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2by Albert Einstein
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29
30Title: Relativity: The Special and General Theory
31
32Author: Albert Einstein
33
34Release Date: February, 2004 [EBook #5001]
35[Yes, we are more than one year ahead of schedule]
36[This file was first posted on April 1, 2002]
37
38Edition: 10
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40Language: English
41
42
43*** START OF THE PROJECT GUTENBERG EBOOK, RELATIVITY ***
44
45
46
47
48ALBERT EINSTEIN REFERENCE ARCHIVE
49
50RELATIVITY: THE SPECIAL AND GENERAL THEORY
51
52BY ALBERT EINSTEIN
53
54
55Written: 1916 (this revised edition: 1924)
56Source: Relativity: The Special and General Theory (1920)
57Publisher: Methuen & Co Ltd
58First Published: December, 1916
59Translated: Robert W. Lawson (Authorised translation)
60Transcription/Markup: Brian Basgen <brian@marxists.org>
61Transcription to text: Gregory B. Newby <gbnewby@petascale.org>
62Thanks to: Einstein Reference Archive (marxists.org)
63The Einstein Reference Archive is online at:
64http://www.marxists.org/reference/archive/einstein/index.htm
65
66Transcriber note: This file is a plain text rendition of HTML.
67Because many equations cannot be presented effectively in plain text,
68images are supplied for many equations and for all figures and tables.
69
70
71CONTENTS
72
73Preface
74
75Part I: The Special Theory of Relativity
76
7701. Physical Meaning of Geometrical Propositions
7802. The System of Co-ordinates
7903. Space and Time in Classical Mechanics
8004. The Galileian System of Co-ordinates
8105. The Principle of Relativity (in the Restricted Sense)
8206. The Theorem of the Addition of Velocities employed in
83Classical Mechanics
8407. The Apparent Incompatability of the Law of Propagation of
85Light with the Principle of Relativity
8608. On the Idea of Time in Physics
8709. The Relativity of Simultaneity
8810. On the Relativity of the Conception of Distance
8911. The Lorentz Transformation
9012. The Behaviour of Measuring-Rods and Clocks in Motion
9113. Theorem of the Addition of Velocities. The Experiment of Fizeau
9214. The Hueristic Value of the Theory of Relativity
9315. General Results of the Theory
9416. Expereince and the Special Theory of Relativity
9517. Minkowski's Four-dimensial Space
96
97
98Part II: The General Theory of Relativity
99
10018. Special and General Principle of Relativity
10119. The Gravitational Field
10220. The Equality of Inertial and Gravitational Mass as an Argument
103for the General Postulate of Relativity
10421. In What Respects are the Foundations of Classical Mechanics
105and of the Special Theory of Relativity Unsatisfactory?
10622. A Few Inferences from the General Principle of Relativity
10723. Behaviour of Clocks and Measuring-Rods on a Rotating Body of
108Reference
10924. Euclidean and non-Euclidean Continuum
11025. Gaussian Co-ordinates
11126. The Space-Time Continuum of the Speical Theory of Relativity
112Considered as a Euclidean Continuum
11327. The Space-Time Continuum of the General Theory of Relativity
114is Not a Eculidean Continuum
11528. Exact Formulation of the General Principle of Relativity
11629. The Solution of the Problem of Gravitation on the Basis of the
117General Principle of Relativity
118
119
120Part III: Considerations on the Universe as a Whole
121
12230. Cosmological Difficulties of Netwon's Theory
12331. The Possibility of a "Finite" and yet "Unbounded" Universe
12432. The Structure of Space According to the General Theory of
125Relativity
126
127
128Appendices:
129
13001. Simple Derivation of the Lorentz Transformation (sup. ch. 11)
13102. Minkowski's Four-Dimensional Space ("World") (sup. ch 17)
13203. The Experimental Confirmation of the General Theory of Relativity
13304. The Structure of Space According to the General Theory of
134Relativity (sup. ch 32)
13505. Relativity and the Problem of Space
136
137Note: The fifth Appendix was added by Einstein at the time of the
138fifteenth re-printing of this book; and as a result is still under
139copyright restrictions so cannot be added without the permission of
140the publisher.
