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It should definitely help, but it doesn鈥檛 help your acne, the health benefits are still worth it.|||In your schooldays most of you who read this book made acquaintance


with the noble building of Euclid%26#039;s geometry, and you remember --


perhaps with more respect than love -- the magnificent structure, on


the lofty staircase of which you were chased about for uncounted hours


by conscientious teachers. By reason of our past experience, you would


certainly regard everyone with disdain who should pronounce even the


most out-of-the-way proposition of this science to be untrue. But


perhaps this feeling of proud certainty would leave you immediately if


some one were to ask you: %26quot;What, then, do you mean by the assertion


that these propositions are true?%26quot; Let us proceed to give this


question a little consideration.





Geometry sets out form certain conceptions such as %26quot;plane,%26quot; %26quot;point,%26quot;


and %26quot;straight line,%26quot; with which we are able to associate more or less


definite ideas, and from certain simple propositions (axioms) which,


in virtue of these ideas, we are inclined to accept as %26quot;true.%26quot; Then,


on the basis of a logical process, the justification of which we feel


ourselves compelled to admit, all remaining propositions are shown to


follow from those axioms, i.e. they are proven. A proposition is then


correct (%26quot;true%26quot;) when it has been derived in the recognised manner


from the axioms. The question of %26quot;truth%26quot; of the individual geometrical


propositions is thus reduced to one of the %26quot;truth%26quot; of the axioms. Now


it has long been known that the last question is not only unanswerable


by the methods of geometry, but that it is in itself entirely without


meaning. We cannot ask whether it is true that only one straight line


goes through two points. We can only say that Euclidean geometry deals


with things called %26quot;straight lines,%26quot; to each of which is ascribed the


property of being uniquely determined by two points situated on it.


The concept %26quot;true%26quot; does not tally with the assertions of pure


geometry, because by the word %26quot;true%26quot; we are eventually in the habit of


designating always the correspondence with a %26quot;real%26quot; object; geometry,


however, is not concerned with the relation of the ideas involved in


it to objects of experience, but only with the logical connection of


these ideas among themselves.





It is not difficult to understand why, in spite of this, we feel


constrained to call the propositions of geometry %26quot;true.%26quot; Geometrical


ideas correspond to more or less exact objects in nature, and these


last are undoubtedly the exclusive cause of the genesis of those


ideas. Geometry ought to refrain from such a course, in order to give


to its structure the largest possible logical unity. The practice, for


example, of seeing in a %26quot;distance%26quot; two marked positions on a


practically rigid body is something which is lodged deeply in our


habit of thought. We are accustomed further to regard three points as


being situated on a straight line, if their apparent positions can be


made to coincide for observation with one eye, under suitable choice


of our place of observation.





If, in pursuance of our habit of thought, we now supplement the


propositions of Euclidean geometry by the single proposition that two


points on a practically rigid body always correspond to the same


distance (line-interval), independently of any changes in position to


which we may subject the body, the propositions of Euclidean geometry


then resolve themselves into propositions on the possible relative


position of practically rigid bodies.* Geometry which has been


supplemented in this way is then to be treated as a branch of physics.


We can now legitimately ask as to the %26quot;truth%26quot; of geometrical


propositions interpreted in this way, since we are justified in asking


whether these propositions are satisfied for those real things we have


associated with the geometrical ideas. In less exact terms we can


express this by saying that by the %26quot;truth%26quot; of a geometrical


proposition in this sense we understand its validity for a


construction with rule and compasses.





Of course the conviction of the %26quot;truth%26quot; of geometrical propositions in


this sense is founded exclusively on rather incomplete experience. For


the present we shall assume the %26quot;truth%26quot; of the geometrical


propositions, then at a later stage (in the general theory of


relativity) we shall see that this %26quot;truth%26quot; is limited, and we shall


consider the extent of its limitation.








Notes





*) It follows that a natural object is associated also with a


straight line. Three points A, B and C on a rigid body thus lie in a


straight line when the points A and C being given, B is chosen such


that the sum of the distances AB and BC is as short as possible. This


incomplete suggestion will suffice for the present purpose.











THE SYSTEM OF CO-ORDINATES








On the basis of the physical interpretation of distance which has been


indicated, we are also in a position to establish the distance between


two points on a rigid body by means of measurements. For this purpose


we require a %26quot; distance %26quot; (rod S) which is to be used once and for


all, and which we employ as a standard measure. If, now, A and B are


two points on a rigid body, we can construct the line joining them


according to the rules of geometry ; then, starting from A, we can


mark off the distance S time after time until we reach B. The number


of these operations required is the numerical measure of the distance


AB. This is the basis of all measurement of length. *





Every description of the scene of an event or of the position of an


object in space is based on the specification of the point on a rigid


body (body of reference) with which that event or object coincides.


This applies not only to scientific description, but also to everyday


life. If I analyse the place specification %26quot; Times Square, New York,%26quot;


**A I arrive at the following result. The earth is the rigid body


to which the specification of place refers; %26quot; Times Square, New York,%26quot;


is a well-defined point, to which a name has been assigned, and with


which the event coincides in space.**B





This primitive method of place specification deals only with places on


the surface of rigid bodies, and is dependent on the existence of


points on this surface which are distinguishable from each other. But


we can free ourselves from both of these limitations without altering


the nature of our specification of position. If, for instance, a cloud


is hovering over Times Square, then we can determine its position


relative to the surface of the earth by erecting a pole


perpendicularly on the Square, so that it reaches the cloud. The


length of the pole measured with the standard measuring-rod, combined


with the specification of the position of the foot of the pole,


supplies us with a complete place specification. On the basis of this


illustration, we are able to see the manner in which a refinement of


the conception of position has been developed.





(a) We imagine the rigid body, to which the place specification is


referred, supplemented in such a manner that the object whose position


we require is reached by. the completed rigid body.





(b) In locating the position of the object, we make use of a number


(here the length of the pole measured with the measuring-rod) instead


of designated points of reference.





(c) We speak of the height of the cloud even when the pole which


reaches the cloud has not been erected. By means of optical


observations of the cloud from different positions on the ground, and


taking into account the properties of the propagation of light, we


determine the length of the pole we should have required in order to


reach the cloud.





From this consideration we see that it will be advantageous if, in the


description of position, it should be possible by means of numerical


measures to make ourselves independent of the existence of marked


positions (possessing names) on the rigid body of reference. In the


physics of measurement this is attained by the application of the


Cartesian system of co-ordinates.





This consists of three plane surfaces perpendicular to each other and


rigidly attached to a rigid body. Referred to a system of


co-ordinates, the scene of any event will be determined (for the main


part) by the specification of the lengths of the three perpendiculars


or co-ordinates (x, y, z) which can be dropped from the scene of the


event to those three plane surfaces. The lengths of these three


perpendiculars can be determined by a series of manipulations with


rigid measuring-rods performed according to the rules and methods laid


down by Euclidean geometry.





In practice, the rigid surfaces which constitute the system of


co-ordinates are generally not available ; furthermore, the magnitudes


of the co-ordinates are not actually determined by constructions with


rigid rods, but by indirect means. If the results of physics and


astronomy are to maintain their clearness, the physical meaning of


specifications of position must always be sought in accordance with


the above considerations. ***





We thus obtain the following result: Every description of events in


space involves the use of a rigid body to which such events have to be


referred. The resulting relationship takes for granted that the laws


of Euclidean geometry hold for %26quot;distances;%26quot; the %26quot;distance%26quot; being


represented physically by means of the convention of two marks on a


rigid body.








Notes





* Here we have assumed that there is nothing left over i.e. that


the measurement gives a whole number. This difficulty is got over by


the use of divided measuring-rods, the introduction of which does not


demand any fundamentally new method.





**A Einstein used %26quot;Potsdamer Platz, Berlin%26quot; in the original text.


In the authorised translation this was supplemented with %26quot;Tranfalgar


Square, London%26quot;. We have changed this to %26quot;Times Square, New York%26quot;, as


this is the most well known/identifiable location to English speakers


in the present day. [Note by the janitor.]





**B It is not necessary here to investigate further the significance


of the expression %26quot;coincidence in space.%26quot; This conception is


sufficiently obvious to ensure that differences of opinion are


scarcely likely to arise as to its applicability in practice.





*** A refinement and modification of these views does not become


necessary until we come to deal with the general theory of relativity,


treated in the second part of this book.











SPACE AND TIME IN CLASSICAL MECHANICS








The purpose of mechanics is to describe how bodies change their


position in space with %26quot;time.%26quot; I should load my conscience with grave


sins against the sacred spirit of lucidity were I to formulate the


aims of mechanics in this way, without serious reflection and detailed


explanations. Let us proceed to disclose these sins.





It is not clear what is to be understood here by %26quot;position%26quot; and


%26quot;space.%26quot; I stand at the window of a railway carriage which is


travelling uniformly, and drop a stone on the embankment, without


throwing it. Then, disregarding the influence of the air resistance, I


see the stone descend in a straight line. A pedestrian who observes


the misdeed from the footpath notices that the stone falls to earth in


a parabolic curve. I now ask: Do the %26quot;positions%26quot; traversed by the


stone lie %26quot;in reality%26quot; on a straight line or on a parabola? Moreover,


what is meant here by motion %26quot;in space%26quot; ? From the considerations of


the previous section the answer is self-evident. In the first place we


entirely shun the vague word %26quot;space,%26quot; of which, we must honestly


acknowledge, we cannot form the slightest conception, and we replace


it by %26quot;motion relative to a practically rigid body of reference.%26quot; The


positions relative to the body of reference (railway carriage or


embankment) have already been defined in detail in the preceding


section. If instead of %26quot; body of reference %26quot; we insert %26quot; system of


co-ordinates,%26quot; which is a useful idea for mathematical description, we


are in a position to say : The stone traverses a straight line


relative to a system of co-ordinates rigidly attached to the carriage,


but relative to a system of co-ordinates rigidly attached to the


ground (embankment) it describes a parabola. With the aid of this


example it is clearly seen that there is no such thing as an


independently existing trajectory (lit. %26quot;path-curve%26quot;*), but only


a trajectory relative to a particular body of reference.





In order to have a complete description of the motion, we must specify


how the body alters its position with time ; i.e. for every point on


the trajectory it must be stated at what time the body is situated


there. These data must be supplemented by such a definition of time


that, in virtue of this definition, these time-values can be regarded


essentially as magnitudes (results of measurements) capable of


observation. If we take our stand on the ground of classical


mechanics, we can satisfy this requirement for our illustration in the


following manner. We imagine two clocks of identical construction ;


the man at the railway-carriage window is holding one of them, and the


man on the footpath the other. Each of the observers determines the


position on his own reference-body occupied by the stone at each tick


of the clock he is holding in his hand. In this connection we have not


taken account of the inaccuracy involved by the finiteness of the


velocity of propagation of light. With this and with a second


difficulty prevailing here we shall have to deal in detail later.








Notes





*) That is, a curve along which the body moves.











THE GALILEIAN SYSTEM OF CO-ORDINATES








As is well known, the fundamental law of the mechanics of


Galilei-Newton, which is known as the law of inertia, can be stated


thus: A body removed sufficiently far from other bodies continues in a


state of rest or of uniform motion in a straight line. This law not


only says something about the motion of the bodies, but it also


indicates the reference-bodies or systems of coordinates, permissible


in mechanics, which can be used in mechanical description. The visible


fixed stars are bodies for which the law of inertia certainly holds to


a high degree of approximation. Now if we use a system of co-ordinates


which is rigidly attached to the earth, then, relative to this system,


every fixed star describes a circle of immense radius in the course of


an astronomical day, a result which is opposed to the statement of the


law of inertia. So that if we adhere to this law we must refer these


motions only to systems of coordinates relative to which the fixed


stars do not move in a circle. A system of co-ordinates of which the


state of motion is such that the law of inertia holds relative to it


is called a %26quot; Galileian system of co-ordinates.%26quot; The laws of the


mechanics of Galflei-Newton can be regarded as valid only for a


Galileian system of co-ordinates.











THE PRINCIPLE OF RELATIVITY


(IN THE RESTRICTED SENSE)








In order to attain the greatest possible clearness, let us return to


our example of the railway carriage supposed to be travelling


uniformly. We call its motion a uniform translation (%26quot;uniform%26quot; because


it is of constant velocity and direction, %26quot; translation %26quot; because


although the carriage changes its position relative to the embankment


yet it does not rotate in so doing). Let us imagine a raven flying


through the air in such a manner that its motion, as observed from the


embankment, is uniform and in a straight line. If we were to observe


the flying raven from the moving railway carriage. we should find that


the motion of the raven would be one of different velocity and


direction, but that it would still be uniform and in a straight line.


Expressed in an abstract manner we may say : If a mass m is moving


uniformly in a straight line with respect to a co-ordinate system K,


then it will also be moving uniformly and in a straight line relative


to a second co-ordinate system K1 provided that the latter is


executing a uniform translatory motion with respect to K. In


accordance with the discussion contained in the preceding section, it


follows that:





If K is a Galileian co-ordinate system. then every other co-ordinate


system K%26#039; is a Galileian one, when, in relation to K, it is in a


condition of uniform motion of translation. Relative to K1 the


mechanical laws of Galilei-Newton hold good exactly as they do with


respect to K.





We advance a step farther in our generalisation when we express the


tenet thus: If, relative to K, K1 is a uniformly moving co-ordinate


system devoid of rotation, then natural phenomena run their course


with respect to K1 according to exactly the same general laws as with


respect to K. This statement is called the principle of relativity (in


the restricted sense).





As long as one was convinced that all natural phenomena were capable


of representation with the help of classical mechanics, there was no


need to doubt the validity of this principle of relativity. But in


view of the more recent development of electrodynamics and optics it


became more and more evident that classical mechanics affords an


insufficient foundation for the physical description of all natural


phenomena. At this juncture the question of the validity of the


principle of relativity became ripe for discussion, and it did not


appear impossible that the answer to this question might be in the


negative.





Nevertheless, there are two general facts which at the outset speak


very much in favour of the validity of the principle of relativity.


Even though classical mechanics does not supply us with a sufficiently


broad basis for the theoretical presentation of all physical


phenomena, still we must grant it a considerable measure of %26quot; truth,%26quot;


since it supplies us with the actual motions of the heavenly bodies


with a delicacy of detail little short of wonderful. The principle of


relativity must therefore apply with great accuracy in the domain of


mechanics. But that a principle of such broad generality should hold


with such exactness in one domain of phenomena, and yet should be


invalid for another, is a priori not very probable.





We now proceed to the second argument, to which, moreover, we shall


return later. If the principle of relativity (in the restricted sense)


does not hold, then the Galileian co-ordinate systems K, K1, K2, etc.,


which are moving uniformly relative to each other, will not be


equivalent for the description of natural phenomena. In this case we


should be constrained to believe that natural laws are capable of


being formulated in a particularly simple manner, and of course only


on condition that, from amongst all possible Galileian co-ordinate


systems, we should have chosen one (K[0]) of a particular state of


motion as our body of reference. We should then be justified (because


of its merits for the description of natural phenomena) in calling


this system %26quot; absolutely at rest,%26quot; and all other Galileian systems K %26quot;


in motion.%26quot; If, for instance, our embankment were the system K[0] then


our railway carriage would be a system K, relative to which less


simple laws would hold than with respect to K[0]. This diminished


simplicity would be due to the fact that the carriage K would be in


motion (i.e.%26quot;really%26quot;)with respect to K[0]. In the general laws of


nature which have been formulated with reference to K, the magnitude


and direction of the velocity of the carriage would necessarily play a


part. We should expect, for instance, that the note emitted by an


organpipe placed with its axis parallel to the direction of travel


would be different from that emitted if the axis of the pipe were


placed perpendicular to this direction.





Now in virtue of its motion in an orbit round the sun, our earth is


comparable with a railway carriage travelling with a velocity of about


30 kilometres per second. If the principle of relativity were not


valid we should therefore expect that the direction of motion of the


earth at any moment would enter into the laws of nature, and also that


physical systems in their behaviour would be dependent on the


orientation in space with respect to the earth. For owing to the


alteration in direction of the velocity of revolution of the earth in


the course of a year, the earth cannot be at rest relative to the


hypothetical system K[0] throughout the whole year. However, the most


careful observations have never revealed such anisotropic properties


in terrestrial physical space, i.e. a physical non-equivalence of


different directions. This is very powerful argument in favour of the


principle of relativity.











THE THEOREM OF THE


ADDITION OF VELOCITIES


EMPLOYED IN CLASSICAL MECHANICS








Let us suppose our old friend the railway carriage to be travelling


along the rails with a constant velocity v, and that a man traverses


the length of the carriage in the direction of travel with a velocity


w. How quickly or, in other words, with what velocity W does the man


advance relative to the embankment during the process ? The only


possible answer seems to result from the following consideration: If


the man were to stand still for a second, he would advance relative to


the embankment through a distance v equal numerically to the velocity


of the carriage. As a consequence of his walking, however, he


traverses an additional distance w relative to the carriage, and hence


also relative to the embankment, in this second, the distance w being


numerically equal to the velocity with which he is walking. Thus in


total be covers the distance W=v+w relative to the embankment in the


second considered. We shall see later that this result, which


expresses the theorem of the addition of velocities employed in


classical mechanics, cannot be maintained ; in other words, the law


that we have just written down does not hold in reality. For the time


being, however, we shall assume its correctness.











THE APPARENT INCOMPATIBILITY OF THE


LAW OF PROPAGATION OF LIGHT WITH THE


PRINCIPLE OF RELATIVITY








There is hardly a simpler law in physics than that according to which


light is propagated in empty space. Every child at school knows, or


believes he knows, that this propagation takes place in straight lines


with a velocity c= 300,000 km./sec. At all events we know with great


exactness that this velocity is the same for all colours, because if


this were not the case, the minimum of emission would not be observed


simultaneously for different colours during the eclipse of a fixed


star by its dark neighbour. By means of similar considerations based


on observa- tions of double stars, the Dutch astronomer De Sitter was


also able to show that the velocity of propagation of light cannot


depend on the velocity of motion of the body emitting the light. The


assumption that this velocity of propagation is dependent on the


direction %26quot;in space%26quot; is in itself improbable.





