #fj4 ,,chapter ,,xi3 ,,is ,,the ,,universe ,,finite8 ,we have been dealing hitherto with matters that must be regarded as acquired scientific results--not that they will never be found to need improvement1 but that further progress must be built upon them1 as ,einstein is built upon ,newton4 ,science does not aim at establishing immutable truths and eternal dogmas3 its aim is to approach truth by successive approximations1 without claiming that at any stage final and complete accuracy has been achieved4 ,there is a difference1 however1 between results which are pretty certainly in the line of advance1 and speculations which may or may not prove to be well founded4 ,some very interesting speculations are connected with the theory of relativity1 and we shall consider certain of them4 ,but it must not be supposed that we are dealing with theories having the same solidity as those with which we have been concerned hitherto4 ,one of the most fascinating of the speculations to which ,i have been alluding is the suggestion that the universe may be of finite extent4 ,two somewhat different finite universes have been constructed1 one by ,einstein1 the other by ,de ,sitter4 ,before considering their differences1 we will discuss what they have in common4 ,there are1 to begin with1 certain reasons for thinking that the total amount of matter in the universe is limited4 ,if this were not the case1 the gravitational effects of enormously distant matter would make the kind of world in which we live impossible4 ,we must therefore suppose that there is some definite number of electrons and protons in the world3 theoretically1 a complete census would be possible4 ,these are all contained within a certain finite region2 whatever space lies outside that region is1 so to speak1 waste1 like unfurnished rooms in a house too large for its inhabitants4 ,this seems futile1 but in former days no one knew of any alternative possibility4 ,it was obviously impossible to conceive of an edge to space1 and therefore1 it was thought1 space must be infinite4 ,non-,euclidean geometry1 however1 showed other possibilities4 ,the surface of a sphere has no boundary1 yet it is not infinite4 ,in traveling round the earth1 we never reach 8the edge of the world10 and yet the earth is not infinite4 ,the surface of the earth is contained in three-dimensional space1 but there is no reason in logic why three-dimensional space should not be constructed on an analogous plan4 ,what we imagine to be straight lines going on for ever will then be like great circles on a sphere3 they will ultimately return to their starting point4 ,there will not be in the universe anything straighter than these great circles2 the ,euclidean straight line may remain as a beautiful dream1 but not as a possibility in the actual world4 ,in particular1 light rays in empty space will travel in what are really great circles4 ,if we could make measurements with sufficient accuracy1 we should be able to infer this state of affairs even from a small part of space1 because the sum of the angles of a triangle would always be greater than two right angles1 and the excess would be proportional to the size of the triangle4 ,the suggestion we have to consider is the suggestion that our universe may be spherical in this sense4 ,the reader must not confuse this suggestion with the non-,euclidean character of space upon which the new law of gravitation depends4 ,the latter is concerned with small regions such as the solar system4 ,the departures from flatness which it notices are like hills and valleys on the surface of the earth1 local irregularities1 not characteristics of the whole4 ,we are now concerned with the possible curvature of the universe as a whole1 not with the occasional ups and downs due to the sun and the stars4 ,it is suggested that on the average1 and in regions remote from matter1 the universe is not quite flat1 but has a slight curvature1 analogous1 in three dimensions1 to the curvature of a sphere in two dimensions4 ,it is important to realize1 in the first place1 that there is not the slightest reason priori why this should not be the case4 ,people unaccustomed to non-,euclidean geometry may feel that1 even if such a thing be logically possible1 the world simply cannot be so odd as all that4 ,we all have a tendency to think that the world must conform to our prejudices4 ,the opposite view involves some effort of thought1 and most people would die sooner than think--in fact1 they do so4 ,but the fact that a spherical universe seems odd to people who have been brought up on ,euclidean prejudices is no evidence that it is impossible4 ,there is no law of nature to the effect that what is taught at school must be true4 ,we cannot therefore dismiss the hypothesis of a spherical universe as in any degree less worthy of examination than any other4 ,we have to ask ourselves the same two questions as we should in any other case1 namely3 7#a7 ,are the facts consistent with this hypothesis8 7#b7 ,is this hypothesis the only one with which the facts are consistent8 ,with regard to the first question1 the answer is undoubtedly in the affirmative4 ,all the known facts are perfectly consistent with the hypothesis of a spherical universe4 ,a very slight modification of the law of gravitation--a modification suggested by ,einstein himself--leads to a spherical space1 without producing any measurable differences in a small region such as the solar system4 ,the known stars are all within a certain distance from us4 ,there is nothing whatever in the stellar universe as we know it to show that space must be infinite4 ,there can therefore be no doubt whatever that1 so far as our present knowledge goes1 the hypothesis of a finite universe may be true4 ,but when we ask whether the hypothesis of a finite universe must be true1 the answer is different4 ,it is obvious1 on general grounds1 that we cannot1 from what we know1 draw conclusive inferences as to the totality of things4 ,a very slight change in the ,newtonian formula for gravitation would prevent masses beyond the limits of the visible universe from having appreciable effects if they existed1 and would therefore destroy our reason for supposing that they do not exist4 ,all arguments as to regions which are too distant to be observed depend upon extending to them the laws which hold in our part of the world1 and upon assuming that there is not1 in these laws1 some inaccuracy which is inappreciable for observable distances1 but fatal to inferences in which very much greater distances are involved4 ,we cannot1 therefore1 say that the universe must be finite4 ,we can say that it may be1 and we can even say a little more than this4 ,we can say that a finite universe fits in better with the laws that hold in the part we know1 and that awkward adjustments of the laws have to be made in order to allow the universe to be infinite4 ,from the point of view of choosing the best framework into which to fit what we know--best1 ,i mean1 from a logico-sthetic point of view--there is no doubt that the hypothesis of a finite universe is preferable4 ,this1 ,i think1 is the extent of what can be said in its favor4 ,let us now see what the two finite universes are like4 ,the difference between them is that in ,einstein's world it is only space that is queer1 whereas in ,de ,sitter's time is queer too4 ,consequently ,einstein's world is less puzzling1 and we will describe it first4 ,in ,einstein's world1 light travels round the whole universe in a time which is supposed to be something like a thousand million years4 ,the odd thing is that all the rays of light which start 7say7 from the sun will meet again1 after their enormous journey1 in the place where the sun was when they started4 ,the case is exactly analogous to that of a number of travelers who set out from ,london to go round the world in great circles1 all traveling at the same rate in different aeroplanes4 ,one starts due north1 passes the ,north ,pole1 then the ,south ,pole1 and finally comes home4 ,another starts due south1 reaches the ,south ,pole first and then the ,north ,pole4 ,another starts westward1 but he must not continue to travel due west1 because then he would not be traveling on a great circle4 ,another starts eastward1 and so on4 ,they all meet in the antipodes of ,london1 and then they all meet again in ,london4 ,now if instead of aeronauts going round the earth you take rays of light going round the universe1 the same sort of thing happens3 they all meet first at the antipodes of their starting point1 and then meet again at their starting point4 ,that means to say that a person who is near the antipodes of the place where the sun was about five hundred million years ago will see what is apparently a body as bright as the sun then was 7except for the small amount of light that has been stopped on the way by opaque bodies71 and having the same shape and size4 ,and a person who is near where the sun was a thousand million years ago will see what is apparently a body just like what the sun was a thousand million years ago4 ,and the same applies to the antipodes of the sun fifteen hundred million years ago1 and to the place of the sun two thousand million years ago1 and so on4 ,this series only ends when it carries us back to a time before the sun existed4 ,but all these suns are only ghosts2 that is to say1 you could pass through them without experiencing resistance1 and they do not exert gravitation4 ,they are1 in fact1 like images in a mirror3 they exist only for the sense of sight1 not for any other sense4 ,it is rather disturbing to reflect that1 if this theory is true1 any number of the objects we see in the heavens may be merely ghosts4 ,they are like ghosts in their habit of revisiting the scenes of their past life4 ,suppose a star had exploded at a certain place1 as stars sometimes will4 ,every thousand million years its ghost would return to the scene of the disaster and explode again in the same place4 ,there is1 however1 considerable doubt whether