141
142
143
144PREFACE
145
146 (December, 1916)
147
148The present book is intended, as far as possible, to give an exact
149insight into the theory of Relativity to those readers who, from a
150general scientific and philosophical point of view, are interested in
151the theory, but who are not conversant with the mathematical apparatus
152of theoretical physics. The work presumes a standard of education
153corresponding to that of a university matriculation examination, and,
154despite the shortness of the book, a fair amount of patience and force
155of will on the part of the reader. The author has spared himself no
156pains in his endeavour to present the main ideas in the simplest and
157most intelligible form, and on the whole, in the sequence and
158connection in which they actually originated. In the interest of
159clearness, it appeared to me inevitable that I should repeat myself
160frequently, without paying the slightest attention to the elegance of
161the presentation. I adhered scrupulously to the precept of that
162brilliant theoretical physicist L. Boltzmann, according to whom
163matters of elegance ought to be left to the tailor and to the cobbler.
164I make no pretence of having withheld from the reader difficulties
165which are inherent to the subject. On the other hand, I have purposely
166treated the empirical physical foundations of the theory in a
167"step-motherly" fashion, so that readers unfamiliar with physics may
168not feel like the wanderer who was unable to see the forest for the
169trees. May the book bring some one a few happy hours of suggestive
170thought!
171
172December, 1916
173A. EINSTEIN
174
175
176
177PART I
178
179THE SPECIAL THEORY OF RELATIVITY
180
181PHYSICAL MEANING OF GEOMETRICAL PROPOSITIONS
182
183
184In your schooldays most of you who read this book made acquaintance
185with the noble building of Euclid's geometry, and you remember --
186perhaps with more respect than love -- the magnificent structure, on
187the lofty staircase of which you were chased about for uncounted hours
188by conscientious teachers. By reason of our past experience, you would
189certainly regard everyone with disdain who should pronounce even the
190most out-of-the-way proposition of this science to be untrue. But
191perhaps this feeling of proud certainty would leave you immediately if
192some one were to ask you: "What, then, do you mean by the assertion
193that these propositions are true?" Let us proceed to give this
194question a little consideration.
195
196Geometry sets out form certain conceptions such as "plane," "point,"
197and "straight line," with which we are able to associate more or less
198definite ideas, and from certain simple propositions (axioms) which,
199in virtue of these ideas, we are inclined to accept as "true." Then,
200on the basis of a logical process, the justification of which we feel
201ourselves compelled to admit, all remaining propositions are shown to
202follow from those axioms, i.e. they are proven. A proposition is then
203correct ("true") when it has been derived in the recognised manner
204from the axioms. The question of "truth" of the individual geometrical
205propositions is thus reduced to one of the "truth" of the axioms. Now
206it has long been known that the last question is not only unanswerable
207by the methods of geometry, but that it is in itself entirely without
208meaning. We cannot ask whether it is true that only one straight line
209goes through two points. We can only say that Euclidean geometry deals
210with things called "straight lines," to each of which is ascribed the
211property of being uniquely determined by two points situated on it.
212The concept "true" does not tally with the assertions of pure
213geometry, because by the word "true" we are eventually in the habit of
214designating always the correspondence with a "real" object; geometry,
215however, is not concerned with the relation of the ideas involved in
216it to objects of experience, but only with the logical connection of
217these ideas among themselves.
218
219It is not difficult to understand why, in spite of this, we feel
220constrained to call the propositions of geometry "true." Geometrical
221ideas correspond to more or less exact objects in nature, and these
222last are undoubtedly the exclusive cause of the genesis of those
223ideas. Geometry ought to refrain from such a course, in order to give
224to its structure the largest possible logical unity. The practice, for
225example, of seeing in a "distance" two marked positions on a
226practically rigid body is something which is lodged deeply in our
227habit of thought. We are accustomed further to regard three points as
228being situated on a straight line, if their apparent positions can be
229made to coincide for observation with one eye, under suitable choice
230of our place of observation.
231
232If, in pursuance of our habit of thought, we now supplement the
233propositions of Euclidean geometry by the single proposition that two
234points on a practically rigid body always correspond to the same
235distance (line-interval), independently of any changes in position to
236which we may subject the body, the propositions of Euclidean geometry
237then resolve themselves into propositions on the possible relative
238position of practically rigid bodies.* Geometry which has been
239supplemented in this way is then to be treated as a branch of physics.
240We can now legitimately ask as to the "truth" of geometrical
241propositions interpreted in this way, since we are justified in asking
242whether these propositions are satisfied for those real things we have
243associated with the geometrical ideas. In less exact terms we can
244express this by saying that by the "truth" of a geometrical
245proposition in this sense we understand its validity for a
246construction with rule and compasses.
247
248Of course the conviction of the "truth" of geometrical propositions in
249this sense is founded exclusively on rather incomplete experience. For
250the present we shall assume the "truth" of the geometrical
251propositions, then at a later stage (in the general theory of
252relativity) we shall see that this "truth" is limited, and we shall
253consider the extent of its limitation.