In short, let us assume that the simple law of the constancy of the


velocity of light c (in vacuum) is justifiably believed by the child


at school. Who would imagine that this simple law has plunged the


conscientiously thoughtful physicist into the greatest intellectual


difficulties? Let us consider how these difficulties arise.





Of course we must refer the process of the propagation of light (and


indeed every other process) to a rigid reference-body (co-ordinate


system). As such a system let us again choose our embankment. We shall


imagine the air above it to have been removed. If a ray of light be


sent along the embankment, we see from the above that the tip of the


ray will be transmitted with the velocity c relative to the


embankment. Now let us suppose that our railway carriage is again


travelling along the railway lines with the velocity v, and that its


direction is the same as that of the ray of light, but its velocity of


course much less. Let us inquire about the velocity of propagation of


the ray of light relative to the carriage. It is obvious that we can


here apply the consideration of the previous section, since the ray of


light plays the part of the man walking along relatively to the


carriage. The velocity w of the man relative to the embankment is here


replaced by the velocity of light relative to the embankment. w is the


required velocity of light with respect to the carriage, and we have





w = c-v.





The velocity of propagation ot a ray of light relative to the carriage


thus comes cut smaller than c.





But this result comes into conflict with the principle of relativity


set forth in Section V. For, like every other general law of


nature, the law of the transmission of light in vacuo [in vacuum]


must, according to the principle of relativity, be the same for the


railway carriage as reference-body as when the rails are the body of


reference. But, from our above consideration, this would appear to be


impossible. If every ray of light is propagated relative to the


embankment with the velocity c, then for this reason it would appear


that another law of propagation of light must necessarily hold with


respect to the carriage -- a result contradictory to the principle of


relativity.





In view of this dilemma there appears to be nothing else for it than


to abandon either the principle of relativity or the simple law of the


propagation of light in vacuo. Those of you who have carefully


followed the preceding discussion are almost sure to expect that we


should retain the principle of relativity, which appeals so


convincingly to the intellect because it is so natural and simple. The


law of the propagation of light in vacuo would then have to be


replaced by a more complicated law conformable to the principle of


relativity. The development of theoretical physics shows, however,


that we cannot pursue this course. The epoch-making theoretical


investigations of H. A. Lorentz on the electrodynamical and optical


phenomena connected with moving bodies show that experience in this


domain leads conclusively to a theory of electromagnetic phenomena, of


which the law of the constancy of the velocity of light in vacuo is a


necessary consequence. Prominent theoretical physicists were theref


ore more inclined to reject the principle of relativity, in spite of


the fact that no empirical data had been found which were


contradictory to this principle.





At this juncture the theory of relativity entered the arena. As a


result of an analysis of the physical conceptions of time and space,


it became evident that in realily there is not the least


incompatibilitiy between the principle of relativity and the law of


propagation of light, and that by systematically holding fast to both


these laws a logically rigid theory could be arrived at. This theory


has been called the special theory of relativity to distinguish it


from the extended theory, with which we shall deal later. In the


following pages we shall present the fundamental ideas of the special


theory of relativity.











ON THE IDEA OF TIME IN PHYSICS








Lightning has struck the rails on our railway embankment at two places


A and B far distant from each other. I make the additional assertion


that these two lightning flashes occurred simultaneously. If I ask you


whether there is sense in this statement, you will answer my question


with a decided %26quot;Yes.%26quot; But if I now approach you with the request to


explain to me the sense of the statement more precisely, you find


after some consideration that the answer to this question is not so


easy as it appears at first sight.





After some time perhaps the following answer would occur to you: %26quot;The


significance of the statement is clear in itself and needs no further


explanation; of course it would require some consideration if I were


to be commissioned to determine by observations whether in the actual


case the two events took place simultaneously or not.%26quot; I cannot be


satisfied with this answer for the following reason. Supposing that as


a result of ingenious considerations an able meteorologist were to


discover that the lightning must always strike the places A and B


simultaneously, then we should be faced with the task of testing


whether or not this theoretical result is in accordance with the


reality. We encounter the same difficulty with all physical statements


in which the conception %26quot; simultaneous %26quot; plays a part. The concept


does not exist for the physicist until he has the possibility of


discovering whether or not it is fulfilled in an actual case. We thus


require a definition of simultaneity such that this definition


supplies us with the method by means of which, in the present case, he


can decide by experiment whether or not both the lightning strokes


occurred simultaneously. As long as this requirement is not satisfied,


I allow myself to be deceived as a physicist (and of course the same


applies if I am not a physicist), when I imagine that I am able to


attach a meaning to the statement of simultaneity. (I would ask the


reader not to proceed farther until he is fully convinced on this


point.)





After thinking the matter over for some time you then offer the


following suggestion with which to test simultaneity. By measuring


along the rails, the connecting line AB should be measured up and an


observer placed at the mid-point M of the distance AB. This observer


should be supplied with an arrangement (e.g. two mirrors inclined at


90^0) which allows him visually to observe both places A and B at the


same time. If the observer perceives the two flashes of lightning at


the same time, then they are simultaneous.





I am very pleased with this suggestion, but for all that I cannot


regard the matter as quite settled, because I feel constrained to


raise the following objection:





%26quot;Your definition would certainly be right, if only I knew that the


light by means of which the observer at M perceives the lightning


flashes travels along the length A arrow M with the same velocity as


along the length B arrow M. But an examination of this supposition


would only be possible if we already had at our disposal the means of


measuring time. It would thus appear as though we were moving here in


a logical circle.%26quot;





After further consideration you cast a somewhat disdainful glance at


me -- and rightly so -- and you declare:





%26quot;I maintain my previous definition nevertheless, because in reality it


assumes absolutely nothing about light. There is only one demand to be


made of the definition of simultaneity, namely, that in every real


case it must supply us with an empirical decision as to whether or not


the conception that has to be defined is fulfilled. That my definition


satisfies this demand is indisputable. That light requires the same


time to traverse the path A arrow M as for the path B arrow M is in


reality neither a supposition nor a hypothesis about the physical


nature of light, but a stipulation which I can make of my own freewill


in order to arrive at a definition of simultaneity.%26quot;





It is clear that this definition can be used to give an exact meaning


not only to two events, but to as many events as we care to choose,


and independently of the positions of the scenes of the events with


respect to the body of reference * (here the railway embankment).


We are thus led also to a definition of %26quot; time %26quot; in physics. For this


purpose we suppose that clocks of identical construction are placed at


the points A, B and C of the railway line (co-ordinate system) and


that they are set in such a manner that the positions of their


pointers are simultaneously (in the above sense) the same. Under these


conditions we understand by the %26quot; time %26quot; of an event the reading


(position of the hands) of that one of these clocks which is in the


immediate vicinity (in space) of the event. In this manner a


time-value is associated with every event which is essentially capable


of observation.





This stipulation contains a further physical hypothesis, the validity


of which will hardly be doubted without empirical evidence to the


contrary. It has been assumed that all these clocks go at the same


rate if they are of identical construction. Stated more exactly: When


two clocks arranged at rest in different places of a reference-body


are set in such a manner that a particular position of the pointers of


the one clock is simultaneous (in the above sense) with the same


position, of the pointers of the other clock, then identical %26quot;


settings %26quot; are always simultaneous (in the sense of the above


definition).








Notes





*) We suppose further, that, when three events A, B and C occur in


different places in such a manner that A is simultaneous with B and B


is simultaneous with C (simultaneous in the sense of the above


definition), then the criterion for the simultaneity of the pair of


events A, C is also satisfied. This assumption is a physical


hypothesis about the the of propagation of light: it must certainly be


fulfilled if we are to maintain the law of the constancy of the


velocity of light in vacuo.











THE RELATIVITY OF SIMULATNEITY








Up to now our considerations have been referred to a particular body


of reference, which we have styled a %26quot; railway embankment.%26quot; We suppose


a very long train travelling along the rails with the constant


velocity v and in the direction indicated in Fig 1. People travelling


in this train will with a vantage view the train as a rigid


reference-body (co-ordinate system); they regard all events in





Fig. 01: file fig01.gif








reference to the train. Then every event which takes place along the


line also takes place at a particular point of the train. Also the


definition of simultaneity can be given relative to the train in


exactly the same way as with respect to the embankment. As a natural


consequence, however, the following question arises :





Are two events (e.g. the two strokes of lightning A and B) which are


simultaneous with reference to the railway embankment also


simultaneous relatively to the train? We shall show directly that the


answer must be in the negative.





When we say that the lightning strokes A and B are simultaneous with


respect to be embankment, we mean: the rays of light emitted at the


places A and B, where the lightning occurs, meet each other at the


mid-point M of the length A arrow B of the embankment. But the events


A and B also correspond to positions A and B on the train. Let M1 be


the mid-point of the distance A arrow B on the travelling train. Just


when the flashes (as judged from the embankment) of lightning occur,


this point M1 naturally coincides with the point M but it moves


towards the right in the diagram with the velocity v of the train. If


an observer sitting in the position M1 in the train did not possess


this velocity, then he would remain permanently at M, and the light


rays emitted by the flashes of lightning A and B would reach him


simultaneously, i.e. they would meet just where he is situated. Now in


reality (considered with reference to the railway embankment) he is


hastening towards the beam of light coming from B, whilst he is riding


on ahead of the beam of light coming from A. Hence the observer will


see the beam of light emitted from B earlier than he will see that


emitted from A. Observers who take the railway train as their


reference-body must therefore come to the conclusion that the


lightning flash B took place earlier than the lightning flash A. We


thus arrive at the important result:





Events which are simultaneous with reference to the embankment are not


simultaneous with respect to the train, and vice versa (relativity of


simultaneity). Every reference-body (co-ordinate system) has its own


particular time ; unless we are told the reference-body to which the


statement of time refers, there is no meaning in a statement of the


time of an event.





Now before the advent of the theory of relativity it had always


tacitly been assumed in physics that the statement of time had an


absolute significance, i.e. that it is independent of the state of


motion of the body of reference. But we have just seen that this


assumption is incompatible with the most natural definition of


simultaneity; if we discard this assumption, then the conflict between


the law of the propagation of light in vacuo and the principle of


relativity (developed in Section 7) disappears.





We were led to that conflict by the considerations of Section 6,


which are now no longer tenable. In that section we concluded that the


man in the carriage, who traverses the distance w per second relative


to the carriage, traverses the same distance also with respect to the


embankment in each second of time. But, according to the foregoing


considerations, the time required by a particular occurrence with


respect to the carriage must not be considered equal to the duration


of the same occurrence as judged from the embankment (as


reference-body). Hence it cannot be contended that the man in walking


travels the distance w relative to the railway line in a time which is


equal to one second as judged from the embankment.





Moreover, the considerations of Section 6 are based on yet a second


assumption, which, in the light of a strict consideration, appears to


be arbitrary, although it was always tacitly made even before the


introduction of the theory of relativity.











ON THE RELATIVITY OF THE CONCEPTION OF DISTANCE








Let us consider two particular points on the train * travelling


along the embankment with the velocity v, and inquire as to their


distance apart. We already know that it is necessary to have a body of


reference for the measurement of a distance, with respect to which


body the distance can be measured up. It is the simplest plan to use


the train itself as reference-body (co-ordinate system). An observer


in the train measures the interval by marking off his measuring-rod in


a straight line (e.g. along the floor of the carriage) as many times


as is necessary to take him from the one marked point to the other.


Then the number which tells us how often the rod has to be laid down


is the required distance.





It is a different matter when the distance has to be judged from the


railway line. Here the following method suggests itself. If we call


A^1 and B^1 the two points on the train whose distance apart is


required, then both of these points are moving with the velocity v


along the embankment. In the first place we require to determine the


points A and B of the embankment which are just being passed by the


two points A^1 and B^1 at a particular time t -- judged from the


embankment. These points A and B of the embankment can be determined


by applying the definition of time given in Section 8. The distance


between these points A and B is then measured by repeated application


of thee measuring-rod along the embankment.





A priori it is by no means certain that this last measurement will


supply us with the same result as the first. Thus the length of the


train as measured from the embankment may be different from that


obtained by measuring in the train itself. This circumstance leads us


to a second objection which must be raised against the apparently


obvious consideration of Section 6. Namely, if the man in the


carriage covers the distance w in a unit of time -- measured from the


train, -- then this distance -- as measured from the embankment -- is


not necessarily also equal to w.








Notes





*) e.g. the middle of the first and of the hundredth carriage.











THE LORENTZ TRANSFORMATION








The results of the last three sections show that the apparent


incompatibility of the law of propagation of light with the principle


of relativity (Section 7) has been derived by means of a


consideration which borrowed two unjustifiable hypotheses from


classical mechanics; these are as follows:





(1) The time-interval (time) between two events is independent of the


condition of motion of the body of reference.





(2) The space-interval (distance) between two points of a rigid body


is independent of the condition of motion of the body of reference.





If we drop these hypotheses, then the dilemma of Section 7


disappears, because the theorem of the addition of velocities derived


in Section 6 becomes invalid. The possibility presents itself that


the law of the propagation of light in vacuo may be compatible with


the principle of relativity, and the question arises: How have we to


modify the considerations of Section 6 in order to remove the


apparent disagreement between these two fundamental results of


experience? This question leads to a general one. In the discussion of


Section 6 we have to do with places and times relative both to the


train and to the embankment. How are we to find the place and time of


an event in relation to the train, when we know the place and time of


the event with respect to the railway embankment ? Is there a


thinkable answer to this question of such a nature that the law of


transmission of light in vacuo does not contradict the principle of


relativity ? In other words : Can we conceive of a relation between


place and time of the individual events relative to both


reference-bodies, such that every ray of light possesses the velocity


of transmission c relative to the embankment and relative to the train


? This question leads to a quite definite positive answer, and to a


perfectly definite transformation law for the space-time magnitudes of


an event when changing over from one body of reference to another.





Before we deal with this, we shall introduce the following incidental


consideration. Up to the present we have only considered events taking


place along the embankment, which had mathematically to assume the


function of a straight line. In the manner indicated in Section 2


we can imagine this reference-body supplemented laterally and in a


vertical direction by means of a framework of rods, so that an event


which takes place anywhere can be localised with reference to this


framework. Fig. 2 Similarly, we can imagine the train travelling with


the velocity v to be continued across the whole of space, so that


every event, no matter how far off it may be, could also be localised


with respect to the second framework. Without committing any


fundamental error, we can disregard the fact that in reality these


frameworks would continually interfere with each other, owing to the


impenetrability of solid bodies. In every such framework we imagine


three surfaces perpendicular to each other marked out, and designated


as %26quot; co-ordinate planes %26quot; (%26quot; co-ordinate system %26quot;). A co-ordinate


system K then corresponds to the embankment, and a co-ordinate system


K%26#039; to the train. An event, wherever it may have taken place, would be


fixed in space with respect to K by the three perpendiculars x, y, z


on the co-ordinate planes, and with regard to time by a time value t.


Relative to K1, the same event would be fixed in respect of space and


time by corresponding values x1, y1, z1, t1, which of course are not


identical with x, y, z, t. It has already been set forth in detail how


these magnitudes are to be regarded as results of physical


measurements.





Obviously our problem can be exactly formulated in the following


manner. What are the values x1, y1, z1, t1, of an event with respect


to K1, when the magnitudes x, y, z, t, of the same event with respect


to K are given ? The relations must be so chosen that the law of the


transmission of light in vacuo is satisfied for one and the same ray


of light (and of course for every ray) with respect to K and K1. For


the relative orientation in space of the co-ordinate systems indicated


in the diagram ([7]Fig. 2), this problem is solved by means of the


equations :





eq. 1: file eq01.gif





y1 = y


z1 = z





eq. 2: file eq02.gif





This system of equations is known as the %26quot; Lorentz transformation.%26quot; *





If in place of the law of transmission of light we had taken as our


basis the tacit assumptions of the older mechanics as to the absolute


character of times and lengths, then instead of the above we should


have obtained the following equations:





x1 = x - vt


y1 = y


z1 = z


t1 = t





This system of equations is often termed the %26quot; Galilei


transformation.%26quot; The Galilei transformation can be obtained from the


Lorentz transformation by substituting an infinitely large value for


the velocity of light c in the latter transformation.





Aided by the following illustration, we can readily see that, in


accordance with the Lorentz transformation, the law of the


transmission of light in vacuo is satisfied both for the


reference-body K and for the reference-body K1. A light-signal is sent


along the positive x-axis, and this light-stimulus advances in


accordance with the equation





x = ct,





i.e. with the velocity c. According to the equations of the Lorentz


transformation, this simple relation between x and t involves a


relation between x1 and t1. In point of fact, if we substitute for x


the value ct in the first and fourth equations of the Lorentz


transformation, we obtain:





eq. 3: file eq03.gif








eq. 4: file eq04.gif





from which, by division, the expression





x1 = ct1





immediately follows. If referred to the system K1, the propagation of


light takes place according to this equation. We thus see that the


velocity of transmission relative to the reference-body K1 is also


equal to c. The same result is obtained for rays of light advancing in


any other direction whatsoever. Of cause this is not surprising, since


the equations of the Lorentz transformation were derived conformably


to this point of view.








Notes





*) A simple derivation of the Lorentz transformation is given in


Appendix I.











THE BEHAVIOUR OF MEASURING-RODS AND CLOCKS IN MOTION








Place a metre-rod in the x1-axis of K1 in such a manner that one end


(the beginning) coincides with the point x1=0 whilst the other end


(the end of the rod) coincides with the point x1=I. What is the length


of the metre-rod relatively to the system K? In order to learn this,


we need only ask where the beginning of the rod and the end of the rod


lie with respect to K at a particular time t of the system K. By means


of the first equation of the Lorentz transformation the values of


these two points at the time t = 0 can be shown to be





eq. 05a: file eq05a.gif








eq. 05b: file eq05b.gif








the distance between the points being eq. 06 .