rays of light could perform the journey with sufficient accuracy to produce a clear image4 ,some would be stopped by matter on the way1 some would be turned out of the straight course by passing near heavy bodies1 as in the eclipse observations described in ,chapter ,,ix1 and for one reason or another their return would not be punctual and exact4 ,there are various reasons for doubting whether ,einstein's universe can be quite right4#ab ,some of these are rather complicated4 ,but there is one objection which is easily appreciated3 in ,einstein's theory1 absolute space and time re-enter by another door4 ,the ghostly sun is formed in the 8place0 where it was a thousand million years ago4 ,both the 8place0 and the period of time are in a sense absolute4 ,we saw as early as ,chapter ,i that 8place0 is a vague and popular notion1 incapable of scientific precision4 ,it seems hardly worth while to go through such a vast intellectual labor if the errors we set out to correct are to reappear at the end4 #ab ,see ,eddington1 ,space1 ,time and ,gravitation1 p4 #afb;ff4 ,de ,sitter's world is even odder than ,einstein's1 because time goes mad as well as space4 ,i despair of explaining1 in non-mathematical language1 the particular form of lunacy with which time is afflicted1 but some of its manifestations can be described4 ,an observer in this world1 if he observes a number of clocks1 each of which is perfectly accurate from its own point of view1 will think that distant clocks are going slow as compared with those in his neighborhood4 ,they will seem to go slower and slower1 until1 at a distance of one quarter of the circumference of the universe1 they will seem to have stopped altogether4 ,that region will seem to our observer a sort of lotus land1 where nothing is ever done4 ,he will not be able to have any cognizance of things farther off1 because no light waves can get across the boundary4 ,not that there is any real boundary3 the people who live in what our observer takes to be lotus land live just as bustling a life as he does1 but get the impression that he is eternally standing still4 ,as a matter of fact1 you would never become aware of the lotus land1 because it would take an infinite time for light to travel from it to you4 ,you could become aware of places just short of it1 but it would remain itself always just beyond your ken4 ,there will not be the ghostly suns of ,einstein's world1 because light cannot travel so far4 ,one of the oddest things about this state of affairs is that empirical evidence for or against it is possible1 and that there is actually some slight evidence in its favor4 ,if all 8clocks0 are slowed down at a great distance from the observer1 this will apply to the periodic motions of atoms1 and therefore to the light which they emit4 ,consequently all rays of light emitted by distant objects ought1 when they reach us1 to look rather more red or less violet than when they started4 ,this can be tested by the spectroscope4 ,we can compare a known line1 as it appears in the spectrum of a spiral nebula1 with the same line as it appears in a terrestrial laboratory4 ,we find1 as a matter of fact1 that in a large majority of spiral nebul there is a considerable displacement of spectral lines towards the red4 ,the spiral nebul are the most distant objects we can see3 ,eddington states that their distances 8may perhaps be of the order of a million light-years40 7,a light-year is the distance light travels in a year47 ,the usual interpretation of a shifting of spectral lines towards the red is that it is a 8,doppler effect10 due to the fact that the source of light is moving away from us4 ,but one would expect to find the nebul just as often moving towards us as moving away from us1 if nothing operated but the law of chances4 ,if the world is such as ,de ,sitter says it is1 the spectral lines of the spiral nebul will be displaced towards the red owing to the slowing down of distant clocks1 even if in fact they are not moving away from us4 ,this1 for what it is worth1 is an argument in favor of ,de ,sitter4 ,the same facts afford another argument in favor of ,de ,sitter1 for another reason4 ,if1 at a given moment1 a body is at rest relatively to the observer1 and at a distance from him1 it will 7in the absence of counteracting causes7 not remain at rest from his point of view1 but will begin to move away from him1 and will continue to move away faster and faster2 the further it is from him1 the more its retreat will be accelerated4 ,for bodies which are not too distant from each other1 gravitation may overcome this tendency2 but as this tendency increases with the distance1 while gravitation diminishes1 we should expect to find very distant bodies receding from us if ,de ,sitter's theory is right4 ,thus we have two reasons for the displacement of spectral lines in spiral nebul3 one1 the slowing down of time2 the other1 the movement away from us which we should expect at distances too great for gravitation to be sensible4 ,however1 it cannot be said that the argument1 on either ground1 is very strong4 ,eddington gives a list of forty-one spiral nebul1 of which five have their spectral lines shifted towards the violet1 not towards the red4 ,thus the material is neither very copious nor quite harmonious4 ,einstein's and ,de ,sitter's hypotheses do not exhaust the possibilities of a finite world3 they are merely the two simplest forms of such a world4 ,there are arguments against each1 and it hardly seems probable that either is quite true4 ,but it does seem probable that something more or less analogous is true4 ,if the universe is finite1 it is theoretically conceivable that there should be a complete inventory of it4 ,we may be coming to the end of what physics can do in the way of stretching the imagination and systematizing the world4 ,the period since ,galileo has been essentially the period of physics1 as the age of the ,greeks was the period of geometry4 ,it may be that physics will lose its attractions through success3 if the fundamental laws of physics come to be fully known1 adventurous and inquiring intellects will turn to other fields4 ,this may alter profoundly the whole texture of human life1 since our present absorption in machinery and industrialism is the reflection in the practical world of the theorist's interest in physical laws4 ,but such speculations are even more rash than those of ,de ,sitter1 and ,i do not wish to lay any stress upon them4 ,,chapter ,,xii3 ,,conventions ,,and ,,natural ,,laws ,one of the most difficult matters in all controversy is to distinguish disputes about words from disputes about facts3 it ought not to be difficult1 but in practice it is4 ,this is quite as true in physics as in other subjects4 ,in the seventeenth century there was a terrific debate as to what 8force0 is2 to us now1 it was obviously a debate as to how the word 8force0 should be defined1 but at the time it was thought to be much more4 ,one of the purposes of the method of tensors1 which is employed in the mathematics of relativity1 is to eliminate what is purely verbal 7in an extended sense7 in physical laws4 ,it is of course obvious that what depends on the choice of co-ordinates is 8verbal0 in the sense concerned4 ,a man punting walks along the boat1 but keeps a constant position with reference to the river bed so long as he does not pick up his pole4 ,the ,lilliputians might debate endlessly whether he is walking or standing still3 the debate would be as to words1 not as to facts4 ,if we choose co-ordinates fixed relatively to the boat1 he is walking2 if we choose co-ordinates fixed relatively to the river bed1 he is standing still4 ,we want to express physical laws in such a way that it shall be obvious when we are expressing the same law by reference to two different systems of co-ordinates1 so that we shall not be misled into supposing we have different laws when we only have one law in different words4 ,this is accomplished by the method of tensors4 ,some laws which seem plausible in one language cannot be translated into another2 these are impossible as laws of nature4 ,the laws that can be translated into any co-ordinate language have certain characteristics3 this is a substantial help in looking for such laws of nature as the theory of relativity can admit to be possible4 ,combined with what we know of the actual motions of bodies1 it enables us to decide what must be the correct expression of the law of gravitation3 logic and experience combine in equal proportions in obtaining this expression4 ,but the problem of arriving at genuine laws of nature is not to be solved by the method of tensors alone2 a good1 deal of careful thought is wanted in addition4 ,some of this has been done1 especially by ,eddington2 much remains to be done4 ,to take a simple illustration3 ,suppose1 as in the hypothesis of the ,fitzgerald contraction1 that lengths in one direction were shorter than in another4 ,let us assume that a foot rule pointing north is only half as long as the same foot rule pointing east1 and that this is equally true of all other bodies4 ,does such an hypothesis have any meaning8 ,if you have a fishing rod fifteen feet long when it is pointing west1 and you then turn it to the north1 it will still measure fifteen feet1 because your foot rule will have shrunk too4 ,it won't 8look0 any shorter1 because your eye will have been affected in the same way4 ,if you are to find out the change1 it cannot be by ordinary measurement2 it must be by some such method as the ,michelson-,morley experiment1 in which the velocity of light is used to measure lengths4 ,then you still have to decide whether it is simpler to suppose a change of length or a change in the velocity of light4 ,the experimental fact would be that light takes longer to traverse what your foot rule declares to be a given distance in one direction than in another--or1 as in the ,michelson-,morley experiment1 that it ought to take longer but doesn't4 ,you can adjust your measures to such a fact in