254
255
256 Notes
257
258*) It follows that a natural object is associated also with a
259straight line. Three points A, B and C on a rigid body thus lie in a
260straight line when the points A and C being given, B is chosen such
261that the sum of the distances AB and BC is as short as possible. This
262incomplete suggestion will suffice for the present purpose.
263
264
265
266THE SYSTEM OF CO-ORDINATES
267
268
269On the basis of the physical interpretation of distance which has been
270indicated, we are also in a position to establish the distance between
271two points on a rigid body by means of measurements. For this purpose
272we require a " distance " (rod S) which is to be used once and for
273all, and which we employ as a standard measure. If, now, A and B are
274two points on a rigid body, we can construct the line joining them
275according to the rules of geometry ; then, starting from A, we can
276mark off the distance S time after time until we reach B. The number
277of these operations required is the numerical measure of the distance
278AB. This is the basis of all measurement of length. *
279
280Every description of the scene of an event or of the position of an
281object in space is based on the specification of the point on a rigid
282body (body of reference) with which that event or object coincides.
283This applies not only to scientific description, but also to everyday
284life. If I analyse the place specification " Times Square, New York,"
285**A I arrive at the following result. The earth is the rigid body
286to which the specification of place refers; " Times Square, New York,"
287is a well-defined point, to which a name has been assigned, and with
288which the event coincides in space.**B
289
290This primitive method of place specification deals only with places on
291the surface of rigid bodies, and is dependent on the existence of
292points on this surface which are distinguishable from each other. But
293we can free ourselves from both of these limitations without altering
294the nature of our specification of position. If, for instance, a cloud
295is hovering over Times Square, then we can determine its position
296relative to the surface of the earth by erecting a pole
297perpendicularly on the Square, so that it reaches the cloud. The
298length of the pole measured with the standard measuring-rod, combined
299with the specification of the position of the foot of the pole,
300supplies us with a complete place specification. On the basis of this
301illustration, we are able to see the manner in which a refinement of
302the conception of position has been developed.
303
304(a) We imagine the rigid body, to which the place specification is
305referred, supplemented in such a manner that the object whose position
306we require is reached by. the completed rigid body.
307
308(b) In locating the position of the object, we make use of a number
309(here the length of the pole measured with the measuring-rod) instead
310of designated points of reference.
311
312(c) We speak of the height of the cloud even when the pole which
313reaches the cloud has not been erected. By means of optical
314observations of the cloud from different positions on the ground, and
315taking into account the properties of the propagation of light, we
316determine the length of the pole we should have required in order to
317reach the cloud.
318
319From this consideration we see that it will be advantageous if, in the
320description of position, it should be possible by means of numerical
321measures to make ourselves independent of the existence of marked
322positions (possessing names) on the rigid body of reference. In the
323physics of measurement this is attained by the application of the
324Cartesian system of co-ordinates.
325
326This consists of three plane surfaces perpendicular to each other and
327rigidly attached to a rigid body. Referred to a system of
328co-ordinates, the scene of any event will be determined (for the main
329part) by the specification of the lengths of the three perpendiculars
330or co-ordinates (x, y, z) which can be dropped from the scene of the
331event to those three plane surfaces. The lengths of these three
332perpendiculars can be determined by a series of manipulations with
333rigid measuring-rods performed according to the rules and methods laid
334down by Euclidean geometry.
335
336In practice, the rigid surfaces which constitute the system of
337co-ordinates are generally not available ; furthermore, the magnitudes
338of the co-ordinates are not actually determined by constructions with
339rigid rods, but by indirect means. If the results of physics and
340astronomy are to maintain their clearness, the physical meaning of
341specifications of position must always be sought in accordance with
342the above considerations. ***
343
344We thus obtain the following result: Every description of events in
345space involves the use of a rigid body to which such events have to be
346referred. The resulting relationship takes for granted that the laws
347of Euclidean geometry hold for "distances;" the "distance" being
348represented physically by means of the convention of two marks on a
349rigid body.
350
351
352 Notes
353
354* Here we have assumed that there is nothing left over i.e. that
355the measurement gives a whole number. This difficulty is got over by
356the use of divided measuring-rods, the introduction of which does not
357demand any fundamentally new method.
358
359**A Einstein used "Potsdamer Platz, Berlin" in the original text.
360In the authorised translation this was supplemented with "Tranfalgar
361Square, London". We have changed this to "Times Square, New York", as
362this is the most well known/identifiable location to English speakers
363in the present day. [Note by the janitor.]