But the metre-rod is moving with the velocity v relative to K. It


therefore follows that the length of a rigid metre-rod moving in the


direction of its length with a velocity v is eq. 06 of a metre.





The rigid rod is thus shorter when in motion than when at rest, and


the more quickly it is moving, the shorter is the rod. For the


velocity v=c we should have eq. 06a ,





and for stiII greater velocities the square-root becomes imaginary.


From this we conclude that in the theory of relativity the velocity c


plays the part of a limiting velocity, which can neither be reached


nor exceeded by any real body.





Of course this feature of the velocity c as a limiting velocity also


clearly follows from the equations of the Lorentz transformation, for


these became meaningless if we choose values of v greater than c.





If, on the contrary, we had considered a metre-rod at rest in the


x-axis with respect to K, then we should have found that the length of


the rod as judged from K1 would have been eq. 06 ;





this is quite in accordance with the principle of relativity which


forms the basis of our considerations.





A Priori it is quite clear that we must be able to learn something


about the physical behaviour of measuring-rods and clocks from the


equations of transformation, for the magnitudes z, y, x, t, are


nothing more nor less than the results of measurements obtainable by


means of measuring-rods and clocks. If we had based our considerations


on the Galileian transformation we should not have obtained a


contraction of the rod as a consequence of its motion.





Let us now consider a seconds-clock which is permanently situated at


the origin (x1=0) of K1. t1=0 and t1=I are two successive ticks of


this clock. The first and fourth equations of the Lorentz


transformation give for these two ticks :





t = 0





and





eq. 07: file eq07.gif





As judged from K, the clock is moving with the velocity v; as judged


from this reference-body, the time which elapses between two strokes


of the clock is not one second, but





eq. 08: file eq08.gif





seconds, i.e. a somewhat larger time. As a consequence of its motion


the clock goes more slowly than when at rest. Here also the velocity c


plays the part of an unattainable limiting velocity.











THEOREM OF THE ADDITION OF VELOCITIES.


THE EXPERIMENT OF FIZEAU








Now in practice we can move clocks and measuring-rods only with


velocities that are small compared with the velocity of light; hence


we shall hardly be able to compare the results of the previous section


directly with the reality. But, on the other hand, these results must


strike you as being very singular, and for that reason I shall now


draw another conclusion from the theory, one which can easily be


derived from the foregoing considerations, and which has been most


elegantly confirmed by experiment.





In Section 6 we derived the theorem of the addition of velocities


in one direction in the form which also results from the hypotheses of


classical mechanics- This theorem can also be deduced readily horn the


Galilei transformation (Section 11). In place of the man walking


inside the carriage, we introduce a point moving relatively to the


co-ordinate system K1 in accordance with the equation





x1 = wt1





By means of the first and fourth equations of the Galilei


transformation we can express x1 and t1 in terms of x and t, and we


then obtain





x = (v + w)t





This equation expresses nothing else than the law of motion of the


point with reference to the system K (of the man with reference to the


embankment). We denote this velocity by the symbol W, and we then


obtain, as in Section 6,





W=v+w A)





But we can carry out this consideration just as well on the basis of


the theory of relativity. In the equation





x1 = wt1 B)





we must then express x1and t1 in terms of x and t, making use of the


first and fourth equations of the Lorentz transformation. Instead of


the equation (A) we then obtain the equation





eq. 09: file eq09.gif








which corresponds to the theorem of addition for velocities in one


direction according to the theory of relativity. The question now


arises as to which of these two theorems is the better in accord with


experience. On this point we axe enlightened by a most important


experiment which the brilliant physicist Fizeau performed more than


half a century ago, and which has been repeated since then by some of


the best experimental physicists, so that there can be no doubt about


its result. The experiment is concerned with the following question.


Light travels in a motionless liquid with a particular velocity w. How


quickly does it travel in the direction of the arrow in the tube T


(see the accompanying diagram, Fig. 3) when the liquid above


mentioned is flowing through the tube with a velocity v ?





In accordance with the principle of relativity we shall certainly have


to take for granted that the propagation of light always takes place


with the same velocity w with respect to the liquid, whether the


latter is in motion with reference to other bodies or not. The


velocity of light relative to the liquid and the velocity of the


latter relative to the tube are thus known, and we require the


velocity of light relative to the tube.





It is clear that we have the problem of Section 6 again before us. The


tube plays the part of the railway embankment or of the co-ordinate


system K, the liquid plays the part of the carriage or of the


co-ordinate system K1, and finally, the light plays the part of the





Figure 03: file fig03.gif








man walking along the carriage, or of the moving point in the present


section. If we denote the velocity of the light relative to the tube


by W, then this is given by the equation (A) or (B), according as the


Galilei transformation or the Lorentz transformation corresponds to


the facts. Experiment * decides in favour of equation (B) derived


from the theory of relativity, and the agreement is, indeed, very


exact. According to recent and most excellent measurements by Zeeman,


the influence of the velocity of flow v on the propagation of light is


represented by formula (B) to within one per cent.





Nevertheless we must now draw attention to the fact that a theory of


this phenomenon was given by H. A. Lorentz long before the statement


of the theory of relativity. This theory was of a purely


electrodynamical nature, and was obtained by the use of particular


hypotheses as to the electromagnetic structure of matter. This


circumstance, however, does not in the least diminish the


conclusiveness of the experiment as a crucial test in favour of the


theory of relativity, for the electrodynamics of Maxwell-Lorentz, on


which the original theory was based, in no way opposes the theory of


relativity. Rather has the latter been developed trom electrodynamics


as an astoundingly simple combination and generalisation of the


hypotheses, formerly independent of each other, on which


electrodynamics was built.








Notes





*) Fizeau found eq. 10 , where eq. 11





is the index of refraction of the liquid. On the other hand, owing to


the smallness of eq. 12 as compared with I,





we can replace (B) in the first place by eq. 13 , or to the same order


of approximation by





eq. 14 , which agrees with Fizeau%26#039;s result.











THE HEURISTIC VALUE OF THE THEORY OF RELATIVITY








Our train of thought in the foregoing pages can be epitomised in the


following manner. Experience has led to the conviction that, on the


one hand, the principle of relativity holds true and that on the other


hand the velocity of transmission of light in vacuo has to be


considered equal to a constant c. By uniting these two postulates we


obtained the law of transformation for the rectangular co-ordinates x,


y, z and the time t of the events which constitute the processes of


nature. In this connection we did not obtain the Galilei


transformation, but, differing from classical mechanics, the Lorentz


transformation.





The law of transmission of light, the acceptance of which is justified


by our actual knowledge, played an important part in this process of


thought. Once in possession of the Lorentz transformation, however, we


can combine this with the principle of relativity, and sum up the


theory thus:





Every general law of nature must be so constituted that it is


transformed into a law of exactly the same form when, instead of the


space-time variables x, y, z, t of the original coordinate system K,


we introduce new space-time variables x1, y1, z1, t1 of a co-ordinate


system K1. In this connection the relation between the ordinary and


the accented magnitudes is given by the Lorentz transformation. Or in


brief : General laws of nature are co-variant with respect to Lorentz


transformations.





This is a definite mathematical condition that the theory of


relativity demands of a natural law, and in virtue of this, the theory


becomes a valuable heuristic aid in the search for general laws of


nature. If a general law of nature were to be found which did not


satisfy this condition, then at least one of the two fundamental


assumptions of the theory would have been disproved. Let us now


examine what general results the latter theory has hitherto evinced.











GENERAL RESULTS OF THE THEORY








It is clear from our previous considerations that the (special) theory


of relativity has grown out of electrodynamics and optics. In these


fields it has not appreciably altered the predictions of theory, but


it has considerably simplified the theoretical structure, i.e. the


derivation of laws, and -- what is incomparably more important -- it


has considerably reduced the number of independent hypothese forming


the basis of theory. The special theory of relativity has rendered the


Maxwell-Lorentz theory so plausible, that the latter would have been


generally accepted by physicists even if experiment had decided less


unequivocally in its favour.





Classical mechanics required to be modified before it could come into


line with the demands of the special theory of relativity. For the


main part, however, this modification affects only the laws for rapid


motions, in which the velocities of matter v are not very small as


compared with the velocity of light. We have experience of such rapid


motions only in the case of electrons and ions; for other motions the


variations from the laws of classical mechanics are too small to make


themselves evident in practice. We shall not consider the motion of


stars until we come to speak of the general theory of relativity. In


accordance with the theory of relativity the kinetic energy of a


material point of mass m is no longer given by the well-known


expression





eq. 15: file eq15.gif





but by the expression





eq. 16: file eq16.gif








This expression approaches infinity as the velocity v approaches the


velocity of light c. The velocity must therefore always remain less


than c, however great may be the energies used to produce the


acceleration. If we develop the expression for the kinetic energy in


the form of a series, we obtain





eq. 17: file eq17.gif








When eq. 18 is small compared with unity, the third of these terms is


always small in comparison with the second,





which last is alone considered in classical mechanics. The first term


mc^2 does not contain the velocity, and requires no consideration if


we are only dealing with the question as to how the energy of a


point-mass; depends on the velocity. We shall speak of its essential


significance later.





The most important result of a general character to which the special


theory of relativity has led is concerned with the conception of mass.


Before the advent of relativity, physics recognised two conservation


laws of fundamental importance, namely, the law of the canservation of


energy and the law of the conservation of mass these two fundamental


laws appeared to be quite independent of each other. By means of the


theory of relativity they have been united into one law. We shall now


briefly consider how this unification came about, and what meaning is


to be attached to it.





The principle of relativity requires that the law of the concervation


of energy should hold not only with reference to a co-ordinate system


K, but also with respect to every co-ordinate system K1 which is in a


state of uniform motion of translation relative to K, or, briefly,


relative to every %26quot; Galileian %26quot; system of co-ordinates. In contrast to


classical mechanics; the Lorentz transformation is the deciding factor


in the transition from one such system to another.





By means of comparatively simple considerations we are led to draw the


following conclusion from these premises, in conjunction with the


fundamental equations of the electrodynamics of Maxwell: A body moving


with the velocity v, which absorbs * an amount of energy E[0] in


the form of radiation without suffering an alteration in velocity in


the process, has, as a consequence, its energy increased by an amount





eq. 19: file eq19.gif





In consideration of the expression given above for the kinetic energy


of the body, the required energy of the body comes out to be





eq. 20: file eq20.gif








Thus the body has the same energy as a body of mass





eq.21: file eq21.gif





moving with the velocity v. Hence we can say: If a body takes up an


amount of energy E[0], then its inertial mass increases by an amount





eq. 22: file eq22.gif








the inertial mass of a body is not a constant but varies according to


the change in the energy of the body. The inertial mass of a system of


bodies can even be regarded as a measure of its energy. The law of the


conservation of the mass of a system becomes identical with the law of


the conservation of energy, and is only valid provided that the system


neither takes up nor sends out energy. Writing the expression for the


energy in the form





eq. 23: file eq23.gif





we see that the term mc^2, which has hitherto attracted our attention,


is nothing else than the energy possessed by the body ** before it


absorbed the energy E[0].





A direct comparison of this relation with experiment is not possible


at the present time (1920; see *** Note, p. 48), owing to the fact that


the changes in energy E[0] to which we can Subject a system are not


large enough to make themselves perceptible as a change in the


inertial mass of the system.





eq. 22: file eq22.gif








is too small in comparison with the mass m, which was present before


the alteration of the energy. It is owing to this circumstance that


classical mechanics was able to establish successfully the


conservation of mass as a law of independent validity.





Let me add a final remark of a fundamental nature. The success of the


Faraday-Maxwell interpretation of electromagnetic action at a distance


resulted in physicists becoming convinced that there are no such


things as instantaneous actions at a distance (not involving an


intermediary medium) of the type of Newton%26#039;s law of gravitation.


According to the theory of relativity, action at a distance with the


velocity of light always takes the place of instantaneous action at a


distance or of action at a distance with an infinite velocity of


transmission. This is connected with the fact that the velocity c


plays a fundamental role in this theory. In Part II we shall see in


what way this result becomes modified in the general theory of


relativity.








Notes





*) E[0] is the energy taken up, as judged from a co-ordinate system


moving with the body.





**) As judged from a co-ordinate system moving with the body.





***[Note] The equation E = mc^2 has been thoroughly proved time and


again since this time.











EXPERIENCE AND THE SPECIAL THEORY OF RELATIVITY








To what extent is the special theory of relativity supported by


experience? This question is not easily answered for the reason


already mentioned in connection with the fundamental experiment of


Fizeau. The special theory of relativity has crystallised out from the


Maxwell-Lorentz theory of electromagnetic phenomena. Thus all facts of


experience which support the electromagnetic theory also support the


theory of relativity. As being of particular importance, I mention


here the fact that the theory of relativity enables us to predict the


effects produced on the light reaching us from the fixed stars. These


results are obtained in an exceedingly simple manner, and the effects


indicated, which are due to the relative motion of the earth with


reference to those fixed stars are found to be in accord with


experience. We refer to the yearly movement of the apparent position


of the fixed stars resulting from the motion of the earth round the


sun (aberration), and to the influence of the radial components of the


relative motions of the fixed stars with respect to the earth on the


colour of the light reaching us from them. The latter effect manifests


itself in a slight displacement of the spectral lines of the light


transmitted to us from a fixed star, as compared with the position of


the same spectral lines when they are produced by a terrestrial source


of light (Doppler principle). The experimental arguments in favour of


the Maxwell-Lorentz theory, which are at the same time arguments in


favour of the theory of relativity, are too numerous to be set forth


here. In reality they limit the theoretical possibilities to such an


extent, that no other theory than that of Maxwell and Lorentz has been


able to hold its own when tested by experience.





But there are two classes of experimental facts hitherto obtained


which can be represented in the Maxwell-Lorentz theory only by the


introduction of an auxiliary hypothesis, which in itself -- i.e.


without making use of the theory of relativity -- appears extraneous.





It is known that cathode rays and the so-called b-rays emitted by


radioactive substances consist of negatively electrified particles


(electrons) of very small inertia and large velocity. By examining the


deflection of these rays under the influence of electric and magnetic


fields, we can study the law of motion of these particles very


exactly.





In the theoretical treatment of these electrons, we are faced with the


difficulty that electrodynamic theory of itself is unable to give an


account of their nature. For since electrical masses of one sign repel


each other, the negative electrical masses constituting the electron


would necessarily be scattered under the influence of their mutual


repulsions, unless there are forces of another kind operating between


them, the nature of which has hitherto remained obscure to us.* If


we now assume that the relative distances between the electrical


masses constituting the electron remain unchanged during the motion of


the electron (rigid connection in the sense of classical mechanics),


we arrive at a law of motion of the electron which does not agree with


experience. Guided by purely formal points of view, H. A. Lorentz was


the first to introduce the hypothesis that the form of the electron


experiences a contraction in the direction of motion in consequence of


that motion. the contracted length being proportional to the


expression





eq. 05: file eq05.gif





This, hypothesis, which is not justifiable by any electrodynamical


facts, supplies us then with that particular law of motion which has


been confirmed with great precision in recent years.





The theory of relativity leads to the same law of motion, without


requiring any special hypothesis whatsoever as to the structure and


the behaviour of the electron. We arrived at a similar conclusion in


Section 13 in connection with the experiment of Fizeau, the result


of which is foretold by the theory of relativity without the necessity


of drawing on hypotheses as to the physical nature of the liquid.





The second class of facts to which we have alluded has reference to


the question whether or not the motion of the earth in space can be


made perceptible in terrestrial experiments. We have already remarked


in Section 5 that all attempts of this nature led to a negative


result. Before the theory of relativity was put forward, it was


difficult to become reconciled to this negative result, for reasons


now to be discussed. The inherited prejudices about time and space did


not allow any doubt to arise as to the prime importance of the


Galileian transformation for changing over from one body of reference


to another. Now assuming that the Maxwell-Lorentz equations hold for a


reference-body K, we then find that they do not hold for a


reference-body K1 moving uniformly with respect to K, if we assume


that the relations of the Galileian transformstion exist between the


co-ordinates of K and K1. It thus appears that, of all Galileian


co-ordinate systems, one (K) corresponding to a particular state of


motion is physically unique. This result was interpreted physically by


regarding K as at rest with respect to a hypothetical 忙ther of space.


On the other hand, all coordinate systems K1 moving relatively to K


were to be regarded as in motion with respect to the 忙ther. To this


motion of K1 against the 忙ther (%26quot;忙ther-drift %26quot; relative to K1) were


attributed the more complicated laws which were supposed to hold


relative to K1. Strictly speaking, such an 忙ther-drift ought also to


be assumed relative to the earth, and for a long time the efforts of


physicists were devoted to attempts to detect the existence of an


忙ther-drift at the earth%26#039;s surface.





In one of the most notable of these attempts Michelson devised a


method which appears as though it must be decisive. Imagine two


mirrors so arranged on a rigid body that the reflecting surfaces face


each other. A ray of light requires a perfectly definite time T to


pass from one mirror to the other and back again, if the whole system


be at rest with respect to the 忙ther. It is found by calculation,


however, that a slightly different time T1 is required for this


process, if the body, together with the mirrors, be moving relatively


to the 忙ther. And yet another point: it is shown by calculation that


for a given velocity v with reference to the 忙ther, this time T1 is


different when the body is moving perpendicularly to the planes of the


mirrors from that resulting when the motion is parallel to these


planes. Although the estimated difference between these two times is


exceedingly small, Michelson and Morley performed an experiment


involving interference in which this difference should have been


clearly detectable. But the experiment gave a negative result -- a


fact very perplexing to physicists. Lorentz and FitzGerald rescued the


theory from this difficulty by assuming that the motion of the body


relative to the 忙ther produces a contraction of the body in the


direction of motion, the amount of contraction being just sufficient


to compensate for the differeace in time mentioned above. Comparison


with the discussion in Section 11 shows that also from the


standpoint of the theory of relativity this solution of the difficulty


was the right one. But on the basis of the theory of relativity the


method of interpretation is incomparably more satisfactory. According


to this theory there is no such thing as a %26quot; specially favoured %26quot;


(unique) co-ordinate system to occasion the introduction of the


忙ther-idea, and hence there can be no 忙ther-drift, nor any experiment


with which to demonstrate it. Here the contraction of moving bodies


follows from the two fundamental principles of the theory, without the


introduction of particular hypotheses ; and as the prime factor


involved in this contraction we find, not the motion in itself, to


which we cannot attach any meaning, but the motion with respect to the


body of reference chosen in the particular case in point. Thus for a


co-ordinate system moving with the earth the mirror system of


Michelson and Morley is not shortened, but it is shortened for a


co-ordinate system which is at rest relatively to the sun.