various ways2 in any way you choose to adopt1 there will be an element of convention4 ,this element of convention survives in the laws that you arrive at after you have made your decision as to measures1 and often it takes subtle and elusive forms4 ,to eliminate the element of convention is1 in fact1 extraordinarily difficult2 the more the subject is studied1 the greater the difficulty is seen to be4 ,a more important example is the question of the size and shape of the electron4 ,we find experimentally that all electrons are the same size1 and that they are symmetrical in all directions4 ,how far is this a genuine fact ascertained by experiment1 and how far is it a result of our conventions of measurement8 ,we have here a number of different comparisons to make3 7#a7 between different directions in regard to one electron at one time2 7#b7 in regard to one electron at different times2 7#c7 in regard to two electrons at the same time4 ,we can then arrive at the comparison of two electrons at different times1 by combining 7#b7 and 7#c74 ,we may dismiss any hypothesis which would affect all electrons equally2 for example1 it would be useless to suppose that in one region of space-time they were all larger than in another4 ,such a change would affect our measuring appliances just as much as the things measured1 and would therefore produce no discoverable phenomena4 ,this is as much as to say that it would be no change at all4 ,but the fact that two electrons have the same mass1 for instance1 cannot be regarded as purely conventional4 ,given sufficient minuteness and accuracy1 we could compare the effects of two different electrons upon a third2 if they were equal under like circumstances1 we should be able to infer equality in a not purely conventional sense4 ,the question of the symmetry of the forces exerted by an electron--i4e41 that these forces depend only upon the distance from the electron1 and not upon the direction--is more complicated4 ,eddington finally comes to the conclusion that this1 too1 is a matter of convention4 ,the argument is difficult and ,i have not fully understood it2 but ,i feel some hesitation in accepting it as valid4 ,eddington describes the process concerned in the more advanced portions of the theory of relativity as 8world-building40 ,the structure to be built is the physical world as we know it2 the economical architect tries to construct it with the smallest possible amount of material4 ,this is a question for logic and mathematics4 ,the greater our technical skill in these two subjects1 the more real building we shall do1 and the less we shall be content with mere heaps of stones4 ,but before we can use in our building the stones that nature provides1 we have to hew them into the right shapes3 this is all part of the process of budding4 ,in order that this may be possible1 the raw material must have some structure 7which we may conceive as analogous to the grain in timber71 but almost any structure will do4 ,by successive mathematical refinements1 we whittle away our initial requirements until they amount to very little4 ,given this necessary minimum of structure in the raw material1 we find that we can construct from it a mathematical expression which will have the properties that are needed for describing the world we perceive--in particular1 the properties of conservation which are characteristic of momentum and energy 7or mass74 ,our raw material consisted merely of events2 but when we find that we can build out of it something which1 as measured1 will seem to be never created or destroyed1 it seems not surprising that we should come to believe in 8bodies40 ,these are really mere mathematical constructions out of events1 but owing to their permanence they are practically important1 and our senses 7which were presumably developed by biological needs7 are adapted for noticing them1 rather than the crude continuum of events which is theoretically more fundamental4 ,from this point of view1 it is astonishing how little of the real world is revealed by physical science3 our knowledge is limited1 not only by the conventional element1 but also by the selectiveness of our perceptual apparatus4 ,we assume that there is an 8interval0 between two events1 in the sense explained in ,chapter ,,vii1 but we no longer assume that we can unambiguously compare the length of an interval in one region with the length of an interval in another4 ,it is assumed by ,weyl1 who introduced this limitation1 that we can compare a number of small intervals which all start from the same point2 also that1 in a very small journey1 our measuring rod will not alter its length much1 so that there will only be a small error if we compare lengths in neighboring places by the usual methods4 ,weyl found that1 by diminishing our assumptions as to interval in this way1 it was possible to bring electromagnetism and gravitation into one system4 ,the mathematics of ,weyl's theory is complicated1 and ,i shall not attempt to explain it4 ,for the present1 ,i am concerned with a different consequence of his theory4 ,if lengths in different regions cannot be compared directly1 there is an element of convention in the indirect comparisons which we actually make4 ,this element will be at first unrecognized1 but will be such as to simplify to the utmost the expression of the laws of nature4 ,in particular1 conditions of symmetry may be entirely created by conventions as to measurement1 and there is no reason to suppose that they represent any property of the real world4 ,the law of gravitation itself1 according to ,eddington1 may be regarded as expressing conventions of measurement4 8,the conventions of measurement10 he says1 8introduce an isotropy#ac and homogeneity into measured space which need not originally have any counterpart in the relation-structure which is being surveyed4 ,this isotropy and homogeneity is exactly expressed by ,einstein's law of gravitation40#ad #ac 8,isotropy0 means being similar in all directions--e4g41 that a foot rule is as long when it points north as when it points east4 #ad ,mathematical ,theory of ,relativity1 p4 #bch4 ,the limitations of knowledge introduced by the selectiveness of our perceptual apparatus may be illustrated by the indestructibility of matter4 ,this has been gradually discovered by experiment1 and seemed a well-founded empirical law of nature4 ,now it turns out that1 from our original space-time continuum1 we can construct a mathematical expression which will have properties causing it to appear indestructible4 ,the statement that matter is indestructible then ceases to be a proposition of physics1 and becomes instead a proposition of linguistics and psychology4 ,as a proposition of linguistics3 8,matter0 is the name of the mathematical expression in question4 ,as a proposition of psychology3 ,our senses are such that we notice what is roughly the mathematical expression in question1 and we are led nearer and nearer to it as we refine upon our crude perceptions by scientific observation4 ,this is much less than physicists used to think they knew about matter4 ,the reader may say3 ,what then is left of physics8 ,what do we really know about the world of matter8 ,here we may distinguish three departments of physics4 ,there is first what is included within the theory of relativity1 generalized as widely as possible4 ,next1 there are laws which cannot be brought within the scope of relativity4 ,thirdly1 there is what may be called geography4 ,let us consider each of these in turn4 ,the theory of relativity1 apart from convention1 tells us that the events in the universe have a four-dimensional order1 and that1 between any two events which are near together in this order1 there is a relation called 8interval10 which is capable of being measured if suitable precautions are taken4 ,we make also an assumption as to what happens when a little measuring rod is carried round a closed circuit in a certain manner2 the consequences of this assumption are such as to make it highly probable that it is true4 ,beyond this1 there is little in the theory of relativity that can be regarded as physical laws4 ,there is a great deal of mathematics1 showing that certain mathematically-constructed quantities must behave like the things we perceive2 and there is a suggestion of a bridge between psychology and physics in the theory that these mathematically-constructed quantities are what our senses are adapted for perceiving4 ,but neither of these things is physics in the strict sense4 ,the part of physics which cannot1 at present1 be brought within the scope of relativity is large and important4 ,there is nothing in relativity to show why there should be electrons and protons2 relativity cannot give any reason why matter should exist in little lumps4 ,with this goes the whole theory of the structure of the atom4 ,the theory of quanta also is quite outside the scope of relativity4 ,relativity is1 in a sense1 the most extreme application of what may be called next-to-next methods4 ,gravitation is no longer regarded as due to the effect of the sun upon a planet1 but as expressing characteristics of the region in which the planet happens to be4 ,distance1 which used to be thought to have a definite meaning however far apart two points might be1 is now only definite for neighboring points4 ,the distance between widely separated places depends upon the route chosen4 ,we may1 it is true1 define the distance as the geodesic distance1 but that can only be estimated by adding up little bits1 that is to say1 by the method we use in estimating the length of a curve4 ,what applies to distance applies equally to the straight line4 ,there is nothing in the actual world having exactly the properties that straight lines were supposed to have2 the nearest approach is the track of a light ray4 ,straight lines have to be replaced by geodesics1 which are defined by what they do at each point1 not all at once1 like ,euclidean straight lines4 ,measurement1 in ,weyl's theory1 suffers the same