Notes





*) The general theory of relativity renders it likely that the


electrical masses of an electron are held together by gravitational


forces.











MINKOWSKI%26#039;S FOUR-DIMENSIONAL SPACE








The non-mathematician is seized by a mysterious shuddering when he


hears of %26quot;four-dimensional%26quot; things, by a feeling not unlike that


awakened by thoughts of the occult. And yet there is no more


common-place statement than that the world in which we live is a


four-dimensional space-time continuum.





Space is a three-dimensional continuum. By this we mean that it is


possible to describe the position of a point (at rest) by means of


three numbers (co-ordinales) x, y, z, and that there is an indefinite


number of points in the neighbourhood of this one, the position of


which can be described by co-ordinates such as x[1], y[1], z[1], which


may be as near as we choose to the respective values of the


co-ordinates x, y, z, of the first point. In virtue of the latter


property we speak of a %26quot; continuum,%26quot; and owing to the fact that there


are three co-ordinates we speak of it as being %26quot; three-dimensional.%26quot;





Similarly, the world of physical phenomena which was briefly called %26quot;


world %26quot; by Minkowski is naturally four dimensional in the space-time


sense. For it is composed of individual events, each of which is


described by four numbers, namely, three space co-ordinates x, y, z,


and a time co-ordinate, the time value t. The%26quot; world%26quot; is in this sense


also a continuum; for to every event there are as many %26quot;neighbouring%26quot;


events (realised or at least thinkable) as we care to choose, the


co-ordinates x[1], y[1], z[1], t[1] of which differ by an indefinitely


small amount from those of the event x, y, z, t originally considered.


That we have not been accustomed to regard the world in this sense as


a four-dimensional continuum is due to the fact that in physics,


before the advent of the theory of relativity, time played a different


and more independent role, as compared with the space coordinates. It


is for this reason that we have been in the habit of treating time as


an independent continuum. As a matter of fact, according to classical


mechanics, time is absolute, i.e. it is independent of the position


and the condition of motion of the system of co-ordinates. We see this


expressed in the last equation of the Galileian transformation (t1 =


t)





The four-dimensional mode of consideration of the %26quot;world%26quot; is natural


on the theory of relativity, since according to this theory time is


robbed of its independence. This is shown by the fourth equation of


the Lorentz transformation:





eq. 24: file eq24.gif








Moreover, according to this equation the time difference Dt1 of two


events with respect to K1 does not in general vanish, even when the


time difference Dt1 of the same events with reference to K vanishes.


Pure %26quot; space-distance %26quot; of two events with respect to K results in %26quot;


time-distance %26quot; of the same events with respect to K. But the


discovery of Minkowski, which was of importance for the formal


development of the theory of relativity, does not lie here. It is to


be found rather in the fact of his recognition that the


four-dimensional space-time continuum of the theory of relativity, in


its most essential formal properties, shows a pronounced relationship


to the three-dimensional continuum of Euclidean geometrical


space.* In order to give due prominence to this relationship,


however, we must replace the usual time co-ordinate t by an imaginary


magnitude eq. 25 proportional to it. Under these conditions, the


natural laws satisfying the demands of the (special) theory of


relativity assume mathematical forms, in which the time co-ordinate


plays exactly the same role as the three space co-ordinates. Formally,


these four co-ordinates correspond exactly to the three space


co-ordinates in Euclidean geometry. It must be clear even to the


non-mathematician that, as a consequence of this purely formal


addition to our knowledge, the theory perforce gained clearness in no


mean measure.





These inadequate remarks can give the reader only a vague notion of


the important idea contributed by Minkowski. Without it the general


theory of relativity, of which the fundamental ideas are developed in


the following pages, would perhaps have got no farther than its long


clothes. Minkowski%26#039;s work is doubtless difficult of access to anyone


inexperienced in mathematics, but since it is not necessary to have a


very exact grasp of this work in order to understand the fundamental


ideas of either the special or the general theory of relativity, I


shall leave it here at present, and revert to it only towards the end


of Part 2.








Notes





*) Cf. the somewhat more detailed discussion in Appendix II.














PART II





THE GENERAL THEORY OF RELATIVITY








SPECIAL AND GENERAL PRINCIPLE OF RELATIVITY








The basal principle, which was the pivot of all our previous


considerations, was the special principle of relativity, i.e. the


principle of the physical relativity of all uniform motion. Let as


once more analyse its meaning carefully.





It was at all times clear that, from the point of view of the idea it


conveys to us, every motion must be considered only as a relative


motion. Returning to the illustration we have frequently used of the


embankment and the railway carriage, we can express the fact of the


motion here taking place in the following two forms, both of which are


equally justifiable :





(a) The carriage is in motion relative to the embankment,


(b) The embankment is in motion relative to the carriage.





In (a) the embankment, in (b) the carriage, serves as the body of


reference in our statement of the motion taking place. If it is simply


a question of detecting or of describing the motion involved, it is in


principle immaterial to what reference-body we refer the motion. As


already mentioned, this is self-evident, but it must not be confused


with the much more comprehensive statement called %26quot;the principle of


relativity,%26quot; which we have taken as the basis of our investigations.





The principle we have made use of not only maintains that we may


equally well choose the carriage or the embankment as our


reference-body for the description of any event (for this, too, is


self-evident). Our principle rather asserts what follows : If we


formulate the general laws of nature as they are obtained from


experience, by making use of





(a) the embankment as reference-body,


(b) the railway carriage as reference-body,





then these general laws of nature (e.g. the laws of mechanics or the


law of the propagation of light in vacuo) have exactly the same form


in both cases. This can also be expressed as follows : For the


physical description of natural processes, neither of the reference


bodies K, K1 is unique (lit. %26quot; specially marked out %26quot;) as compared


with the other. Unlike the first, this latter statement need not of


necessity hold a priori; it is not contained in the conceptions of %26quot;


motion%26quot; and %26quot; reference-body %26quot; and derivable from them; only


experience can decide as to its correctness or incorrectness.





Up to the present, however, we have by no means maintained the


equivalence of all bodies of reference K in connection with the


formulation of natural laws. Our course was more on the following


Iines. In the first place, we started out from the assumption that


there exists a reference-body K, whose condition of motion is such


that the Galileian law holds with respect to it : A particle left to


itself and sufficiently far removed from all other particles moves


uniformly in a straight line. With reference to K (Galileian


reference-body) the laws of nature were to be as simple as possible.


But in addition to K, all bodies of reference K1 should be given


preference in this sense, and they should be exactly equivalent to K


for the formulation of natural laws, provided that they are in a state


of uniform rectilinear and non-rotary motion with respect to K ; all


these bodies of reference are to be regarded as Galileian


reference-bodies. The validity of the principle of relativity was


assumed only for these reference-bodies, but not for others (e.g.


those possessing motion of a different kind). In this sense we speak


of the special principle of relativity, or special theory of


relativity.





In contrast to this we wish to understand by the %26quot;general principle of


relativity%26quot; the following statement : All bodies of reference K, K1,


etc., are equivalent for the description of natural phenomena


(formulation of the general laws of nature), whatever may be their


state of motion. But before proceeding farther, it ought to be pointed


out that this formulation must be replaced later by a more abstract


one, for reasons which will become evident at a later stage.





Since the introduction of the special principle of relativity has been


justified, every intellect which strives after generalisation must


feel the temptation to venture the step towards the general principle


of relativity. But a simple and apparently quite reliable


consideration seems to suggest that, for the present at any rate,


there is little hope of success in such an attempt; Let us imagine


ourselves transferred to our old friend the railway carriage, which is


travelling at a uniform rate. As long as it is moving unifromly, the


occupant of the carriage is not sensible of its motion, and it is for


this reason that he can without reluctance interpret the facts of the


case as indicating that the carriage is at rest, but the embankment in


motion. Moreover, according to the special principle of relativity,


this interpretation is quite justified also from a physical point of


view.





If the motion of the carriage is now changed into a non-uniform


motion, as for instance by a powerful application of the brakes, then


the occupant of the carriage experiences a correspondingly powerful


jerk forwards. The retarded motion is manifested in the mechanical


behaviour of bodies relative to the person in the railway carriage.


The mechanical behaviour is different from that of the case previously


considered, and for this reason it would appear to be impossible that


the same mechanical laws hold relatively to the non-uniformly moving


carriage, as hold with reference to the carriage when at rest or in


uniform motion. At all events it is clear that the Galileian law does


not hold with respect to the non-uniformly moving carriage. Because of


this, we feel compelled at the present juncture to grant a kind of


absolute physical reality to non-uniform motion, in opposition to the


general principle of relatvity. But in what follows we shall soon see


that this conclusion cannot be maintained.











THE GRAVITATIONAL FIELD








%26quot;If we pick up a stone and then let it go, why does it fall to the


ground ?%26quot; The usual answer to this question is: %26quot;Because it is


attracted by the earth.%26quot; Modern physics formulates the answer rather


differently for the following reason. As a result of the more careful


study of electromagnetic phenomena, we have come to regard action at a


distance as a process impossible without the intervention of some


intermediary medium. If, for instance, a magnet attracts a piece of


iron, we cannot be content to regard this as meaning that the magnet


acts directly on the iron through the intermediate empty space, but we


are constrained to imagine -- after the manner of Faraday -- that the


magnet always calls into being something physically real in the space


around it, that something being what we call a %26quot;magnetic field.%26quot; In


its turn this magnetic field operates on the piece of iron, so that


the latter strives to move towards the magnet. We shall not discuss


here the justification for this incidental conception, which is indeed


a somewhat arbitrary one. We shall only mention that with its aid


electromagnetic phenomena can be theoretically represented much more


satisfactorily than without it, and this applies particularly to the


transmission of electromagnetic waves. The effects of gravitation also


are regarded in an analogous manner.





The action of the earth on the stone takes place indirectly. The earth


produces in its surrounding a gravitational field, which acts on the


stone and produces its motion of fall. As we know from experience, the


intensity of the action on a body dimishes according to a quite


definite law, as we proceed farther and farther away from the earth.


From our point of view this means : The law governing the properties


of the gravitational field in space must be a perfectly definite one,


in order correctly to represent the diminution of gravitational action


with the distance from operative bodies. It is something like this:


The body (e.g. the earth) produces a field in its immediate


neighbourhood directly; the intensity and direction of the field at


points farther removed from the body are thence determined by the law


which governs the properties in space of the gravitational fields


themselves.





In contrast to electric and magnetic fields, the gravitational field


exhibits a most remarkable property, which is of fundamental


importance for what follows. Bodies which are moving under the sole


influence of a gravitational field receive an acceleration, which does


not in the least depend either on the material or on the physical


state of the body. For instance, a piece of lead and a piece of wood


fall in exactly the same manner in a gravitational field (in vacuo),


when they start off from rest or with the same initial velocity. This


law, which holds most accurately, can be expressed in a different form


in the light of the following consideration.





According to Newton%26#039;s law of motion, we have





(Force) = (inertial mass) x (acceleration),





where the %26quot;inertial mass%26quot; is a characteristic constant of the


accelerated body. If now gravitation is the cause of the acceleration,


we then have





(Force) = (gravitational mass) x (intensity of the gravitational


field),





where the %26quot;gravitational mass%26quot; is likewise a characteristic constant


for the body. From these two relations follows:





eq. 26: file eq26.gif








If now, as we find from experience, the acceleration is to be


independent of the nature and the condition of the body and always the


same for a given gravitational field, then the ratio of the


gravitational to the inertial mass must likewise be the same for all


bodies. By a suitable choice of units we can thus make this ratio


equal to unity. We then have the following law: The gravitational mass


of a body is equal to its inertial law.





It is true that this important law had hitherto been recorded in


mechanics, but it had not been interpreted. A satisfactory


interpretation can be obtained only if we recognise the following fact


: The same quality of a body manifests itself according to


circumstances as %26quot; inertia %26quot; or as %26quot; weight %26quot; (lit. %26quot; heaviness %26#039;). In


the following section we shall show to what extent this is actually


the case, and how this question is connected with the general


postulate of relativity.














THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS


AS AN ARGUMENT FOR THE GENERAL POSTULE OF RELATIVITY








We imagine a large portion of empty space, so far removed from stars


and other appreciable masses, that we have before us approximately the


conditions required by the fundamental law of Galilei. It is then


possible to choose a Galileian reference-body for this part of space


(world), relative to which points at rest remain at rest and points in


motion continue permanently in uniform rectilinear motion. As


reference-body let us imagine a spacious chest resembling a room with


an observer inside who is equipped with apparatus. Gravitation


naturally does not exist for this observer. He must fasten himself


with strings to the floor, otherwise the slightest impact against the


floor will cause him to rise slowly towards the ceiling of the room.





To the middle of the lid of the chest is fixed externally a hook with


rope attached, and now a %26quot; being %26quot; (what kind of a being is immaterial


to us) begins pulling at this with a constant force. The chest


together with the observer then begin to move %26quot;upwards%26quot; with a


uniformly accelerated motion. In course of time their velocity will


reach unheard-of values -- provided that we are viewing all this from


another reference-body which is not being pulled with a rope.





But how does the man in the chest regard the Process ? The


acceleration of the chest will be transmitted to him by the reaction


of the floor of the chest. He must therefore take up this pressure by


means of his legs if he does not wish to be laid out full length on


the floor. He is then standing in the chest in exactly the same way as


anyone stands in a room of a home on our earth. If he releases a body


which he previously had in his land, the accelertion of the chest will


no longer be transmitted to this body, and for this reason the body


will approach the floor of the chest with an accelerated relative


motion. The observer will further convince himself that the


acceleration of the body towards the floor of the chest is always of


the same magnitude, whatever kind of body he may happen to use for the


experiment.





Relying on his knowledge of the gravitational field (as it was


discussed in the preceding section), the man in the chest will thus


come to the conclusion that he and the chest are in a gravitational


field which is constant with regard to time. Of course he will be


puzzled for a moment as to why the chest does not fall in this


gravitational field. just then, however, he discovers the hook in the


middle of the lid of the chest and the rope which is attached to it,


and he consequently comes to the conclusion that the chest is


suspended at rest in the gravitational field.





Ought we to smile at the man and say that he errs in his conclusion ?


I do not believe we ought to if we wish to remain consistent ; we must


rather admit that his mode of grasping the situation violates neither


reason nor known mechanical laws. Even though it is being accelerated


with respect to the %26quot;Galileian space%26quot; first considered, we can


nevertheless regard the chest as being at rest. We have thus good


grounds for extending the principle of relativity to include bodies of


reference which are accelerated with respect to each other, and as a


result we have gained a powerful argument for a generalised postulate


of relativity.





We must note carefully that the possibility of this mode of


interpretation rests on the fundamental property of the gravitational


field of giving all bodies the same acceleration, or, what comes to


the same thing, on the law of the equality of inertial and


gravitational mass. If this natural law did not exist, the man in the


accelerated chest would not be able to interpret the behaviour of the


bodies around him on the supposition of a gravitational field, and he


would not be justified on the grounds of experience in supposing his


reference-body to be %26quot; at rest.%26quot;





Suppose that the man in the chest fixes a rope to the inner side of


the lid, and that he attaches a body to the free end of the rope. The


result of this will be to strech the rope so that it will hang %26quot;


vertically %26quot; downwards. If we ask for an opinion of the cause of


tension in the rope, the man in the chest will say: %26quot;The suspended


body experiences a downward force in the gravitational field, and this


is neutralised by the tension of the rope ; what determines the


magnitude of the tension of the rope is the gravitational mass of the


suspended body.%26quot; On the other hand, an observer who is poised freely


in space will interpret the condition of things thus : %26quot; The rope must


perforce take part in the accelerated motion of the chest, and it


transmits this motion to the body attached to it. The tension of the


rope is just large enough to effect the acceleration of the body. That


which determines the magnitude of the tension of the rope is the


inertial mass of the body.%26quot; Guided by this example, we see that our


extension of the principle of relativity implies the necessity of the


law of the equality of inertial and gravitational mass. Thus we have


obtained a physical interpretation of this law.





From our consideration of the accelerated chest we see that a general


theory of relativity must yield important results on the laws of


gravitation. In point of fact, the systematic pursuit of the general


idea of relativity has supplied the laws satisfied by the


gravitational field. Before proceeding farther, however, I must warn


the reader against a misconception suggested by these considerations.


A gravitational field exists for the man in the chest, despite the


fact that there was no such field for the co-ordinate system first


chosen. Now we might easily suppose that the existence of a


gravitational field is always only an apparent one. We might also


think that, regardless of the kind of gravitational field which may be


present, we could always choose another reference-body such that no


gravitational field exists with reference to it. This is by no means


true for all gravitational fields, but only for those of quite special


form. It is, for instance, impossible to choose a body of reference


such that, as judged from it, the gravitational field of the earth (in


its entirety) vanishes.





We can now appreciate why that argument is not convincing, which we


brought forward against the general principle of relativity at theend


of Section 18. It is certainly true that the observer in the


railway carriage experiences a jerk forwards as a result of the


application of the brake, and that he recognises, in this the


non-uniformity of motion (retardation) of the carriage. But he is


compelled by nobody to refer this jerk to a %26quot; real %26quot; acceleration


(retardation) of the carriage. He might also interpret his experience


thus: %26quot; My body of reference (the carriage) remains permanently at


rest. With reference to it, however, there exists (during the period


of application of the brakes) a gravitational field which is directed


forwards and which is variable with respect to time. Under the


influence of this field, the embankment together with the earth moves


non-uniformly in such a manner that their original velocity in the


backwards direction is continuously reduced.%26quot;











IN WHAT RESPECTS ARE THE FOUNDATIONS OF CLASSICAL MECHANICS AND OF THE


SPECIAL THEORY OF RELATIVITY UNSATISFACTORY?








We have already stated several times that classical mechanics starts


out from the following law: Material particles sufficiently far


removed from other material particles continue to move uniformly in a


straight line or continue in a state of rest. We have also repeatedly


emphasised that this fundamental law can only be valid for bodies of


reference K which possess certain unique states of motion, and which


are in uniform translational motion relative to each other. Relative


to other reference-bodies K the law is not valid. Both in classical


mechanics and in the special theory of relativity we therefore


differentiate between reference-bodies K relative to which the


recognised %26quot; laws of nature %26quot; can be said to hold, and


reference-bodies K relative to which these laws do not hold.





But no person whose mode of thought is logical can rest satisfied with


this condition of things. He asks : %26quot; How does it come that certain


reference-bodies (or their states of motion) are given priority over


other reference-bodies (or their states of motion) ? What is the


reason for this Preference? In order to show clearly what I mean by


this question, I shall make use of a comparison.





I am standing in front of a gas range. Standing alongside of each


other on the range are two pans so much alike that one may be mistaken


for the other. Both are half full of water. I notice that steam is


being emitted continuously from the one pan, but not from the other. I


am surprised at this, even if I have never seen either a gas range or


a pan before. But if I now notice a luminous something of bluish


colour under the first pan but not under the other, I cease to be


astonished, even if I have never before seen a gas flame. For I can


only say that this bluish something will cause the emission of the


steam, or at least possibly it may do so. If, however, I notice the


bluish something in neither case, and if I observe that the one


continuously emits steam whilst the other does not, then I shall


remain astonished and dissatisfied until I have discovered some


circumstance to which I can attribute the different behaviour of the


two pans.





Analogously, I seek in vain for a real something in classical


mechanics (or in the special theory of relativity) to which I can


attribute the different behaviour of bodies considered with respect to


the reference systems K and K1.* Newton saw this objection and


attempted to invalidate it, but without success. But E. Mach recognsed


it most clearly of all, and because of this objection he claimed that


mechanics must be placed on a new basis. It can only be got rid of by


means of a physics which is conformable to the general principle of


relativity, since the equations of such a theory hold for every body


of reference, whatever may be its state of motion.








Notes





*) The objection is of importance more especially when the state of


motion of the reference-body is of such a nature that it does not


require any external agency for its maintenance, e.g. in the case when


the reference-body is rotating uniformly.











A FEW INFERENCES FROM THE GENERAL PRINCIPLE OF RELATIVITY








The considerations of Section 20 show that the general principle of


relativity puts us in a position to derive properties of the


gravitational field in a purely theoretical manner. Let us suppose,


for instance, that we know the space-time %26quot; course %26quot; for any natural


process whatsoever, as regards the manner in which it takes place in


the Galileian domain relative to a Galileian body of reference K. By


means of purely theoretical operations (i.e. simply by calculation) we


are then able to find how this known natural process appears, as seen


from a reference-body K1 which is accelerated relatively to K. But


since a gravitational field exists with respect to this new body of


reference K1, our consideration also teaches us how the gravitational


field influences the process studied.





For example, we learn that a body which is in a state of uniform


rectilinear motion with respect to K (in accordance with the law of


Galilei) is executing an accelerated and in general curvilinear motion


with respect to the accelerated reference-body K1 (chest). This


acceleration or curvature corresponds to the influence on the moving


body of the gravitational field prevailing relatively to K. It is


known that a gravitational field influences the movement of bodies in


this way, so that our consideration supplies us with nothing


essentially new.





However, we obtain a new result of fundamental importance when we


carry out the analogous consideration for a ray of light. With respect


to the Galileian reference-body K, such a ray of light is transmitted


rectilinearly with the velocity c. It can easily be shown that the


path of the same ray of light is no longer a straight line when we


consider it with reference to the accelerated chest (reference-body


K1). From this we conclude, that, in general, rays of light are


propagated curvilinearly in gravitational fields. In two respects this


result is of great importance.





In the first place, it can be compared with the reality. Although a


detailed examination of the question shows that the curvature of light


rays required by the general theory of relativity is only exceedingly


small for the gravitational fields at our disposal in practice, its


estimated magnitude for light rays passing the sun at grazing


incidence is nevertheless 1.7 seconds of arc. This ought to manifest


itself in the following way. As seen from the earth, certain fixed


stars appear to be in the neighbourhood of the sun, and are thus


capable of observation during a total eclipse of the sun. At such


times, these stars ought to appear to be displaced outwards from the


sun by an amount indicated above, as compared with their apparent


position in the sky when the sun is situated at another part of the


heavens. The examination of the correctness or otherwise of this


deduction is a problem of the greatest importance, the early solution


of which is to be expected of astronomers.[2]*





In the second place our result shows that, according to the general


theory of relativity, the law of the constancy of the velocity of


light in vacuo, which constitutes one of the two fundamental


assumptions in the special theory of relativity and to which we have


already frequently referred, cannot claim any unlimited validity. A


curvature of rays of light can only take place when the velocity of


propagation of light varies with position. Now we might think that as


a consequence of this, the special theory of relativity and with it


the whole theory of relativity would be laid in the dust. But in


reality this is not the case. We can only conclude that the special


theory of relativity cannot claim an unlinlited domain of validity ;


its results hold only so long as we are able to disregard the


influences of gravitational fields on the phenomena (e.g. of light).





Since it has often been contended by opponents of the theory of


relativity that the special theory of relativity is overthrown by the


general theory of relativity, it is perhaps advisable to make the


facts of the case clearer by means of an appropriate comparison.


Before the development of electrodynamics the laws of electrostatics


were looked upon as the laws of electricity. At the present time we


know that electric fields can be derived correctly from electrostatic


considerations only for the case, which is never strictly realised, in


which the electrical masses are quite at rest relatively to each


other, and to the co-ordinate system. Should we be justified in saying


that for this reason electrostatics is overthrown by the


field-equations of Maxwell in electrodynamics ? Not in the least.


Electrostatics is contained in electrodynamics as a limiting case ;


the laws of the latter lead directly to those of the former for the


case in which the fields are invariable with regard to time. No fairer


destiny could be allotted to any physical theory, than that it should


of itself point out the way to the introduction of a more


comprehensive theory, in which it lives on as a limiting case.





In the example of the transmission of light just dealt with, we have


seen that the general theory of relativity enables us to derive


theoretically the influence of a gravitational field on the course of


natural processes, the Iaws of which are already known when a


gravitational field is absent. But the most attractive problem, to the


solution of which the general theory of relativity supplies the key,


concerns the investigation of the laws satisfied by the gravitational


field itself. Let us consider this for a moment.





We are acquainted with space-time domains which behave (approximately)


in a %26quot; Galileian %26quot; fashion under suitable choice of reference-body,


i.e. domains in which gravitational fields are absent. If we now refer


such a domain to a reference-body K1 possessing any kind of motion,


then relative to K1 there exists a gravitational field which is


variable with respect to space and time.[3]** The character of this


field will of course depend on the motion chosen for K1. According to


the general theory of relativity, the general law of the gravitational


field must be satisfied for all gravitational fields obtainable in


this way. Even though by no means all gravitationial fields can be


produced in this way, yet we may entertain the hope that the general


law of gravitation will be derivable from such gravitational fields of


a special kind. This hope has been realised in the most beautiful


manner. But between the clear vision of this goal and its actual


realisation it was necessary to surmount a serious difficulty, and as


this lies deep at the root of things, I dare not withhold it from the


reader. We require to extend our ideas of the space-time continuum


still farther.








Notes





*) By means of the star photographs of two expeditions equipped by


a Joint Committee of the Royal and Royal Astronomical Societies, the


existence of the deflection of light demanded by theory was first


confirmed during the solar eclipse of 29th May, 1919. (Cf. Appendix


III.)





**) This follows from a generalisation of the discussion in


Section 20











BEHAVIOUR OF CLOCKS AND MEASURING-RODS ON A ROTATING BODY OF REFERENCE








Hitherto I have purposely refrained from speaking about the physical


interpretation of space- and time-data in the case of the general


theory of relativity. As a consequence, I am guilty of a certain


slovenliness of treatment, which, as we know from the special theory


of relativity, is far from being unimportant and pardonable. It is now


high time that we remedy this defect; but I would mention at the


outset, that this matter lays no small claims on the patience and on


the power of abstraction of the reader.





We start off again from quite special cases, which we have frequently


used before. Let us consider a space time domain in which no


gravitational field exists relative to a reference-body K whose state


of motion has been suitably chosen. K is then a Galileian


reference-body as regards the domain considered, and the results of


the special theory of relativity hold relative to K. Let us supposse


the same domain referred to a second body of reference K1, which is


rotating uniformly with respect to K. In order to fix our ideas, we


shall imagine K1 to be in the form of a plane circular disc, which


rotates uniformly in its own plane about its centre. An observer who


is sitting eccentrically on the disc K1 is sensible of a force which


acts outwards in a radial direction, and which would be interpreted as


an effect of inertia (centrifugal force) by an observer who was at


rest with respect to the original reference-body K. But the observer


on the disc may regard his disc as a reference-body which is %26quot; at rest


%26quot; ; on the basis of the general principle of relativity he is


justified in doing this. The force acting on himself, and in fact on


all other bodies which are at rest relative to the disc, he regards as


the effect of a gravitational field. Nevertheless, the


space-distribution of this gravitational field is of a kind that would


not be possible on Newton%26#039;s theory of gravitation.* But since the


observer believes in the general theory of relativity, this does not


disturb him; he is quite in the right when he believes that a general


law of gravitation can be formulated- a law which not only explains


the motion of the stars correctly, but also the field of force


experienced by himself.





The observer performs experiments on his circular disc with clocks and


measuring-rods. In doing so, it is his intention to arrive at exact


definitions for the signification of time- and space-data with


reference to the circular disc K1, these definitions being based on


his observations. What will be his experience in this enterprise ?





To start with, he places one of two identically constructed clocks at


the centre of the circular disc, and the other on the edge of the


disc, so that they are at rest relative to it. We now ask ourselves


whether both clocks go at the same rate from the standpoint of the


non-rotating Galileian reference-body K. As judged from this body, the


clock at the centre of the disc has no velocity, whereas the clock at


the edge of the disc is in motion relative to K in consequence of the


rotation. According to a result obtained in Section 12, it follows


that the latter clock goes at a rate permanently slower than that of


the clock at the centre of the circular disc, i.e. as observed from K.


It is obvious that the same effect would be noted by an observer whom


we will imagine sitting alongside his clock at the centre of the


circular disc. Thus on our circular disc, or, to make the case more


general, in every gravitational field, a clock will go more quickly or


less quickly, according to the position in which the clock is situated


(at rest). For this reason it is not possible to obtain a reasonable


definition of time with the aid of clocks which are arranged at rest


with respect to the body of reference. A similar difficulty presents


itself when we attempt to apply our earlier definition of simultaneity


in such a case, but I do not wish to go any farther into this


question.





Moreover, at this stage the definition of the space co-ordinates also


presents insurmountable difficulties. If the observer applies his


standard measuring-rod (a rod which is short as compared with the


radius of the disc) tangentially to the edge of the disc, then, as


judged from the Galileian system, the length of this rod will be less


than I, since, according to Section 12, moving bodies suffer a


shortening in the direction of the motion. On the other hand, the


measaring-rod will not experience a shortening in length, as judged


from K, if it is applied to the disc in the direction of the radius.


If, then, the observer first measures the circumference of the disc


with his measuring-rod and then the diameter of the disc, on dividing


the one by the other, he will not obtain as quotient the familiar


number p = 3.14 . . ., but a larger number,[4]** whereas of course,


for a disc which is at rest with respect to K, this operation would


yield p exactly. This proves that the propositions of Euclidean


geometry cannot hold exactly on the rotating disc, nor in general in a


gravitational field, at least if we attribute the length I to the rod


in all positions and in every orientation. Hence the idea of a


straight line also loses its meaning. We are therefore not in a


position to define exactly the co-ordinates x, y, z relative to the


disc by means of the method used in discussing the special theory, and


as long as the co- ordinates and times of events have not been


defined, we cannot assign an exact meaning to the natural laws in


which these occur.





Thus all our previous conclusions based on general relativity would


appear to be called in question. In reality we must make a subtle


detour in order to be able to apply the postulate of general


relativity exactly. I shall prepare the reader for this in the


following paragraphs.








Notes





*) The field disappears at the centre of the disc and increases


proportionally to the distance from the centre as we proceed outwards.





**) Throughout this consideration we have to use the Galileian


(non-rotating) system K as reference-body, since we may only assume


the validity of the results of the special theory of relativity


relative to K (relative to K1 a gravitational field prevails).











EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM








The surface of a marble table is spread out in front of me. I can get


from any one point on this table to any other point by passing


continuously from one point to a %26quot; neighbouring %26quot; one, and repeating


this process a (large) number of times, or, in other words, by going


from point to point without executing %26quot;jumps.%26quot; I am sure the reader


will appreciate with sufficient clearness what I mean here by %26quot;


neighbouring %26quot; and by %26quot; jumps %26quot; (if he is not too pedantic). We


express this property of the surface by describing the latter as a


continuum.





Let us now imagine that a large number of little rods of equal length


have been made, their lengths being small compared with the dimensions


of the marble slab. When I say they are of equal length, I mean that


one can be laid on any other without the ends overlapping. We next lay


four of these little rods on the marble slab so that they constitute a


quadrilateral figure (a square), the diagonals of which are equally


long. To ensure the equality of the diagonals, we make use of a little


testing-rod. To this square we add similar ones, each of which has one


rod in common with the first. We proceed in like manner with each of


these squares until finally the whole marble slab is laid out with


squares. The arrangement is such, that each side of a square belongs


to two squares and each corner to four squares.





It is a veritable wonder that we can carry out this business without


getting into the greatest difficulties. We only need to think of the


following. If at any moment three squares meet at a corner, then two


sides of the fourth square are already laid, and, as a consequence,


the arrangement of the remaining two sides of the square is already


completely determined. But I am now no longer able to adjust the


quadrilateral so that its diagonals may be equal. If they are equal of


their own accord, then this is an especial favour of the marble slab


and of the little rods, about which I can only be thankfully


surprised. We must experience many such surprises if the construction


is to be successful.





If everything has really gone smoothly, then I say that the points of


the marble slab constitute a Euclidean continuum with respect to the


little rod, which has been used as a %26quot; distance %26quot; (line-interval). By


choosing one corner of a square as %26quot; origin%26quot; I can characterise every


other corner of a square with reference to this origin by means of two


numbers. I only need state how many rods I must pass over when,


starting from the origin, I proceed towards the %26quot; right %26quot; and then %26quot;


upwards,%26quot; in order to arrive at the corner of the square under


consideration. These two numbers are then the %26quot; Cartesian co-ordinates


%26quot; of this corner with reference to the %26quot; Cartesian co-ordinate system%26quot;


which is determined by the arrangement of little rods.





By making use of the following modification of this abstract


experiment, we recognise that there must also be cases in which the


experiment would be unsuccessful. We shall suppose that the rods %26quot;


expand %26quot; by in amount proportional to the increase of temperature. We


heat the central part of the marble slab, but not the periphery, in


which case two of our little rods can still be brought into


coincidence at every position on the table. But our construction of


squares must necessarily come into disorder during the heating,


because the little rods on the central region of the table expand,


whereas those on the outer part do not.





With reference to our little rods -- defined as unit lengths -- the


marble slab is no longer a Euclidean continuum, and we are also no


longer in the position of defining Cartesian co-ordinates directly


with their aid, since the above construction can no longer be carried


out. But since there are other things which are not influenced in a


similar manner to the little rods (or perhaps not at all) by the


temperature of the table, it is possible quite naturally to maintain


the point of view that the marble slab is a %26quot; Euclidean continuum.%26quot;


This can be done in a satisfactory manner by making a more subtle


stipulation about the measurement or the comparison of lengths.





But if rods of every kind (i.e. of every material) were to behave in


the same way as regards the influence of temperature when they are on


the variably heated marble slab, and if we had no other means of


detecting the effect of temperature than the geometrical behaviour of


our rods in experiments analogous to the one described above, then our


best plan would be to assign the distance one to two points on the


slab, provided that the ends of one of our rods could be made to


coincide with these two points ; for how else should we define the


distance without our proceeding being in the highest measure grossly


arbitrary ? The method of Cartesian coordinates must then be


discarded, and replaced by another which does not assume the validity


of Euclidean geometry for rigid bodies.* The reader will notice


that the situation depicted here corresponds to the one brought about


by the general postitlate of relativity (Section 23).








Notes





*) Mathematicians have been confronted with our problem in the


following form. If we are given a surface (e.g. an ellipsoid) in


Euclidean three-dimensional space, then there exists for this surface


a two-dimensional geometry, just as much as for a plane surface. Gauss


undertook the task of treating this two-dimensional geometry from


first principles, without making use of the fact that the surface


belongs to a Euclidean continuum of three dimensions. If we imagine


constructions to be made with rigid rods in the surface (similar to


that above with the marble slab), we should find that different laws


hold for these from those resulting on the basis of Euclidean plane


geometry. The surface is not a Euclidean continuum with respect to the


rods, and we cannot define Cartesian co-ordinates in the surface.


Gauss indicated the principles according to which we can treat the


geometrical relationships in the surface, and thus pointed out the way


to the method of Riemman of treating multi-dimensional, non-Euclidean


continuum. Thus it is that mathematicians long ago solved the formal


problems to which we are led by the general postulate of relativity.