fate4 ,we can only use a measuring rod to give lengths in one place3 when we move it to another region1 there is no knowing how it will alter4 ,we do assume1 however1 that1 if it alters1 it alters bit by bit1 gradually1 continuously1 and not by sudden jumps4 ,perhaps this assumption is unjustified4 ,it belongs to the general outlook of relativity1 which is that of continuity4 ,no doubt it is owing to this outlook that relativity is unable to account for the discontinuities in physics1 such as quanta1 electrons and protons4 ,perhaps relativity will conquer these domains when it learns to dispense with the assumption of continuity4 ,finally we come to geography1 in which ,i include history4 ,the separation of history from geography rests upon the separation of time from space2 when we amalgamate the two in space-time1 we need one word to describe the combination of geography and history4 ,for the sake of simplicity1 ,i shall use the one word geography in this extended sense4 ,geography1 in this sense1 includes everything that1 as a matter of crude fact1 distinguishes one part of space-time from another4 ,one part is occupied by the sun1 one by the earth2 the intermediate regions contain light waves1 but no matter 7apart from a very little here and there74 ,there is a certain degree of theoretical connection between different geographical facts2 to establish this is the purpose of physical laws4 ,it is thought that a sufficient knowledge of the geographical facts of the solar system throughout any finite time1 however short1 would enable an ideally competent physicist to predict the future of the solar system so long as it remained remote from other stars4 ,we are already in a position to calculate the large facts about the solar system backwards and forwards for vast periods of time4 ,but in all such calculations we need a basis of crude fact4 ,the facts are interconnected1 but facts can only be inferred from other facts1 not from general laws alone4 ,thus the facts of geography have a certain independent status in physics4 ,no amount of physical laws will enable us to infer a physical fact unless we know other facts as data for our inference4 ,and here when ,i speak of 8facts0 ,i am thinking of particular facts of geography1 in the extended sense in which ,i am using the term4 ,in the theory of relativity1 we are concerned with structure1 not with the material of which the structure is composed4 ,in geography1 on the other hand1 the material is relevant4 ,if there is to be any difference between one place and another1 there must either be differences between the material in one place and that in another1 or places where there is material and places where there is none4 ,the former of these alternatives seems the more satisfactory4 ,we might try to say3 ,there are electrons and protons1 and the rest is empty4 ,but in the 8empty0 regions there are light waves1 so that we cannot say nothing happens in them4 ,some people maintain that the light waves take place in the ether1 others are content to say simply that they take place2 but in any case events are occurring where there are light waves4 ,that is all that we can really say for the places where there is matter1 since matter has turned out to be a mathematical construction built out of events4 ,we may say1 therefore1 that there are events everywhere in space-time1 but they must be of a somewhat different kind according as we are dealing with a region where there is an electron or proton or with the sort of region we should ordinarily call empty4 ,but as to the intrinsic nature of these events we can know nothing1 except when they happen to be events in our own lives4 ,our own perceptions and feelings must be part of the crude material of events which physics arranges into a pattern--or rather1 which physics finds to be arranged in a pattern4 ,as regards events which do not form part of our own lives1 physics tells us the pattern of them1 but is quite unable to tell us what they are like in themselves4 ,nor does it seem possible that this should be discovered by any other method4 ,,chapter ,,xiii3 ,,the ,,abolition ,,of 8,,force0 ,in the ,newtonian system1 bodies under the action of no forces move in straight lines with uniform velocity2 when bodies do not move in this way1 their change of motion is ascribed to a 8force40 ,some forces seem intelligible to our imagination3 those exerted by a rope or string1 by bodies colliding1 or by any kind of obvious pushing or pulling4 ,as explained in an earlier chapter1 our apparent imaginative understanding of these processes is quite fallacious2 all that it really means is that past experience enables us to foresee more or less what is going to happen without the need of mathematical calculations4 ,but the 8forces0 involved in gravitation and in the less familiar forms of electrical action do not seem in this way 8natural0 to our imagination4 ,it seems odd that the earth can float in the void3 the natural thing to suppose is that it must fall4 ,that is why it has to be supported on an elephant1 and the elephant on a tortoise1 according to some early speculators4 ,the ,newtonian theory1 in addition to action at a distance1 introduced two other imaginative novelties4 ,the first was1 that gravitation is not always and essentially directed what we should call 8downwards10 i4e41 towards the center of the earth4 ,the second was1 that a body going round and round in a circle with uniform velocity is not 8moving uniformly0 in the sense in which that phrase is applied to the motion of bodies under no forces1 but is perpetually being turned out of the straight course towards the center of the circle1 which requires a force pulling it in that direction4 ,hence ,newton arrived at the view that the planets are attracted to the sun by a force1 which is called gravitation4 ,this whole point of view1 as we have seen1 is superseded by relativity4 ,there are no longer such things as 8straight lines0 in the old geometrical sense4 ,there are 8straightest lines10 or geodesics1 but these involve time as well as space4 ,a light ray passing through the solar system does not describe the same orbit as a comet1 from a geometrical point of view2 nevertheless each moves in a geodesic4 ,the whole imaginative picture is changed4 ,a poet might say that water runs down hill because it is attracted to the sea1 but a physicist or an ordinary mortal would say that it moves as it does1 at each point1 because of the nature of the ground at that point1 without regard to what lies ahead of it4 ,just as the sea does not cause the water to run towards it1 so the sun does not cause the planets to move round it4 ,the planets move round the sun because that is the easiest thing to do--in the technical sense of 8least action40 ,it is the easiest thing to do because of the nature of the region in which they are1 not because of an influence emanating from the sun4 ,the supposed necessity of attributing gravitation to a 8force0 attracting the planets towards the sun has arisen from the determination to preserve ,euclidean geometry at all costs4 ,if we suppose that our space is ,euclidean1 when in fact it is not1 we shall have to call in physics to rectify the errors of our geometry4 ,we shall find bodies not moving in what we insist upon regarding as straight lines1 and we shall demand a cause for this behavior4 ,eddington has stated this matter with admirable lucidity4 ,he supposes a physicist who has assumed the formula for interval which is used in the special theory of relativity--a formula which still supposes that the observer's space is ,euclidean4 ,he continues3 ,since intervals can be compared by experimental methods1 he ought soon to discover that his 7formula for the interval7 cannot be reconciled with observational results1 and so realize his mistake4 ,but the mind does not so readily get rid of an obsession4 ,it is more likely that our observer will continue in his opinion1 and attribute the discrepancy of the observations to some influence which is present and affects the behavior of his test-bodies4 ,he will1 so to speak1 introduce a supernatural agency which he can blame for the consequences of his mistake4444 ,the name given to any agency which causes deviation from uniform motion in a straight line is force according to the ,newtonian definition of force4 ,hence the agency invoked through our observer's mistake is described as a 8field of force40444 ,a field of force represents the discrepancy between the natural geometry of a co-ordinate system and the abstract geometry arbitrarily ascribed to it4#ae #ae ,mathematical ,theory of ,relativity1 pp4 #cg-ch4 ,italics in the original4 ,if people were to learn to conceive the world in the new way1 without the old notion of 8force10 it would alter not only their physical imagination1 but probably also their morals and politics4 ,the latter effect would be quite illogical1 but is none the less probable on that account4 ,in ,newton's theory of the solar system1 the sun seems like a monarch whose behests the planets have to obey4 ,in ,einstein's world there is more individualism and less government than in ,newton's4 ,there is also far less hustle3 we have seen that laziness is the fundamental law of ,einstein's universe4 ,the word 8dynamic0 has come to mean1 in newspaper language1 8energetic and forceful02 but if it meant 8illustrating the principles of dynamics10 it ought to be applied to the people in hot climates who sit under banana trees waiting for the fruit to drop into their mouths4 ,i hope that journalists1 in future1 when they speak of a 8dynamic personality10 will mean a person who does what is least trouble at the moment1 without thinking of remote consequences4 ,if ,i can contribute to this result1 ,i shall not have written in vain4 ,it has been customary for people to draw arguments from the laws of nature as to what we ought to do4 ,such arguments seem to me a mistake3 to imitate nature may be merely slavish4 ,but if nature1 as portrayed by ,einstein1 is to be our model1 it would seem that the anarchists will have the best of the argument4 ,the physical universe is orderly1 not because there is a central government1 but because every body minds its own business4 ,no two particles of matter ever come into contact2 when they get too close1 they both move off4 ,if a man were had up for knocking another man down1 he would be scientifically correct in pleading that he had never touched him4 ,what happened was that there was a hill in space-time in the region of the other man's nose1 and it fell down the hill4 ,the abolition of 8force0 seems to be connected with the substitution of sight for touch as the source of physical ideas1 as explained in ,chapter ,i4 ,when an image in a looking glass moves1 we do not think that something has pushed it4 ,in places where there are two large mirrors opposite to each other1 you may see innumerable reflections of the same object4 ,suppose a gentleman in a top-hat is standing between the mirrors1 there may be twenty or thirty top-hats in the reflections4 ,suppose now somebody comes and knocks off the gentleman's hat with a stick3 all the other twenty or thirty top-hats will tumble down at the same moment4 ,we think that a force is needed to knock off the 8real0 top-hat1 but we think the remaining twenty or thirty tumble off1 so to speak1 of themselves1 or out of a mere passion for imitation4 ,let us try to think out this matter a little more seriously4 ,obviously something happens when an image in a looking glass moves4 ,from the point of view of sight1 the event seems just as real as if it were not in a mirror4 ,but nothing has happened from the point of view of touch or hearing4 ,when the 8real0 top-hat falls1 it makes a noise2 the twenty or thirty reflections fall without a sound4 ,if it falls on your toe1 you feel it2 but we believe that the twenty or thirty people in the mirrors feel nothing1 though top-hats fall on their toes too4 ,but all this is equally true of the astronomical world4 ,it makes no noise1 because sound cannot travel across a vacuum4 ,so far as we know1 it causes no 8feelings10 because there is no one on the spot to 8feel0 it4 ,the astronomical world1 therefore1 seems hardly more 8real0 or 8solid0 than the world in the looking glass1 and has just as little need of 8force0 to make it move4 ,the reader may feel that ,i am indulging in idle sophistry4 8,after all10 he may say1 8the image in the mirror is the reflection of something solid1 and the top-hat in the mirror only falls off because of the force applied to the real top-hat4 ,the top-hat in the mirror cannot indulge in behavior of its own2 it has to copy the real one4 ,this shows how different the image is from the sun and the planets1 because they are not obliged to be perpetually imitating a prototype4 ,so you had better give up pretending that an image is just as real as one of the heavenly bodies40 ,there is1 of course1 some truth in this2 the point is to discover exactly what truth4 ,in the first place1 images are not 8imaginary40 ,when you see an image1 certain perfectly real light waves reach your eye2 and if you hang a cloth over the mirror1 these light waves cease to exist4 ,there is1 however1 a purely optical difference between an 8image0 and a 8real0 thing4 ,the optical difference is bound up with this question of imitation4 ,when you hang a cloth over the mirror1 it makes no difference to the 8real0 object2 but when you move the 8real0 object away1 the image vanishes also4 ,this makes us say that the light rays which make the image are only reflected at the surface of the mirror1 and do not really come from a point behind it1 but from the 8real0 object4 ,we have here an example of a general principle of great importance4 ,most of the events in the world are not isolated occurrences1 but members of groups of more or less similar events1 which are such that each group is connected in an assignable manner with a certain small region of space-time4 ,this is the case with the light rays which make us see both the object and its reflection in the mirror3 they all emanate from the object as a center4 ,if you put an opaque globe round the object at a certain distance1 the object and its reflection are invisible at any point outside the globe4 ,we have seen that gravitation1 although no longer regarded as an action at a distance1 is still connected with a center3 there is1 so to speak1 a hill symmetrically arranged about its summit1 and the summit is the place where we conceive the body to be which is connected with the gravitational field we are considering4 ,for simplicity1 common sense lumps together all the events which form one group in the above sense4 ,when two people see the same object1 two different events occur1 but they are events belonging to one group and connected with the same center4 ,just the same applies when two people 7as we say7 hear the same noise4 ,and so the reflection in a mirror is less 8real0 than the object reflected1 even from an optical point of view1 because light rays do not spread in all directions from the place where the image seems to be1 but only in directions in front of the mirror1 and only so long as the object reflected remains in position4 ,this illustrates the usefulness of grouping connected events about a center in the way we have been considering4 ,when we examine the changes in such a group of objects1 we find that they are of two kinds3 there are those which affect only some member of the group1 and those which make connected alterations in all the members of the group4 ,if you put a candle in front of a mirror1 and then hang black cloth over the mirror1 you alter only the reflection of the candle as seen from various places4 ,if you shut your eyes1 you alter its appearance to you1 but not its appearance elsewhere4 ,if you put a red globe round it at a distance of a foot1 you alter its appearance at any distance greater than a foot1 but not at any distance less than a foot4 ,in all these cases1 you do not regard the candle itself as having changed2 in fact1 in all of them1 you find that there are groups of changes connected with a different center or with a number of different centers4 ,when you shut your eyes1 for instance1 your eyes1 not the candle1 look different to any other observer3 the center of the changes that occur is in your eyes4 ,but when you blow out the candle1 its appearance everywhere is changed2 in this case you say that the change has happened to the candle4 ,the changes that happen to an object are those that affect the whole group of events which center about the object4 ,all this is only an interpretation of common sense1 and an attempt to explain what we mean by saying that the image of the candle in the mirror is less 8real0 than the candle4 ,there is no connected group of events situated all round the place where the image seems to be1 and changes in the image center about the candle1 not about a point behind the mirror4 ,this gives a perfectly verifiable meaning to the statement that the image is 8only0 a reflection4 ,and at the same time it enables us to regard the heavenly bodies1 although we can only see and not touch them1 as more 8real0 than an image in a looking glass4 ,we can now begin to interpret the common sense notion of one body having an 8effect0 upon another1 which we must do if we are really to understand what is meant by the abolition of 8force40 ,suppose you come into a dark room and switch on the electric light3 the appearance of everything in the room is changed4 ,since everything in the room is visible because it reflects the electric light1 this case is really analogous to that of the image in the mirror2 the electric light is the center from which all the changes emanate4 ,in this case1 the 8effect0 is explained by what we have already said4 ,the more important case is when the effect is a movement4 ,suppose you let loose a tiger in the middle of a ,bank ,holiday crowd3 they would all move1 and the tiger would be the center of their various movements4 ,a person who could see the people but not the tiger would infer that there was something repulsive at that point4 ,we say in this case that the tiger has an effect upon the people1 and we might describe the tiger's action upon them as of the nature of a repulsive force4 ,we know1 however1 that they fly because of something which happens to them1 not merely because the tiger is where he is4 ,they fly because they can see and hear him1 that is to say1 because certain waves reach their eyes and ears4 ,if these waves could be made to reach them without there being any tiger1 they would fly just as fast1 because the neighborhood would seem to them just as unpleasant4 ,let us now apply similar considerations to the sun's gravitation4 ,the 8force0 exerted by the sun only differs from that exerted by the tiger in being attractive instead of repulsive4 ,instead of acting through waves of light or sound1 the sun acquires its apparent power through the fact that there are modifications of space-time all round the sun4 ,like the noise of the tiger1 they are more intense near their source2 as we travel away they grow less and less4 ,to say that the sun 8causes0 these modifications of space-time is to add nothing to our knowledge4 ,what we know is that the modifications proceed according to a certain rule1 and that they are grouped symmetrically about the sun as center4 ,the language of cause and effect adds only a number of quite irrelevant imaginings1 connected with will1 muscular tension1 and such matters4 ,what we can more or less ascertain is merely the formula according to which space-time is modified by the presence of gravitating matter4 ,more correctly3 we can ascertain what kind of space-time is the presence of gravitating matter4 ,when space-time is not accurately ,euclidean in a certain region1 but has a non-,euclidean character which grows more and more marked as we approach a certain center1 and when1 further1 the departure from ,euclid obeys a certain law1 we describe this state of