GAUSSIAN CO-ORDINATES








According to Gauss, this combined analytical and geometrical mode of


handling the problem can be arrived at in the following way. We


imagine a system of arbitrary curves (see Fig. 4) drawn on the surface


of the table. These we designate as u-curves, and we indicate each of


them by means of a number. The Curves u= 1, u= 2 and u= 3 are drawn in


the diagram. Between the curves u= 1 and u= 2 we must imagine an


infinitely large number to be drawn, all of which correspond to real


numbers lying between 1 and 2. fig. 04 We have then a system of


u-curves, and this %26quot;infinitely dense%26quot; system covers the whole surface


of the table. These u-curves must not intersect each other, and


through each point of the surface one and only one curve must pass.


Thus a perfectly definite value of u belongs to every point on the


surface of the marble slab. In like manner we imagine a system of


v-curves drawn on the surface. These satisfy the same conditions as


the u-curves, they are provided with numbers in a corresponding


manner, and they may likewise be of arbitrary shape. It follows that a


value of u and a value of v belong to every point on the surface of


the table. We call these two numbers the co-ordinates of the surface


of the table (Gaussian co-ordinates). For example, the point P in the


diagram has the Gaussian co-ordinates u= 3, v= 1. Two neighbouring


points P and P1 on the surface then correspond to the co-ordinates





P: u,v





P1: u + du, v + dv,





where du and dv signify very small numbers. In a similar manner we may


indicate the distance (line-interval) between P and P1, as measured


with a little rod, by means of the very small number ds. Then


according to Gauss we have





ds2 = g[11]du2 + 2g[12]dudv = g[22]dv2





where g[11], g[12], g[22], are magnitudes which depend in a perfectly


definite way on u and v. The magnitudes g[11], g[12] and g[22],


determine the behaviour of the rods relative to the u-curves and


v-curves, and thus also relative to the surface of the table. For the


case in which the points of the surface considered form a Euclidean


continuum with reference to the measuring-rods, but only in this case,


it is possible to draw the u-curves and v-curves and to attach numbers


to them, in such a manner, that we simply have :





ds2 = du2 + dv2





Under these conditions, the u-curves and v-curves are straight lines


in the sense of Euclidean geometry, and they are perpendicular to each


other. Here the Gaussian coordinates are samply Cartesian ones. It is


clear that Gauss co-ordinates are nothing more than an association of


two sets of numbers with the points of the surface considered, of such


a nature that numerical values differing very slightly from each other


are associated with neighbouring points %26quot; in space.%26quot;





So far, these considerations hold for a continuum of two dimensions.


But the Gaussian method can be applied also to a continuum of three,


four or more dimensions. If, for instance, a continuum of four


dimensions be supposed available, we may represent it in the following


way. With every point of the continuum, we associate arbitrarily four


numbers, x[1], x[2], x[3], x[4], which are known as %26quot; co-ordinates.%26quot;


Adjacent points correspond to adjacent values of the coordinates. If a


distance ds is associated with the adjacent points P and P1, this


distance being measurable and well defined from a physical point of


view, then the following formula holds:





ds2 = g[11]dx[1]^2 + 2g[12]dx[1]dx[2] . . . . g[44]dx[4]^2,





where the magnitudes g[11], etc., have values which vary with the


position in the continuum. Only when the continuum is a Euclidean one


is it possible to associate the co-ordinates x[1] . . x[4]. with the


points of the continuum so that we have simply





ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.





In this case relations hold in the four-dimensional continuum which


are analogous to those holding in our three-dimensional measurements.





However, the Gauss treatment for ds2 which we have given above is not


always possible. It is only possible when sufficiently small regions


of the continuum under consideration may be regarded as Euclidean


continua. For example, this obviously holds in the case of the marble


slab of the table and local variation of temperature. The temperature


is practically constant for a small part of the slab, and thus the


geometrical behaviour of the rods is almost as it ought to be


according to the rules of Euclidean geometry. Hence the imperfections


of the construction of squares in the previous section do not show


themselves clearly until this construction is extended over a


considerable portion of the surface of the table.





We can sum this up as follows: Gauss invented a method for the


mathematical treatment of continua in general, in which %26quot;


size-relations %26quot; (%26quot; distances %26quot; between neighbouring points) are


defined. To every point of a continuum are assigned as many numbers


(Gaussian coordinates) as the continuum has dimensions. This is done


in such a way, that only one meaning can be attached to the


assignment, and that numbers (Gaussian coordinates) which differ by an


indefinitely small amount are assigned to adjacent points. The


Gaussian coordinate system is a logical generalisation of the


Cartesian co-ordinate system. It is also applicable to non-Euclidean


continua, but only when, with respect to the defined %26quot;size%26quot; or


%26quot;distance,%26quot; small parts of the continuum under consideration behave


more nearly like a Euclidean system, the smaller the part of the


continuum under our notice.











THE SPACE-TIME CONTINUUM OF THE SPEICAL THEORY OF RELATIVITY CONSIDERED AS A


EUCLIDEAN CONTINUUM








We are now in a position to formulate more exactly the idea of


Minkowski, which was only vaguely indicated in Section 17. In


accordance with the special theory of relativity, certain co-ordinate


systems are given preference for the description of the


four-dimensional, space-time continuum. We called these %26quot; Galileian


co-ordinate systems.%26quot; For these systems, the four co-ordinates x, y,


z, t, which determine an event or -- in other words, a point of the


four-dimensional continuum -- are defined physically in a simple


manner, as set forth in detail in the first part of this book. For the


transition from one Galileian system to another, which is moving


uniformly with reference to the first, the equations of the Lorentz


transformation are valid. These last form the basis for the derivation


of deductions from the special theory of relativity, and in themselves


they are nothing more than the expression of the universal validity of


the law of transmission of light for all Galileian systems of


reference.





Minkowski found that the Lorentz transformations satisfy the following


simple conditions. Let us consider two neighbouring events, the


relative position of which in the four-dimensional continuum is given


with respect to a Galileian reference-body K by the space co-ordinate


differences dx, dy, dz and the time-difference dt. With reference to a


second Galileian system we shall suppose that the corresponding


differences for these two events are dx1, dy1, dz1, dt1. Then these


magnitudes always fulfil the condition*





dx2 + dy2 + dz2 - c^2dt2 = dx1 2 + dy1 2 + dz1 2 - c^2dt1 2.





The validity of the Lorentz transformation follows from this


condition. We can express this as follows: The magnitude





ds2 = dx2 + dy2 + dz2 - c^2dt2,





which belongs to two adjacent points of the four-dimensional


space-time continuum, has the same value for all selected (Galileian)


reference-bodies. If we replace x, y, z, sq. rt. -I . ct , by x[1],


x[2], x[3], x[4], we also obtaill the result that





ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.





is independent of the choice of the body of reference. We call the


magnitude ds the %26quot; distance %26quot; apart of the two events or


four-dimensional points.





Thus, if we choose as time-variable the imaginary variable sq. rt. -I


. ct instead of the real quantity t, we can regard the space-time


contintium -- accordance with the special theory of relativity -- as a


%26quot;, Euclidean %26quot; four-dimensional continuum, a result which follows from


the considerations of the preceding section.








Notes





*) Cf. Appendixes I and 2. The relations which are derived


there for the co-ordlnates themselves are valid also for co-ordinate


differences, and thus also for co-ordinate differentials (indefinitely


small differences).











THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF REALTIIVTY IS NOT A


ECULIDEAN CONTINUUM








In the first part of this book we were able to make use of space-time


co-ordinates which allowed of a simple and direct physical


interpretation, and which, according to Section 26, can be regarded


as four-dimensional Cartesian co-ordinates. This was possible on the


basis of the law of the constancy of the velocity of tight. But


according to Section 21 the general theory of relativity cannot


retain this law. On the contrary, we arrived at the result that


according to this latter theory the velocity of light must always


depend on the co-ordinates when a gravitational field is present. In


connection with a specific illustration in Section 23, we found


that the presence of a gravitational field invalidates the definition


of the coordinates and the ifine, which led us to our objective in the


special theory of relativity.





In view of the resuIts of these considerations we are led to the


conviction that, according to the general principle of relativity, the


space-time continuum cannot be regarded as a Euclidean one, but that


here we have the general case, corresponding to the marble slab with


local variations of temperature, and with which we made acquaintance


as an example of a two-dimensional continuum. Just as it was there


impossible to construct a Cartesian co-ordinate system from equal


rods, so here it is impossible to build up a system (reference-body)


from rigid bodies and clocks, which shall be of such a nature that


measuring-rods and clocks, arranged rigidly with respect to one


another, shaIll indicate position and time directly. Such was the


essence of the difficulty with which we were confronted in Section


23.





But the considerations of Sections 25 and 26 show us the way to


surmount this difficulty. We refer the fourdimensional space-time


continuum in an arbitrary manner to Gauss co-ordinates. We assign to


every point of the continuum (event) four numbers, x[1], x[2], x[3],


x[4] (co-ordinates), which have not the least direct physical


significance, but only serve the purpose of numbering the points of


the continuum in a definite but arbitrary manner. This arrangement


does not even need to be of such a kind that we must regard x[1],


x[2], x[3], as %26quot;space%26quot; co-ordinates and x[4], as a %26quot; time %26quot;


co-ordinate.





The reader may think that such a description of the world would be


quite inadequate. What does it mean to assign to an event the


particular co-ordinates x[1], x[2], x[3], x[4], if in themselves these


co-ordinates have no significance ? More careful consideration shows,


however, that this anxiety is unfounded. Let us consider, for


instance, a material point with any kind of motion. If this point had


only a momentary existence without duration, then it would to


described in space-time by a single system of values x[1], x[2], x[3],


x[4]. Thus its permanent existence must be characterised by an


infinitely large number of such systems of values, the co-ordinate


values of which are so close together as to give continuity;


corresponding to the material point, we thus have a (uni-dimensional)


line in the four-dimensional continuum. In the same way, any such


lines in our continuum correspond to many points in motion. The only


statements having regard to these points which can claim a physical


existence are in reality the statements about their encounters. In our


mathematical treatment, such an encounter is expressed in the fact


that the two lines which represent the motions of the points in


question have a particular system of co-ordinate values, x[1], x[2],


x[3], x[4], in common. After mature consideration the reader will


doubtless admit that in reality such encounters constitute the only


actual evidence of a time-space nature with which we meet in physical


statements.





When we were describing the motion of a material point relative to a


body of reference, we stated nothing more than the encounters of this


point with particular points of the reference-body. We can also


determine the corresponding values of the time by the observation of


encounters of the body with clocks, in conjunction with the


observation of the encounter of the hands of clocks with particular


points on the dials. It is just the same in the case of


space-measurements by means of measuring-rods, as a litttle


consideration will show.





The following statements hold generally : Every physical description


resolves itself into a number of statements, each of which refers to


the space-time coincidence of two events A and B. In terms of Gaussian


co-ordinates, every such statement is expressed by the agreement of


their four co-ordinates x[1], x[2], x[3], x[4]. Thus in reality, the


description of the time-space continuum by means of Gauss co-ordinates


completely replaces the description with the aid of a body of


reference, without suffering from the defects of the latter mode of


description; it is not tied down to the Euclidean character of the


continuum which has to be represented.











EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY








We are now in a position to replace the pro. visional formulation of


the general principle of relativity given in Section 18 by an exact


formulation. The form there used, %26quot;All bodies of reference K, K1,


etc., are equivalent for the description of natural phenomena


(formulation of the general laws of nature), whatever may be their


state of motion,%26quot; cannot be maintained, because the use of rigid


reference-bodies, in the sense of the method followed in the special


theory of relativity, is in general not possible in space-time


description. The Gauss co-ordinate system has to take the place of the


body of reference. The following statement corresponds to the


fundamental idea of the general principle of relativity: %26quot;All Gaussian


co-ordinate systems are essentially equivalent for the formulation of


the general laws of nature.%26quot;





We can state this general principle of relativity in still another


form, which renders it yet more clearly intelligible than it is when


in the form of the natural extension of the special principle of


relativity. According to the special theory of relativity, the


equations which express the general laws of nature pass over into


equations of the same form when, by making use of the Lorentz


transformation, we replace the space-time variables x, y, z, t, of a


(Galileian) reference-body K by the space-time variables x1, y1, z1,


t1, of a new reference-body K1. According to the general theory of


relativity, on the other hand, by application of arbitrary


substitutions of the Gauss variables x[1], x[2], x[3], x[4], the


equations must pass over into equations of the same form; for every


transformation (not only the Lorentz transformation) corresponds to


the transition of one Gauss co-ordinate system into another.





If we desire to adhere to our %26quot;old-time%26quot; three-dimensional view of


things, then we can characterise the development which is being


undergone by the fundamental idea of the general theory of relativity


as follows : The special theory of relativity has reference to


Galileian domains, i.e. to those in which no gravitational field


exists. In this connection a Galileian reference-body serves as body


of reference, i.e. a rigid body the state of motion of which is so


chosen that the Galileian law of the uniform rectilinear motion of


%26quot;isolated%26quot; material points holds relatively to it.





Certain considerations suggest that we should refer the same Galileian


domains to non-Galileian reference-bodies also. A gravitational field


of a special kind is then present with respect to these bodies (cf.


Sections 20 and 23).





In gravitational fields there are no such things as rigid bodies with


Euclidean properties; thus the fictitious rigid body of reference is


of no avail in the general theory of relativity. The motion of clocks


is also influenced by gravitational fields, and in such a way that a


physical definition of time which is made directly with the aid of


clocks has by no means the same degree of plausibility as in the


special theory of relativity.





For this reason non-rigid reference-bodies are used, which are as a


whole not only moving in any way whatsoever, but which also suffer


alterations in form ad lib. during their motion. Clocks, for which the


law of motion is of any kind, however irregular, serve for the


definition of time. We have to imagine each of these clocks fixed at a


point on the non-rigid reference-body. These clocks satisfy only the


one condition, that the %26quot;readings%26quot; which are observed simultaneously


on adjacent clocks (in space) differ from each other by an


indefinitely small amount. This non-rigid reference-body, which might


appropriately be termed a %26quot;reference-mollusc%26quot;, is in the main


equivalent to a Gaussian four-dimensional co-ordinate system chosen


arbitrarily. That which gives the %26quot;mollusc%26quot; a certain


comprehensibility as compared with the Gauss co-ordinate system is the


(really unjustified) formal retention of the separate existence of the


space co-ordinates as opposed to the time co-ordinate. Every point on


the mollusc is treated as a space-point, and every material point


which is at rest relatively to it as at rest, so long as the mollusc


is considered as reference-body. The general principle of relativity


requires that all these molluscs can be used as reference-bodies with


equal right and equal success in the formulation of the general laws


of nature; the laws themselves must be quite independent of the choice


of mollusc.





The great power possessed by the general principle of relativity lies


in the comprehensive limitation which is imposed on the laws of nature


in consequence of what we have seen above.











THE SOLUTION OF THE PROBLEM OF GRAVITATION ON THE BASIS OF THE GENERAL


PRINCIPLE OF RELATIVITY








If the reader has followed all our previous considerations, he will


have no further difficulty in understanding the methods leading to the


solution of the problem of gravitation.





We start off on a consideration of a Galileian domain, i.e. a domain


in which there is no gravitational field relative to the Galileian


reference-body K. The behaviour of measuring-rods and clocks with


reference to K is known from the special theory of relativity,


likewise the behaviour of %26quot;isolated%26quot; material points; the latter move


uniformly and in straight lines.





Now let us refer this domain to a random Gauss coordinate system or to


a %26quot;mollusc%26quot; as reference-body K1. Then with respect to K1 there is a


gravitational field G (of a particular kind). We learn the behaviour


of measuring-rods and clocks and also of freely-moving material points


with reference to K1 simply by mathematical transformation. We


interpret this behaviour as the behaviour of measuring-rods, docks and


material points tinder the influence of the gravitational field G.


Hereupon we introduce a hypothesis: that the influence of the


gravitational field on measuringrods, clocks and freely-moving


material points continues to take place according to the same laws,


even in the case where the prevailing gravitational field is not


derivable from the Galfleian special care, simply by means of a


transformation of co-ordinates.





The next step is to investigate the space-time behaviour of the


gravitational field G, which was derived from the Galileian special


case simply by transformation of the coordinates. This behaviour is


formulated in a law, which is always valid, no matter how the


reference-body (mollusc) used in the description may be chosen.





This law is not yet the general law of the gravitational field, since


the gravitational field under consideration is of a special kind. In


order to find out the general law-of-field of gravitation we still


require to obtain a generalisation of the law as found above. This can


be obtained without caprice, however, by taking into consideration the


following demands:





(a) The required generalisation must likewise satisfy the general


postulate of relativity.





(b) If there is any matter in the domain under consideration, only its


inertial mass, and thus according to Section 15 only its energy is


of importance for its etfect in exciting a field.





(c) Gravitational field and matter together must satisfy the law of


the conservation of energy (and of impulse).





Finally, the general principle of relativity permits us to determine


the influence of the gravitational field on the course of all those


processes which take place according to known laws when a


gravitational field is absent i.e. which have already been fitted into


the frame of the special theory of relativity. In this connection we


proceed in principle according to the method which has already been


explained for measuring-rods, clocks and freely moving material


points.





The theory of gravitation derived in this way from the general


postulate of relativity excels not only in its beauty ; nor in


removing the defect attaching to classical mechanics which was brought


to light in Section 21; nor in interpreting the empirical law of


the equality of inertial and gravitational mass ; but it has also


already explained a result of observation in astronomy, against which


classical mechanics is powerless.





If we confine the application of the theory to the case where the


gravitational fields can be regarded as being weak, and in which all


masses move with respect to the coordinate system with velocities


which are small compared with the velocity of light, we then obtain as


a first approximation the Newtonian theory. Thus the latter theory is


obtained here without any particular assumption, whereas Newton had to


introduce the hypothesis that the force of attraction between mutually


attracting material points is inversely proportional to the square of


the distance between them. If we increase the accuracy of the


calculation, deviations from the theory of Newton make their


appearance, practically all of which must nevertheless escape the test


of observation owing to their smallness.