affairs briefly by saying that there is gravitating matter at the center4 ,but this is only a compendious account of what we know4 ,what we know is about the places where the gravitating matter is not1 not about the place where it is4 ,the language of cause and effect 7of which 8force0 is a particular case7 is thus merely a convenient shorthand for certain purposes2 it does not represent anything that is genuinely to be found in the physical world4 ,and how about matter8 ,is matter also no more than a convenient shorthand8 ,this question1 however1 being a large one1 demands a separate chapter4 ,,chapter ,,xiv3 ,,what ,,is ,,matter8 ,the question 8,what is matter80 is of the kind that is asked by metaphysicians1 and answered in vast books of incredible obscurity4 ,but ,i am not asking the question as metaphysician3 ,i am asking it as a person who wants to find out what is the moral of modern physics1 and more especially of the theory of relativity4 ,it is obvious from what we have learned of that theory that matter cannot be conceived quite as it used to be4 ,i think we can now say more or less what the new conception must be4 ,there were two traditional conceptions of matter1 both of which have had advocates ever since scientific speculation began4 ,there were the atomists1 who thought that matter consisted of tiny lumps which could never be divided2 these were supposed to hit each other and then bounce off in various ways4 ,after ,newton1 they were no longer supposed actually to come into contact with each other1 but to attract and repel each other1 and move in orbits round each other4 ,then there were those who thought that there is matter of some kind everywhere1 and that a true vacuum is impossible4 ,descartes held this view1 and attributed the motions of the planets to vortices in the ether4 ,the ,newtonian theory of gravitation caused the view that there is matter everywhere to fall into discredit1 the more so as light was thought by ,newton and his disciples to be due to actual particles traveling from the source of the light4 ,but when this view of light was disproved1 and it was shown that light consisted of waves1 the ether was revived so that there should be something to undulate4 ,the ether became still more respectable when it was found to play the same part in electromagnetic phenomena as in the propagation of light4 ,it was even hoped that atoms might turn out to be a mode of motion of the ether4 ,at this stage1 the atomic view of matter was1 on the whole1 getting the worst of it4 ,leaving relativity aside for the moment1 modern physics has provided proof of the atomic structure of ordinary matter1 while not disproving the arguments in favor of the ether1 to which no such structure is attributed4 ,the result was a sort of compromise between the two views1 the one applying to what was called 8gross0 matter1 the other to the ether4 ,there can be no doubt about electrons and protons1 though1 as we shall see shortly1 they need not be conceived as atoms were conceived traditionally4 ,as for the ether1 its status is very curious3 many physicists still maintain that1 without it1 the propagation of light and other electromagnetic waves would be inconceivable1 but except in this way it is difficult to see what purpose it serves4 ,the truth is1 ,i think1 that relativity demands the abandonment of the old conception of 8matter10 which is infected by the metaphysics associated with 8substance10 and represents a point of view not really necessary in dealing with phenomena4 ,this is what we must now investigate4 ,in the old view1 a piece of matter was something which survived all through time1 while never being at more than one place at a given time4 ,this way of looking at things is obviously connected with the complete separation of space and time in which people formerly believed4 ,when we substitute space-time for space and time1 we shall naturally expect to derive the physical world from constituents which are as limited in time as in space4 ,such constituents are what we call 8events40 ,an event does not persist and move1 like the traditional piece of matter2 it merely exists for its little moment and then ceases4 ,a piece of matter will thus be resolved into a series of events4 ,just as1 in the old view1 an extended body was composed of a number of particles1 so1 now1 each particle1 being extended in time1 must be regarded as composed of what we may call 8event-particles40 ,the whole series of these events makes up the whole history of the particle1 and the particle is regarded as being its history1 not some metaphysical entity to which the events happen4 ,this view is rendered necessary by the fact that relativity compels us to place time and space more on a level than they were in the older physics4 ,this abstract requirement must be brought into relation with the known facts of the physical world4 ,now what are the known facts8 ,let us take it as conceded that light consists of waves traveling with the received velocity4 ,we then know a great deal about what goes on in the parts of space-time where there is no matter2 we know1 that is to say1 that there are periodic occurrences 7light waves7 obeying certain laws4 ,these light waves start from atoms1 and the modern theory of the structure of the atoms enables us to know a great deal about the circumstances under which they start1 and the reasons which determine their wave lengths4 ,we can find out not only how one light wave travels1 but how its source moves relatively to ourselves4 ,but when ,i say this ,i am assuming that we can recognise a source of light as the same at two slightly different times4 ,this is1 however1 the very thing which had to be investigated4 ,we saw1 in the preceding chapter1 how a group of connected events can be formed1 all related to each other by a law1 and all ranged about a center in space-time4 ,such a group of events will be the arrival1 at various places1 of the light waves emitted by a brief flash of light4 ,we do not need to suppose that anything particular is happening at the center2 certainly we do not need to suppose that we know what is happening there4 ,what we know is that1 as a matter of geometry1 the group of events in question are ranged about a center1 like widening ripples on a pool when a fly has touched it4 ,we can hypothetically invent an occurrence which is to have happened at the center1 and set forth the laws by which the consequent disturbance is transmitted4 ,this hypothetical occurrence will then appear to common sense as the 8cause0 of the disturbance4 ,it will also count as one event in the biography of the particle of matter which is supposed to occupy the center of the disturbance4 ,now we find not only that one light wave travels outward from a center according to a certain law1 but also that1 in general1 it is followed by other closely similar light waves4 ,the sun1 for example1 does not change its appearance suddenly2 even if a cloud passes across it during a high wind1 the transition is gradual1 though swift4 ,in this way a group of occurrences connected with a center at one point of space-time is brought into relation with other very similar groups whose centers are at neighboring points of space-time4 ,for each of these other groups common sense invents similar hypothetical occurrences to occupy their centers1 and says that all these hypothetical occurrences are part of one history2 that is to say1 it invents a hypothetical 8particle0 to which the hypothetical occurrences are to have occurred4 ,it is only by this double use of hypothesis1 perfectly unnecessary in each case1 that we arrive at anything that can be called 8matter0 in the old sense of the word4 ,if we are to avoid unnecessary hypotheses1 we shall say that an electron at a given moment is the various disturbances in the surrounding medium which1 in ordinary language1 would be said to be 8caused0 by it4 ,but we shall not take these disturbances at what is1 for us1 the moment in question1 since that would make them depend upon the observer2 we shall instead travel outward from the electron with the velocity of light1 and take the disturbance we find in each place as we reach it4 ,the closely similar set of disturbances1 with very nearly the same center1 which is found existing slightly earlier or slightly later1 will be defined as being the electron at a slightly earlier or slightly later moment4 ,in this way1 we preserve all the laws of physics1 without having recourse to unnecessary hypotheses or inferred entities1 and we remain in harmony with the general principle of economy which has enabled the theory of relativity to clear away so much useless lumber4 ,common sense imagines that when it sees a table it sees a table4 ,this is a gross delusion4 ,when common sense sees a table1 certain light waves reach its eyes1 and these are of a sort which1 in its previous experience1 has been associated with certain sensations of touch1 as well as with other people's testimony that they also saw the table4 ,but none of this ever brought us to the table itself4 ,the light waves caused occurrences in our eyes1 and these caused occurrences in the optic nerve1 and these in turn caused occurrences in the brain4 ,any one of these1 happening without the usual preliminaries1 would have caused us to have the sensations we call 8seeing the table10 even if there had been no table4 7,of course1 if matter in general is to be interpreted as a group of occurrences1 this must apply also to the eye1 the optic nerve1 and the brain47 ,as to the sense of touch when we press the table with our fingers1 that is an electric disturbance in the electrons and protons of our finger tips1 produced1 according to modern physics1 by the proximity of the electrons and protons in the table4 ,if the same disturbance in our finger tips arose in any other way1 we should have the same sensations1 in spite of there being no table4 ,the testimony of others is obviously a second-hand affair4 ,a witness in a law court1 if asked whether he had seen some occurrence1 would not be allowed to reply that he believed so because of the testimony of others to that effect4 ,in any case1 testimony consists of sound waves and demands psychological as well as physical interpretation2 its connection with the object is therefore very indirect4 ,for all these reasons1 when we say that a man 8sees a table10 we use a highly abbreviated form of expression1 concealing complicated and difficult inferences1 the validity of which may well be open to question4 ,but we are in danger of becoming entangled in psychological questions1 which we must avoid if we can4 ,let us therefore return to the purely physical point of view4 ,what ,i wish to suggest may be put as follows4 ,everything that occurs elsewhere1 owing to the existence of an electron1 can be explored experimentally1 at least in theory1 unless it occurs in certain concealed ways4 ,but what occurs within the electron 7if anything occurs there7 it is absolutely impossible to know3 there is no conceivable apparatus by which we could obtain even a glimpse of it4 ,an electron is known by its 8effects40 ,but the word 8effects0 belongs to a view of causation which will not fit modern physics1 and in particular will not fit relativity4 ,all that we have a right to say is that certain groups of occurrences happen together1 that is to say1 in neighboring parts of space-time4 ,a given observer will regard one member of the group as earlier than the other1 but another observer may judge the time order differently4 ,and even when the time order is the same for all observers1 all that we really have is a connection between the two events1 which works equally backwards and forwards4 ,it is not true that the past determines the future in some sense other than that in which the future determines the past3 the apparent difference is only due to our ignorance1 because we know less about the future than about the past4 ,this is a mere accident3 there might be beings who would remember the future and have to infer the past4 ,the feelings of such beings in these matters would be the exact opposite of our own1 but no more fallacious4 ,the moral of this is that1 if an electron is only known by its 8effects10 there is no reason to suppose that anything exists except the 8effects40 ,in so far as these 8effects0 consist of light waves and other electromagnetic disturbances1 we may say that what is called 8empty space0 consists of regions where these disturbances are propagated freely4 ,every such disturbance1 we find1 has a center1 and when we get very near the center 7though still at a finite distance from it7 we find that the law of propagation of the disturbance ceases to be valid4 ,this region within which the law does not hold is called 8matter02 it will be an electron or proton according to circumstances4 ,the region so defined is found to move relatively to other such regions1 and its movements follow the known laws of dynamics4 ,so far1 this theory provides for electromagnetic phenomena and the motions of matter2 and it does so without assuming that 8matter0 is anything but systems of electromagnetic phenomena4 ,in order to carry out the theory fully1 it would no doubt be necessary to introduce many complications4 ,but it seems fairly clear that all the facts and laws of physics can be interpreted without assuming that 8matter0 is anything more than groups of events1 each event being of the sort which we should naturally regard as 8caused0 by the matter in question4 ,this does not involve any change in the symbols or formul of physics3 it is merely a question of interpretation of the symbols4 ,this latitude in interpretation is a characteristic of mathematical physics4 ,what we know is certain very abstract logical relations1 which we express in mathematical formul2 we know also that1 at certain points1 we arrive at results which are capable of being tested experimentally4 ,take1 for example1 the eclipse observations by which ,einstein's theory as to the bending of light was established4 ,the actual observation consisted in the careful measurement of certain distances on certain photographic plates4 ,the formul which were to be verified were concerned with the course of light in passing near the sun4 ,although the part of these formul which gives the observed result must always be interpreted in the same way1 the other part of them may be capable of a great variety of interpretations4 ,the formul giving the motions of the planets are almost exactly the same in ,einstein's theory as in ,newton's1 but the meaning of the formul is quite different4 ,it may be said generally that1 in the mathematical treatment of nature1 we can be far more certain that our formul are approximately correct than we can be as to the correctness of this or that interpretation of them4 ,and so in the case with which this chapter is concerned3 the question as to the nature of an electron or a proton is by no means answered when we know all that mathematical physics has to say as to the laws of its motion and the laws of its interaction with the environment4 ,a definite and conclusive answer to our question is not possible just because a variety of answers are compatible with the truth of mathematical physics4 ,nevertheless some answers are preferable to others1 because some have a greater probability in their favor4 ,we have been seeking1 in this chapter1 to define matter so that there must be such a thing if the formul of physics are true4 ,if we had made our definition such as to secure that a particle of matter should be what one thinks of as substantial1 a hard1 definite lump1 we should not have been sure that any such thing exists4 ,that is why our definition1 though it may seem complicated1 is preferable from the point of view of logical economy and scientific caution4 ,,chapter ,,xv3 ,,philosophical ,,consequences ,the philosophical consequences of relativity are neither so great nor so startling as is sometimes thought4 ,it throws very little light on time-honored controversies1 such as that between realism and idealism4 ,some people think that it supports ,kant's view that space and time are 8subjective0 and are 8forms of intuition40 ,i think such people have been misled by the way in which writers on relativity speak of 8the observer40 ,it is natural to suppose that the observer is a human being1 or at least a mind2 but he is just as likely to be a photographic plate or a clock4 ,that is to say1 the odd results as to the difference between one 8point of view0 and another are concerned with 8point of view0 in a sense applicable to physical instruments just as much as to people with perceptions4 ,the 8subjectivity0 concerned in the theory of relativity is a physical subjectivity1 which would exist equally if there were no such things as minds or senses in the world4 ,moreover1 it is a strictly limited subjectivity4 ,the theory does not say that everything is relative2 on the contrary1 it gives a technique for distinguishing what is relative from what belongs to a physical occurrence in its own right4 ,if we are going to say that the theory supports ,kant about space and time1 we shall have to say that it refutes him about space-time4 ,in my view1 neither statement is correct4 ,i see no reason why1 on such issues1 philosophers should not all stick to the views they previously held4 ,there were no conclusive arguments on either side before1 and there are none now2 to hold either view shows a dogmatic rather than a scientific temper4 ,nevertheless1 when the ideas involved in ,einstein's work have become familiar1 as they will when they are taught in schools1 certain changes in our habits of thought are likely to result1 and to have great importance in the long run4 ,one thing which emerges is that physics tells us much less about the physical world than we thought it did4 ,almost all the 8great principles0 of traditional physics turn out to be like the 8great law0 that there are always three feet to a yard2 others turn out to be downright false4 ,the conservation of mass may serve to illustrate both these misfortunes to which a 8law0 is liable4 ,mass used to be defined as 8quantity of matter10 and as far as experiment showed it was never increased or diminished4 ,but with the greater accuracy of modern measurements1 curious things were found to happen4 ,in the first place1 the mass as measured was found to increase with the velocity2 this kind of mass was found to be really the same thing as energy4 ,this kind of mass is not constant for a given body1 but the total amount of it in the universe is conserved1 or at least obeys a law very closely analogous to conservation4 ,this law itself1 however1 is to be regarded as a truism1 of the nature of the 8law0 that there are three feet to a yard2 it results from our methods of measurement1 and does not express a genuine property of matter4 ,the other kind of mass1 which we may call 8proper mass10 is that which is found to be the mass by an observer moving with the body4 ,this is the ordinary terrestrial case1 where the body we are weighing is not flying through the air4 ,the 8proper mass0 of a body is very nearly constant1 but not quite1 and the total amount of 8proper mass0 in the world is not quite constant4 ,one would suppose that if you have four one-pound weights1 and you put them all together into the scales1 they will together weigh four pounds4 ,this is a fond delusion3 they weigh rather less1 though not enough less to be discovered by even the most careful measurements4 ,in the case of four hydrogen atoms1 however1 when they are put together to make one helium atom1 the defect is noticeable2 the helium atom weighs measurably less than four separate hydrogen atoms4 ,broadly speaking1 traditional physics has collapsed into two portions1 truisms and geography4 ,there are1 however1 newer portions of physics1 such as the theory of quanta1 which do not come under this head1 but appear to give genuine knowledge of laws reached by experiment4 ,the world which the theory of relativity