We must draw attention here to one of these deviations. According to


Newton%26#039;s theory, a planet moves round the sun in an ellipse, which


would permanently maintain its position with respect to the fixed


stars, if we could disregard the motion of the fixed stars themselves


and the action of the other planets under consideration. Thus, if we


correct the observed motion of the planets for these two influences,


and if Newton%26#039;s theory be strictly correct, we ought to obtain for the


orbit of the planet an ellipse, which is fixed with reference to the


fixed stars. This deduction, which can be tested with great accuracy,


has been confirmed for all the planets save one, with the precision


that is capable of being obtained by the delicacy of observation


attainable at the present time. The sole exception is Mercury, the


planet which lies nearest the sun. Since the time of Leverrier, it has


been known that the ellipse corresponding to the orbit of Mercury,


after it has been corrected for the influences mentioned above, is not


stationary with respect to the fixed stars, but that it rotates


exceedingly slowly in the plane of the orbit and in the sense of the


orbital motion. The value obtained for this rotary movement of the


orbital ellipse was 43 seconds of arc per century, an amount ensured


to be correct to within a few seconds of arc. This effect can be


explained by means of classical mechanics only on the assumption of


hypotheses which have little probability, and which were devised


solely for this purponse.





On the basis of the general theory of relativity, it is found that the


ellipse of every planet round the sun must necessarily rotate in the


manner indicated above ; that for all the planets, with the exception


of Mercury, this rotation is too small to be detected with the


delicacy of observation possible at the present time ; but that in the


case of Mercury it must amount to 43 seconds of arc per century, a


result which is strictly in agreement with observation.





Apart from this one, it has hitherto been possible to make only two


deductions from the theory which admit of being tested by observation,


to wit, the curvature of light rays by the gravitational field of the


sun,*x and a displacement of the spectral lines of light reaching


us from large stars, as compared with the corresponding lines for


light produced in an analogous manner terrestrially (i.e. by the same


kind of atom).** These two deductions from the theory have both


been confirmed.








Notes





*) First observed by Eddington and others in 1919. (Cf. Appendix


III, pp. 126-129).





**) Established by Adams in 1924. (Cf. p. 132)














PART III





CONSIDERATIONS ON THE UNIVERSE AS A WHOLE








COSMOLOGICAL DIFFICULTIES OF NEWTON%26#039;S THEORY








Part from the difficulty discussed in Section 21, there is a second


fundamental difficulty attending classical celestial mechanics, which,


to the best of my knowledge, was first discussed in detail by the


astronomer Seeliger. If we ponder over the question as to how the


universe, considered as a whole, is to be regarded, the first answer


that suggests itself to us is surely this: As regards space (and time)


the universe is infinite. There are stars everywhere, so that the


density of matter, although very variable in detail, is nevertheless


on the average everywhere the same. In other words: However far we


might travel through space, we should find everywhere an attenuated


swarm of fixed stars of approrimately the same kind and density.





This view is not in harmony with the theory of Newton. The latter


theory rather requires that the universe should have a kind of centre


in which the density of the stars is a maximum, and that as we proceed


outwards from this centre the group-density of the stars should


diminish, until finally, at great distances, it is succeeded by an


infinite region of emptiness. The stellar universe ought to be a


finite island in the infinite ocean of space.*





This conception is in itself not very satisfactory. It is still less


satisfactory because it leads to the result that the light emitted by


the stars and also individual stars of the stellar system are


perpetually passing out into infinite space, never to return, and


without ever again coming into interaction with other objects of


nature. Such a finite material universe would be destined to become


gradually but systematically impoverished.





In order to escape this dilemma, Seeliger suggested a modification of


Newton%26#039;s law, in which he assumes that for great distances the force


of attraction between two masses diminishes more rapidly than would


result from the inverse square law. In this way it is possible for the


mean density of matter to be constant everywhere, even to infinity,


without infinitely large gravitational fields being produced. We thus


free ourselves from the distasteful conception that the material


universe ought to possess something of the nature of a centre. Of


course we purchase our emancipation from the fundamental difficulties


mentioned, at the cost of a modification and complication of Newton%26#039;s


law which has neither empirical nor theoretical foundation. We can


imagine innumerable laws which would serve the same purpose, without


our being able to state a reason why one of them is to be preferred to


the others ; for any one of these laws would be founded just as little


on more general theoretical principles as is the law of Newton.








Notes





*) Proof -- According to the theory of Newton, the number of %26quot;lines


of force%26quot; which come from infinity and terminate in a mass m is


proportional to the mass m. If, on the average, the Mass density p[0]


is constant throughout tithe universe, then a sphere of volume V will


enclose the average man p[0]V. Thus the number of lines of force


passing through the surface F of the sphere into its interior is


proportional to p[0] V. For unit area of the surface of the sphere the


number of lines of force which enters the sphere is thus proportional


to p[0] V/F or to p[0]R. Hence the intensity of the field at the


surface would ultimately become infinite with increasing radius R of


the sphere, which is impossible.











THE POSSIBILITY OF A %26quot;FINITE%26quot; AND YET %26quot;UNBOUNDED%26quot; UNIVERSE








But speculations on the structure of the universe also move in quite


another direction. The development of non-Euclidean geometry led to


the recognition of the fact, that we can cast doubt on the


infiniteness of our space without coming into conflict with the laws


of thought or with experience (Riemann, Helmholtz). These questions


have already been treated in detail and with unsurpassable lucidity by


Helmholtz and Poincar茅, whereas I can only touch on them briefly here.





In the first place, we imagine an existence in two dimensional space.


Flat beings with flat implements, and in particular flat rigid


measuring-rods, are free to move in a plane. For them nothing exists


outside of this plane: that which they observe to happen to themselves


and to their flat %26quot; things %26quot; is the all-inclusive reality of their


plane. In particular, the constructions of plane Euclidean geometry


can be carried out by means of the rods e.g. the lattice construction,


considered in Section 24. In contrast to ours, the universe of


these beings is two-dimensional; but, like ours, it extends to


infinity. In their universe there is room for an infinite number of


identical squares made up of rods, i.e. its volume (surface) is


infinite. If these beings say their universe is %26quot; plane,%26quot; there is


sense in the statement, because they mean that they can perform the


constructions of plane Euclidean geometry with their rods. In this


connection the individual rods always represent the same distance,


independently of their position.





Let us consider now a second two-dimensional existence, but this time


on a spherical surface instead of on a plane. The flat beings with


their measuring-rods and other objects fit exactly on this surface and


they are unable to leave it. Their whole universe of observation


extends exclusively over the surface of the sphere. Are these beings


able to regard the geometry of their universe as being plane geometry


and their rods withal as the realisation of %26quot; distance %26quot; ? They cannot


do this. For if they attempt to realise a straight line, they will


obtain a curve, which we %26quot; three-dimensional beings %26quot; designate as a


great circle, i.e. a self-contained line of definite finite length,


which can be measured up by means of a measuring-rod. Similarly, this


universe has a finite area that can be compared with the area, of a


square constructed with rods. The great charm resulting from this


consideration lies in the recognition of the fact that the universe of


these beings is finite and yet has no limits.





But the spherical-surface beings do not need to go on a world-tour in


order to perceive that they are not living in a Euclidean universe.


They can convince themselves of this on every part of their %26quot; world,%26quot;


provided they do not use too small a piece of it. Starting from a


point, they draw %26quot; straight lines %26quot; (arcs of circles as judged in


three dimensional space) of equal length in all directions. They will


call the line joining the free ends of these lines a %26quot; circle.%26quot; For a


plane surface, the ratio of the circumference of a circle to its


diameter, both lengths being measured with the same rod, is, according


to Euclidean geometry of the plane, equal to a constant value p, which


is independent of the diameter of the circle. On their spherical


surface our flat beings would find for this ratio the value





eq. 27: file eq27.gif





i.e. a smaller value than p, the difference being the more


considerable, the greater is the radius of the circle in comparison


with the radius R of the %26quot; world-sphere.%26quot; By means of this relation


the spherical beings can determine the radius of their universe (%26quot;


world %26quot;), even when only a relatively small part of their worldsphere


is available for their measurements. But if this part is very small


indeed, they will no longer be able to demonstrate that they are on a


spherical %26quot; world %26quot; and not on a Euclidean plane, for a small part of


a spherical surface differs only slightly from a piece of a plane of


the same size.





Thus if the spherical surface beings are living on a planet of which


the solar system occupies only a negligibly small part of the


spherical universe, they have no means of determining whether they are


living in a finite or in an infinite universe, because the %26quot; piece of


universe %26quot; to which they have access is in both cases practically


plane, or Euclidean. It follows directly from this discussion, that


for our sphere-beings the circumference of a circle first increases


with the radius until the %26quot; circumference of the universe %26quot; is


reached, and that it thenceforward gradually decreases to zero for


still further increasing values of the radius. During this process the


area of the circle continues to increase more and more, until finally


it becomes equal to the total area of the whole %26quot; world-sphere.%26quot;





Perhaps the reader will wonder why we have placed our %26quot; beings %26quot; on a


sphere rather than on another closed surface. But this choice has its


justification in the fact that, of all closed surfaces, the sphere is


unique in possessing the property that all points on it are


equivalent. I admit that the ratio of the circumference c of a circle


to its radius r depends on r, but for a given value of r it is the


same for all points of the %26quot; worldsphere %26quot;; in other words, the %26quot;


world-sphere %26quot; is a %26quot; surface of constant curvature.%26quot;





To this two-dimensional sphere-universe there is a three-dimensional


analogy, namely, the three-dimensional spherical space which was


discovered by Riemann. its points are likewise all equivalent. It


possesses a finite volume, which is determined by its %26quot;radius%26quot;


(2p2R3). Is it possible to imagine a spherical space? To imagine a


space means nothing else than that we imagine an epitome of our %26quot;


space %26quot; experience, i.e. of experience that we can have in the


movement of %26quot; rigid %26quot; bodies. In this sense we can imagine a spherical


space.





Suppose we draw lines or stretch strings in all directions from a


point, and mark off from each of these the distance r with a


measuring-rod. All the free end-points of these lengths lie on a


spherical surface. We can specially measure up the area (F) of this


surface by means of a square made up of measuring-rods. If the


universe is Euclidean, then F = 4pR2 ; if it is spherical, then F is


always less than 4pR2. With increasing values of r, F increases from


zero up to a maximum value which is determined by the %26quot; world-radius,%26quot;


but for still further increasing values of r, the area gradually


diminishes to zero. At first, the straight lines which radiate from


the starting point diverge farther and farther from one another, but


later they approach each other, and finally they run together again at


a %26quot;counter-point%26quot; to the starting point. Under such conditions they


have traversed the whole spherical space. It is easily seen that the


three-dimensional spherical space is quite analogous to the


two-dimensional spherical surface. It is finite (i.e. of finite


volume), and has no bounds.





It may be mentioned that there is yet another kind of curved space: %26quot;


elliptical space.%26quot; It can be regarded as a curved space in which the


two %26quot; counter-points %26quot; are identical (indistinguishable from each


other). An elliptical universe can thus be considered to some extent


as a curved universe possessing central symmetry.





It follows from what has been said, that closed spaces without limits


are conceivable. From amongst these, the spherical space (and the


elliptical) excels in its simplicity, since all points on it are


equivalent. As a result of this discussion, a most interesting


question arises for astronomers and physicists, and that is whether


the universe in which we live is infinite, or whether it is finite in


the manner of the spherical universe. Our experience is far from being


sufficient to enable us to answer this question. But the general


theory of relativity permits of our answering it with a moduate degree


of certainty, and in this connection the difficulty mentioned in


Section 30 finds its solution.











THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY








According to the general theory of relativity, the geometrical


properties of space are not independent, but they are determined by


matter. Thus we can draw conclusions about the geometrical structure


of the universe only if we base our considerations on the state of the


matter as being something that is known. We know from experience that,


for a suitably chosen co-ordinate system, the velocities of the stars


are small as compared with the velocity of transmission of light. We


can thus as a rough approximation arrive at a conclusion as to the


nature of the universe as a whole, if we treat the matter as being at


rest.





We already know from our previous discussion that the behaviour of


measuring-rods and clocks is influenced by gravitational fields, i.e.


by the distribution of matter. This in itself is sufficient to exclude


the possibility of the exact validity of Euclidean geometry in our


universe. But it is conceivable that our universe differs only


slightly from a Euclidean one, and this notion seems all the more


probable, since calculations show that the metrics of surrounding


space is influenced only to an exceedingly small extent by masses even


of the magnitude of our sun. We might imagine that, as regards


geometry, our universe behaves analogously to a surface which is


irregularly curved in its individual parts, but which nowhere departs


appreciably from a plane: something like the rippled surface of a


lake. Such a universe might fittingly be called a quasi-Euclidean


universe. As regards its space it would be infinite. But calculation


shows that in a quasi-Euclidean universe the average density of matter


would necessarily be nil. Thus such a universe could not be inhabited


by matter everywhere ; it would present to us that unsatisfactory


picture which we portrayed in Section 30.





If we are to have in the universe an average density of matter which


differs from zero, however small may be that difference, then the


universe cannot be quasi-Euclidean. On the contrary, the results of


calculation indicate that if matter be distributed uniformly, the


universe would necessarily be spherical (or elliptical). Since in


reality the detailed distribution of matter is not uniform, the real


universe will deviate in individual parts from the spherical, i.e. the


universe will be quasi-spherical. But it will be necessarily finite.


In fact, the theory supplies us with a simple connection * between


the space-expanse of the universe and the average density of matter in


it.








Notes





*) For the radius R of the universe we obtain the equation





eq. 28: file eq28.gif





The use of the C.G.S. system in this equation gives 2/k = 1^.08.10^27;


p is the average density of the matter and k is a constant connected


with the Newtonian constant of gravitation.











APPENDIX I





SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION


(SUPPLEMENTARY TO SECTION 11)








For the relative orientation of the co-ordinate systems indicated in


Fig. 2, the x-axes of both systems pernumently coincide. In the


present case we can divide the problem into parts by considering first


only events which are localised on the x-axis. Any such event is


represented with respect to the co-ordinate system K by the abscissa x


and the time t, and with respect to the system K1 by the abscissa x%26#039;


and the time t%26#039;. We require to find x%26#039; and t%26#039; when x and t are given.





A light-signal, which is proceeding along the positive axis of x, is


transmitted according to the equation





x = ct





or





x - ct = 0 . . . (1).





Since the same light-signal has to be transmitted relative to K1 with


the velocity c, the propagation relative to the system K1 will be


represented by the analogous formula





x%26#039; - ct%26#039; = O . . . (2)





Those space-time points (events) which satisfy (x) must also satisfy


(2). Obviously this will be the case when the relation





(x%26#039; - ct%26#039;) = l (x - ct) . . . (3).





is fulfilled in general, where l indicates a constant ; for, according


to (3), the disappearance of (x - ct) involves the disappearance of


(x%26#039; - ct%26#039;).





If we apply quite similar considerations to light rays which are being


transmitted along the negative x-axis, we obtain the condition





(x%26#039; + ct%26#039;) = 碌(x + ct) . . . (4).





By adding (or subtracting) equations (3) and (4), and introducing for


convenience the constants a and b in place of the constants l and 碌,


where





eq. 29: file eq29.gif





and





eq. 30: file eq30.gif





we obtain the equations





eq. 31: file eq31.gif





We should thus have the solution of our problem, if the constants a


and b were known. These result from the following discussion.





For the origin of K1 we have permanently x%26#039; = 0, and hence according


to the first of the equations (5)





eq. 32: file eq32.gif





If we call v the velocity with which the origin of K1 is moving


relative to K, we then have





eq. 33: file eq33.gif





The same value v can be obtained from equations (5), if we calculate


the velocity of another point of K1 relative to K, or the velocity


(directed towards the negative x-axis) of a point of K with respect to


K%26#039;. In short, we can designate v as the relative velocity of the two


systems.





Furthermore, the principle of relativity teaches us that, as judged


from K, the length of a unit measuring-rod which is at rest with


reference to K1 must be exactly the same as the length, as judged from


K%26#039;, of a unit measuring-rod which is at rest relative to K. In order


to see how the points of the x-axis appear as viewed from K, we only


require to take a %26quot; snapshot %26quot; of K1 from K; this means that we have


to insert a particular value of t (time of K), e.g. t = 0. For this


value of t we then obtain from the first of the equations (5)





x%26#039; = ax





Two points of the x%26#039;-axis which are separated by the distance Dx%26#039; = I


when measured in the K1 system are thus separated in our instantaneous


photograph by the distance





eq. 34: file eq34.gif





But if the snapshot be taken from K%26#039;(t%26#039; = 0), and if we eliminate t


from the equations (5), taking into account the expression (6), we


obtain





eq. 35: file eq35.gif





From this we conclude that two points on the x-axis separated by the


distance I (relative to K) will be represented on our snapshot by the


distance





eq. 36: file eq36.gif





But from what has been said, the two snapshots must be identical;


hence Dx in (7) must be equal to Dx%26#039; in (7a), so that we obtain





eq. 37: file eq37.gif





The equations (6) and (7b) determine the constants a and b. By


inserting the values of these constants in (5), we obtain the first


and the fourth of the equations given in Section 11.





eq. 38: file eq38.gif





Thus we have obtained the Lorentz transformation for events on the


x-axis. It satisfies the condition





x%26#039;2 - c^2t%26#039;2 = x2 - c^2t2 . . . (8a).





The extension of this result, to include events which take place


outside the x-axis, is obtained by retaining equations (8) and


supplementing them by the relations





eq. 39: file eq39.gif





In this way we satisfy the postulate of the constancy of the velocity


of light in vacuo for rays of light of arbitrary direction, both for


the system K and for the system K%26#039;. This may be shown in the following


manner.





We suppose a light-signal sent out from the origin of K at the time t


= 0. It will be propagated according to the equation





eq. 40: file eq40.gif





or, if we square this equation, according to the equation





x2 + y2 + z2 = c^2t2 = 0 . . . (10).





It is required by the law of propagation of light, in conjunction with


the postulate of relativity, that the transmission of the signal in


question should take place -- as judged from K1 -- in accordance with


the corresponding formula





r%26#039; = ct%26#039;





or,





x%26#039;2 + y%26#039;2 + z%26#039;2 - c^2t%26#039;2 = 0 . . . (10a).





In order that equation (10a) may be a consequence of equation (10), we


must have





x%26#039;2 + y%26#039;2 + z%26#039;2 - c^2t%26#039;2 = s (x2 + y2 + z2 - c^2t2) (11).





Since equation (8a) must hold for points on the x-axis, we thus have s


= I. It is easily seen that the Lorentz transformation really


satisfies equation (11) for s = I; for (11) is a consequence of (8a)


and (9), and hence also of (8) and (9). We have thus derived the


Lorentz transformation.





The Lorentz transformation represented by (8) and (9) still requires


to be generalised. Obviously it is immaterial whether the axes of K1


be chosen so that they are spatially parallel to those of K. It is


also not essential that the velocity of translation of K1 with respect


to K should be in the direction of the x-axis. A simple consideration


shows that we are able to construct the Lorentz transformation in this


general sense from two kinds of transformations, viz. from Lorentz


transformations in the special sense and from purely spatial


transformations. which corresponds to the replacement of the


rectangular co-ordinate system by a new system with its axes pointing


in other directions.





Mathematically, we can characterise the generalised Lorentz


transformation thus :





It expresses x%26#039;, y%26#039;, x%26#039;, t%26#039;, in terms of linear homogeneous functions


of x, y, x, t, of such a kind that the relation





x%26#039;2 + y%26#039;2 + z%26#039;2 - c^2t%26#039;2 = x2 + y2 + z2 - c^2t2 (11a).





is satisficd identically. That is to say: If we substitute their


expressions in x, y, x, t, in place of x%26#039;, y%26#039;, x%26#039;, t%26#039;, on the


left-hand side, then the left-hand side of (11a) agrees with the


right-hand side.











APPENDIX II





MINKOWSKI%26#039;S FOUR-DIMENSIONAL SPACE (%26quot;WORLD%26quot;)


(SUPPLEMENTARY TO SECTION 17)








We can characterise the Lorentz transformation still more simply if we


introduce the imaginary eq. 25 in place of t, as time-variable. If, in


accordance with this, we insert





x[1] = x


x[2] = y


x[3] = z


x[4] = eq. 25





and similarly for the accented system K1, then the condition which is


identically satisfied by the transformation can be expressed thus :





x[1]%26#039;2 + x[2]%26#039;2 + x[3]%26#039;2 + x[4]%26#039;2 = x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2


(12).





That is, by the afore-mentioned choice of %26quot; coordinates,%26quot; (11a) [see


the end of Appendix II] is transformed into this equation.





We see from (12) that the imaginary time co-ordinate x[4], enters into


the condition of transformation in exactly the same way as the space


co-ordinates x[1], x[2], x[3]. It is due to this fact that, according


to the theory of relativity, the %26quot; time %26quot;x[4], enters into natural


laws in the same form as the space co ordinates x[1], x[2], x[3].





A four-dimensional continuum described by the %26quot;co-ordinates%26quot; x[1],


x[2], x[3], x[4], was called %26quot;world%26quot; by Minkowski, who also termed a


point-event a %26quot; world-point.%26quot; From a %26quot;happening%26quot; in three-dimensional


space, physics becomes, as it were, an %26quot; existence %26quot; in the


four-dimensional %26quot; world.%26quot;





This four-dimensional %26quot; world %26quot; bears a close similarity to the


three-dimensional %26quot; space %26quot; of (Euclidean) analytical geometry. If we


introduce into the latter a new Cartesian co-ordinate system (x%26#039;[1],


x%26#039;[2], x%26#039;[3]) with the same origin, then x%26#039;[1], x%26#039;[2], x%26#039;[3], are


linear homogeneous functions of x[1], x[2], x[3] which identically


satisfy the equation





x%26#039;[1]^2 + x%26#039;[2]^2 + x%26#039;[3]^2 = x[1]^2 + x[2]^2 + x[3]^2





The analogy with (12) is a complete one. We can regard Minkowski%26#039;s %26quot;


world %26quot; in a formal manner as a four-dimensional Euclidean space (with


an imaginary time coordinate) ; the Lorentz transformation corresponds


to a %26quot; rotation %26quot; of the co-ordinate system in the fourdimensional %26quot;


world.%26quot;











APPENDIX III





THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY








From a systematic theoretical point of view, we may imagine the


process of evolution of an empirical science to be a continuous


process of induction. Theories are evolved and are expressed in short


compass as statements of a large number of individual observations in


the form of empirical laws, from which the general laws can be


ascertained by comparison. Regarded in this way, the development of a


science bears some resemblance to the compilation of a classified


catalogue. It is, as it were, a purely empirical enterprise.





But this point of view by no means embraces the whole of the actual


process ; for it slurs over the important part played by intuition and


deductive thought in the development of an exact science. As soon as a


science has emerged from its initial stages, theoretical advances are


no longer achieved merely by a process of arrangement. Guided by


empirical data, the investigator rather develops a system of thought


which, in general, is built up logically from a small number of


fundamental assumptions, the so-called axioms. We call such a system


of thought a theory. The theory finds the justification for its


existence in the fact that it correlates a large number of single


observations, and it is just here that the %26quot; truth %26quot; of the theory


lies.





Corresponding to the same complex of empirical data, there may be


several theories, which differ from one another to a considerable


extent. But as regards the deductions from the theories which are


capable of being tested, the agreement between the theories may be so


complete that it becomes difficult to find any deductions in which the


two theories differ from each other. As an example, a case of general


interest is available in the province of biology, in the Darwinian


theory of the development of species by selection in the struggle for


existence, and in the theory of development which is based on the


hypothesis of the hereditary transmission of acquired characters.





We have another instance of far-reaching agreement between the


deductions from two theories in Newtonian mechanics on the one hand,


and the general theory of relativity on the other. This agreement goes


so far, that up to the preseat we have been able to find only a few


deductions from the general theory of relativity which are capable of


investigation, and to which the physics of pre-relativity days does


not also lead, and this despite the profound difference in the


fundamental assumptions of the two theories. In what follows, we shall


again consider these important deductions, and we shall also discuss


the empirical evidence appertaining to them which has hitherto been


obtained.





(a) Motion of the Perihelion of Mercury





According to Newtonian mechanics and Newton%26#039;s law of gravitation, a


planet which is revolving round the sun would describe an ellipse


round the latter, or, more correctly, round the common centre of


gravity of the sun and the planet. In such a system, the sun, or the


common centre of gravity, lies in one of the foci of the orbital


ellipse in such a manner that, in the course of a planet-year, the


distance sun-planet grows from a minimum to a maximum, and then


decreases again to a minimum. If instead of Newton%26#039;s law we insert a


somewhat different law of attraction into the calculation, we find


that, according to this new law, the motion would still take place in


such a manner that the distance sun-planet exhibits periodic


variations; but in this case the angle described by the line joining


sun and planet during such a period (from perihelion--closest


proximity to the sun--to perihelion) would differ from 360^0. The line


of the orbit would not then be a closed one but in the course of time


it would fill up an annular part of the orbital plane, viz. between


the circle of least and the circle of greatest distance of the planet


from the sun.





According also to the general theory of relativity, which differs of


course from the theory of Newton, a small variation from the


Newton-Kepler motion of a planet in its orbit should take place, and


in such away, that the angle described by the radius sun-planet


between one perhelion and the next should exceed that corresponding to


one complete revolution by an amount given by





eq. 41: file eq41.gif





(N.B. -- One complete revolution corresponds to the angle 2p in the


absolute angular measure customary in physics, and the above


expression giver the amount by which the radius sun-planet exceeds


this angle during the interval between one perihelion and the next.)


In this expression a represents the major semi-axis of the ellipse, e


its eccentricity, c the velocity of light, and T the period of


revolution of the planet. Our result may also be stated as follows :


According to the general theory of relativity, the major axis of the


ellipse rotates round the sun in the same sense as the orbital motion


of the planet. Theory requires that this rotation should amount to 43


seconds of arc per century for the planet Mercury, but for the other


Planets of our solar system its magnitude should be so small that it


would necessarily escape detection. *





In point of fact, astronomers have found that the theory of Newton


does not suffice to calculate the observed motion of Mercury with an


exactness corresponding to that of the delicacy of observation


attainable at the present time. After taking account of all the


disturbing influences exerted on Mercury by the remaining planets, it


was found (Leverrier: 1859; and Newcomb: 1895) that an unexplained


perihelial movement of the orbit of Mercury remained over, the amount


of which does not differ sensibly from the above mentioned +43 seconds


of arc per century. The uncertainty of the empirical result amounts to


a few seconds only.





(b) Deflection of Light by a Gravitational Field





In Section 22 it has been already mentioned that according to the


general theory of relativity, a ray of light will experience a


curvature of its path when passing through a gravitational field, this


curvature being similar to that experienced by the path of a body


which is projected through a gravitational field. As a result of this


theory, we should expect that a ray of light which is passing close to


a heavenly body would be deviated towards the latter. For a ray of


light which passes the sun at a distance of D sun-radii from its


centre, the angle of deflection (a) should amount to





eq. 42: file eq42.gif





It may be added that, according to the theory, half of Figure 05 this


deflection is produced by the Newtonian field of attraction of the


sun, and the other half by the geometrical modification (%26quot; curvature


%26quot;) of space caused by the sun.





This result admits of an experimental test by means of the


photographic registration of stars during a total eclipse of the sun.


The only reason why we must wait for a total eclipse is because at


every other time the atmosphere is so strongly illuminated by the


light from the sun that the stars situated near the sun%26#039;s disc are


invisible. The predicted effect can be seen clearly from the


accompanying diagram. If the sun (S) were not present, a star which is


practically infinitely distant would be seen in the direction D[1], as


observed front the earth. But as a consequence of the deflection of


light from the star by the sun, the star will be seen in the direction


D[2], i.e. at a somewhat greater distance from the centre of the sun


than corresponds to its real position.





In practice, the question is tested in the following way. The stars in


the neighbourhood of the sun are photographed during a solar eclipse.


In addition, a second photograph of the same stars is taken when the


sun is situated at another position in the sky, i.e. a few months


earlier or later. As compared whh the standard photograph, the


positions of the stars on the eclipse-photograph ought to appear


displaced radially outwards (away from the centre of the sun) by an


amount corresponding to the angle a.





We are indebted to the [British] Royal Society and to the Royal


Astronomical Society for the investigation of this important


deduction. Undaunted by the [first world] war and by difficulties of


both a material and a psychological nature aroused by the war, these


societies equipped two expeditions -- to Sobral (Brazil), and to the


island of Principe (West Africa) -- and sent several of Britain%26#039;s most


celebrated astronomers (Eddington, Cottingham, Crommelin, Davidson),


in order to obtain photographs of the solar eclipse of 29th May, 1919.


The relative discrepancies to be expected between the stellar


photographs obtained during the eclipse and the comparison photographs


amounted to a few hundredths of a millimetre only. Thus great accuracy


was necessary in making the adjustments required for the taking of the


photographs, and in their subsequent measurement.





The results of the measurements confirmed the theory in a thoroughly


satisfactory manner. The rectangular components of the observed and of


the calculated deviations of the stars (in seconds of arc) are set


forth in the following table of results :





Table 01: file table01.gif





(c) Displacement of Spectral Lines Towards the Red





In Section 23 it has been shown that in a system K1 which is in


rotation with regard to a Galileian system K, clocks of identical


construction, and which are considered at rest with respect to the


rotating reference-body, go at rates which are dependent on the


positions of the clocks. We shall now examine this dependence


quantitatively. A clock, which is situated at a distance r from the


centre of the disc, has a velocity relative to K which is given by





V = wr





where w represents the angular velocity of rotation of the disc K1


with respect to K. If v[0], represents the number of ticks of the


clock per unit time (%26quot; rate %26quot; of the clock) relative to K when the


clock is at rest, then the %26quot; rate %26quot; of the clock (v) when it is moving


relative to K with a velocity V, but at rest with respect to the disc,


will, in accordance with Section 12, be given by





eq. 43: file eq43.gif





or with sufficient accuracy by





eq. 44: file eq44.gif





This expression may also be stated in the following form:





eq. 45: file eq45.gif





If we represent the difference of potential of the centrifugal force


between the position of the clock and the centre of the disc by f,


i.e. the work, considered negatively, which must be performed on the


unit of mass against the centrifugal force in order to transport it


from the position of the clock on the rotating disc to the centre of


the disc, then we have





eq. 46: file eq46.gif





From this it follows that





eq. 47: file eq47.gif





In the first place, we see from this expression that two clocks of


identical construction will go at different rates when situated at


different distances from the centre of the disc. This result is aiso


valid from the standpoint of an observer who is rotating with the


disc.





Now, as judged from the disc, the latter is in a gravititional field


of potential f, hence the result we have obtained will hold quite


generally for gravitational fields. Furthermore, we can regard an atom


which is emitting spectral lines as a clock, so that the following


statement will hold:





An atom absorbs or emits light of a frequency which is dependent on


the potential of the gravitational field in which it is situated.





The frequency of an atom situated on the surface of a heavenly body


will be somewhat less than the frequency of an atom of the same


element which is situated in free space (or on the surface of a


smaller celestial body).





Now f = - K (M/r), where K is Newton%26#039;s constant of gravitation, and M


is the mass of the heavenly body. Thus a displacement towards the red


ought to take place for spectral lines produced at the surface of


stars as compared with the spectral lines of the same element produced


at the surface of the earth, the amount of this displacement being





eq. 48: file eq48.gif





For the sun, the displacement towards the red predicted by theory


amounts to about two millionths of the wave-length. A trustworthy


calculation is not possible in the case of the stars, because in


general neither the mass M nor the radius r are known.





It is an open question whether or not this effect exists, and at the


present time (1920) astronomers are working with great zeal towards


the solution. Owing to the smallness of the effect in the case of the


sun, it is difficult to form an opinion as to its existence. Whereas


Grebe and Bachem (Bonn), as a result of their own measurements and


those of Evershed and Schwarzschild on the cyanogen bands, have placed


the existence of the effect almost beyond doubt, while other


investigators, particularly St. John, have been led to the opposite


opinion in consequence of their measurements.





Mean displacements of lines towards the less refrangible end of the


spectrum are certainly revealed by statistical investigations of the


fixed stars ; but up to the present the examination of the available


data does not allow of any definite decision being arrived at, as to


whether or not these displacements are to be referred in reality to


the effect of gravitation. The results of observation have been


collected together, and discussed in detail from the standpoint of the


question which has been engaging our attention here, in a paper by E.


Freundlich entitled %26quot;Zur Pr眉fung der allgemeinen


Relativit%26amp;umlaut;ts-Theorie%26quot; (Die Naturwissenschaften, 1919, No. 35,


p. 520: Julius Springer, Berlin).





At all events, a definite decision will be reached during the next few


years. If the displacement of spectral lines towards the red by the


gravitational potential does not exist, then the general theory of


relativity will be untenable. On the other hand, if the cause of the


displacement of spectral lines be definitely traced to the


gravitational potential, then the study of this displacement will


furnish us with important information as to the mass of the heavenly


bodies. [5][A]








Notes





*) Especially since the next planet Venus has an orbit that is


almost an exact circle, which makes it more difficult to locate the


perihelion with precision.





The displacentent of spectral lines towards the red end of the


spectrum was definitely established by Adams in 1924, by observations


on the dense companion of Sirius, for which the effect is about thirty


times greater than for the Sun. R.W.L. -- translator











APPENDIX IV





THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY


(SUPPLEMENTARY TO SECTION 32)








Since the publication of the first edition of this little book, our


knowledge about the structure of space in the large (%26quot; cosmological


problem %26quot;) has had an important development, which ought to be


mentioned even in a popular presentation of the subject.





My original considerations on the subject were based on two


hypotheses:





(1) There exists an average density of matter in the whole of space


which is everywhere the same and different from zero.





(2) The magnitude (%26quot; radius %26quot;) of space is independent of time.





Both these hypotheses proved to be consistent, according to the


general theory of relativity, but only after a hypothetical term was


added to the field equations, a term which was not required by the


theory as such nor did it seem natural from a theoretical point of


view (%26quot; cosmological term of the field equations %26quot;).





Hypothesis (2) appeared unavoidable to me at the time, since I thought


that one would get into bottomless speculations if one departed from


it.





However, already in the %26#039;twenties, the Russian mathematician Friedman


showed that a different hypothesis was natural from a purely


theoretical point of view. He realized that it was possible to


preserve hypothesis (1) without introducing the less natural


cosmological term into the field equations of gravitation, if one was


ready to drop hypothesis (2). Namely, the original field equations


admit a solution in which the %26quot; world radius %26quot; depends on time


(expanding space). In that sense one can say, according to Friedman,


that the theory demands an expansion of space.





A few years later Hubble showed, by a special investigation of the


extra-galactic nebulae (%26quot; milky ways %26quot;), that the spectral lines


emitted showed a red shift which increased regularly with the distance


of the nebulae. This can be interpreted in regard to our present


knowledge only in the sense of Doppler%26#039;s principle, as an expansive


motion of the system of stars in the large -- as required, according


to Friedman, by the field equations of gravitation. Hubble%26#039;s discovery


can, therefore, be considered to some extent as a confirmation of the


theory.





There does arise, however, a strange difficulty. The interpretation of


the galactic line-shift discovered by Hubble as an expansion (which


can hardly be doubted from a theoretical point of view), leads to an


origin of this expansion which lies %26quot; only %26quot; about 10^9 years ago,


while physical astronomy makes it appear likely that the development


of individual stars and systems of stars takes considerably longer. It


is in no way known how this incongruity is to be overcome.





I further want to rernark that the theory of expanding space, together


with the empirical data of astronomy, permit no decision to be reached


about the finite or infinite character of (three-dimensional) space,


while the original %26quot; static %26quot; hypothesis of space yielded the closure


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