presents to our imagination is not so much a world of 8things0 in 8motion0 as a world of events4 ,it is true that there are still electrons and protons which persist1 but these 7as we saw in the preceding chapter7 are really to be conceived as strings of connected events1 like the successive notes of a song4 ,it is events that are the stuff of relativity physics4 ,between two events which are not too remote from each other there is1 in the general theory as in the special theory1 a measurable relation called 8interval10 which appears to be the physical reality of which lapse of time and distance in space are two more or less confused representations4 ,between two distant events1 there is not any one definite interval4 ,but there is one way of moving from one event to another which makes the sum of all the little intervals along the route greater than by any other route4 ,this route is called a 8geodesic10 and it is the route which a body will choose if left to itself4 ,the whole of relativity physics is a much more step-by-step matter than the physics and geometry of former days4 ,euclid's straight lines have to be replaced by light rays1 which do not quite come up to ,euclid's standard of straightness when they pass near the sun or any other very heavy body4 ,the sum of the angles of a triangle is still thought to be two right angles in very remote regions of empty space1 but not where there is matter in the neighborhood4 ,we1 who cannot leave the earth1 are incapable of reaching a place where ,euclid is true4 ,propositions which used to be proved by reasoning have now become either conventions1 or merely approximate truths verified by observation4 ,it is a curious fact--of which relativity is not the only illustration--that1 as reasoning improves1 its claims to the power of proving facts grow less and less4 ,logic used to be thought to teach us how to draw inferences2 now1 it teaches us rather how not to draw inferences4 ,animals and children are terribly prone to inference3 a horse is surprised beyond measure if you take an unusual turning4 ,when men began to reason1 they tried to justify the inferences that they had drawn unthinkingly in earlier days4 ,a great deal of bad philosophy and bad science resulted from this propensity4 8,great principles10 such as the 8uniformity of nature10 the 8law of universal causation10 and so on1 are attempts to bolster up our belief that what has often happened before will happen again1 which is no better founded than the horse's belief that you will take the turning you usually take4 ,it is not altogether easy to see what is to replace these pseudo-principles in the practice of science2 but perhaps the theory of relativity gives us a glimpse of the kind of thing we may expect4 ,causation1 in the old sense1 no longer has a place in theoretical physics4 ,there is1 of course1 something else which takes its place1 but the substitute appears to have a better empirical foundation than the old principle which it has superseded4 ,the collapse of the notion of one all-embracing time1 in which all events throughout the universe can be dated1 must in the long run affect our views as to cause and effect1 evolution1 and many other matters4 ,for instance1 the question whether1 on the whole1 there is progress in the universe1 may depend upon our choice of a measure of time4 ,if we choose one out of a number of equally good clocks1 we may find that the universe is progressing as fast as the most optimistic ,american thinks it is2 if we choose another equally good clock1 we may find that the universe is going from bad to worse as fast as the most melancholy ,slav could imagine4 ,thus optimism and pessimism are neither true nor false1 but depend upon the choice of clocks4 ,the effect of this upon a certain type of emotion is devastating4 ,the poet speaks of ,one far-off divine event ,to which the whole creation moves4 ,but if the event is sufficiently far off1 and the creation moves sufficiently quickly1 some parts will judge that the event has already happened1 while others will judge that it is still in the future4 ,this spoils the poetry4 ,the second line ought to be3 ,to which some parts of the creation move1 while others move away from it4 ,but this won't do4 ,i suggest that an emotion which can be destroyed by a little mathematics is neither very genuine nor very valuable4 ,but this line of argument would lead to a criticism of the ,victorian ,age1 which lies outside my theme4 ,what we know about the physical world1 ,i repeat1 is much more abstract1 than was formerly supposed4 ,between bodies there are occurrences1 such as light waves2 of the laws of these occurrences1 we know something--just so much as can be expressed in mathematical formul--but of their nature we know nothing4 ,of the bodies themselves1 as we saw in the preceding chapter1 we know so little that we cannot even be sure that they are anything3 they may be merely groups of events in other places1 those events which we should naturally regard as their effects4 ,we naturally interpret the world pictorially2 that is to say1 we imagine that what goes on is more or less like what we see4 ,but in fact this likeness can only extend to certain formal logical properties expressing structure1 so that all we can know is certain general characteristics of its changes4 ,perhaps an illustration may make the matter clear4 ,between a piece of orchestral music as played1 and the same piece of music as printed in the score1 there is a certain resemblance1 which may be described as a resemblance in structure4 ,the resemblance is of such a sort that1 when you know the rules1 you can infer the music from the score or the score from the music4 ,but suppose you had been stone deaf from birth1 but had lived among musical people4 ,you could understand1 if you had learned to speak and to do lip-reading1 that the musical scores represented something quite different from themselves in intrinsic quality1 though similar in structure4#af ,the value of music would be completely unimaginable to you1 but you could infer all its mathematical characteristics1 since they are the same as those of the score4 ,now our knowledge of nature is something like this4 ,we can read the scores1 and infer just so much as our stone-deaf person could have inferred about music4 ,but we have not the advantages which he derived from association with musical people4 ,we cannot know whether the music represented by the scores is beautiful or hideous2 perhaps1 in the last analysis1 we cannot be quite sure that the scores represent anything but themselves4 ,but this is a doubt which the physicist1 in his professional capacity1 cannot permit himself to entertain4 #af ,for the definition of 8structure10 see the present author's ,introduction to ,mathematical ,philosophy4 ,assuming the utmost that can be claimed for physics1 it does not tell us what it is that changes1 or what are its various states2 it only tells us such things as that changes follow each other periodically1 or spread with a certain speed4 ,even now we are probably not at the end of the process of stripping away what is merely imagination1 in order to reach the core of true scientific knowledge4 ,the theory of relativity has accomplished a very great deal in this respect1 and in doing so has taken us nearer and nearer to bare structure1 which is the mathematician's goal--not because it is the only thing in which he is interested as a human being1 but because it is the only thing that he can express in mathematical formul4 ,but far as we have traveled in the direction of abstraction1 it may be that we shall have to travel further still4 ,in the preceding chapter1 ,i suggested what may be called a minimum definition of matter1 that is to say1 one in which matter has1 so to speak1 as little 8substance0 as is compatible with the truth of physics4 ,in adopting a definition of this kind1 we are playing for safety3 our tenuous matter will exist1 even if something more beefy also exists4 ,we tried to make our definition of matter1 like ,isabella's gruel in ,jane ,austen1 8thin1 but not too thin40 ,we shall1 however1 fall into error if we assert positively that matter is nothing more than this4 ,leibniz thought that a piece of matter is really a colony of souls4 ,there is nothing to show that he was wrong1 though there is also nothing to show that he was right3 we know no more about it either way than we do about the flora and fauna of ,mars4 ,to the non-mathematical mind1 the abstract character of our physical knowledge may seem unsatisfactory4 ,from an artistic or imaginative point of view1 it is perhaps regrettable1 but from a practical point of view it is of no consequence4 ,abstraction1 difficult as it is1 is the source of practical power4 ,a financier1 whose dealings with the world are more abstract than those of any other 8practical0 man1 is also more powerful than any other practical man4 ,he can deal in wheat or cotton without needing ever to have seen either3 all he needs to know is whether they will go up or down4 ,this is abstract mathematical knowledge1 at least as compared to the knowledge of the agriculturist4 ,similarly the physicist1 who knows nothing of matter except certain laws of its movements1 nevertheless knows enough to enable him to manipulate it4 ,after working through whole strings of equations1 in which the symbols stand for things whose intrinsic nature can never be known to us1 he arrives at last at a result which can be interpreted in terms of our own perceptions1 and utilized to bring about desired effects in our own lives4 ,what we know about matter1 abstract and schematic as it is1 is enough1 in principle1 to tell us the rules according to which it produces perceptions and feelings in ourselves2 and it is upon these rules that the practical uses of physics depend4 ,the final conclusion is that we know very little1 and yet it is astonishing that we know so much1 and still more astonishing that so little knowledge can give us so much power4 ,,the ,,end