364
365**B It is not necessary here to investigate further the significance
366of the expression "coincidence in space." This conception is
367sufficiently obvious to ensure that differences of opinion are
368scarcely likely to arise as to its applicability in practice.
369
370*** A refinement and modification of these views does not become
371necessary until we come to deal with the general theory of relativity,
372treated in the second part of this book.
373
374
375
376SPACE AND TIME IN CLASSICAL MECHANICS
377
378
379The purpose of mechanics is to describe how bodies change their
380position in space with "time." I should load my conscience with grave
381sins against the sacred spirit of lucidity were I to formulate the
382aims of mechanics in this way, without serious reflection and detailed
383explanations. Let us proceed to disclose these sins.
384
385It is not clear what is to be understood here by "position" and
386"space." I stand at the window of a railway carriage which is
387travelling uniformly, and drop a stone on the embankment, without
388throwing it. Then, disregarding the influence of the air resistance, I
389see the stone descend in a straight line. A pedestrian who observes
390the misdeed from the footpath notices that the stone falls to earth in
391a parabolic curve. I now ask: Do the "positions" traversed by the
392stone lie "in reality" on a straight line or on a parabola? Moreover,
393what is meant here by motion "in space" ? From the considerations of
394the previous section the answer is self-evident. In the first place we
395entirely shun the vague word "space," of which, we must honestly
396acknowledge, we cannot form the slightest conception, and we replace
397it by "motion relative to a practically rigid body of reference." The
398positions relative to the body of reference (railway carriage or
399embankment) have already been defined in detail in the preceding
400section. If instead of " body of reference " we insert " system of
401co-ordinates," which is a useful idea for mathematical description, we
402are in a position to say : The stone traverses a straight line
403relative to a system of co-ordinates rigidly attached to the carriage,
404but relative to a system of co-ordinates rigidly attached to the
405ground (embankment) it describes a parabola. With the aid of this
406example it is clearly seen that there is no such thing as an
407independently existing trajectory (lit. "path-curve"*), but only
408a trajectory relative to a particular body of reference.
409
410In order to have a complete description of the motion, we must specify
411how the body alters its position with time ; i.e. for every point on
412the trajectory it must be stated at what time the body is situated
413there. These data must be supplemented by such a definition of time
414that, in virtue of this definition, these time-values can be regarded
415essentially as magnitudes (results of measurements) capable of
416observation. If we take our stand on the ground of classical
417mechanics, we can satisfy this requirement for our illustration in the
418following manner. We imagine two clocks of identical construction ;
419the man at the railway-carriage window is holding one of them, and the
420man on the footpath the other. Each of the observers determines the
421position on his own reference-body occupied by the stone at each tick
422of the clock he is holding in his hand. In this connection we have not
423taken account of the inaccuracy involved by the finiteness of the
424velocity of propagation of light. With this and with a second
425difficulty prevailing here we shall have to deal in detail later.
426
427
428 Notes
429
430*) That is, a curve along which the body moves.
431
432
433
434THE GALILEIAN SYSTEM OF CO-ORDINATES
435
436
437As is well known, the fundamental law of the mechanics of
438Galilei-Newton, which is known as the law of inertia, can be stated
439thus: A body removed sufficiently far from other bodies continues in a
440state of rest or of uniform motion in a straight line. This law not
441only says something about the motion of the bodies, but it also
442indicates the reference-bodies or systems of coordinates, permissible
443in mechanics, which can be used in mechanical description. The visible
444fixed stars are bodies for which the law of inertia certainly holds to
445a high degree of approximation. Now if we use a system of co-ordinates
446which is rigidly attached to the earth, then, relative to this system,
447every fixed star describes a circle of immense radius in the course of
448an astronomical day, a result which is opposed to the statement of the
449law of inertia. So that if we adhere to this law we must refer these
450motions only to systems of coordinates relative to which the fixed
451stars do not move in a circle. A system of co-ordinates of which the
452state of motion is such that the law of inertia holds relative to it
453is called a " Galileian system of co-ordinates." The laws of the
454mechanics of Galflei-Newton can be regarded as valid only for a
455Galileian system of co-ordinates.
456
457
458
459THE PRINCIPLE OF RELATIVITY
460(IN THE RESTRICTED SENSE)
461
462
463In order to attain the greatest possible clearness, let us return to
464our example of the railway carriage supposed to be travelling
465uniformly. We call its motion a uniform translation ("uniform" because
466it is of constant velocity and direction, " translation " because
467although the carriage changes its position relative to the embankment
468yet it does not rotate in so doing). Let us imagine a raven flying
469through the air in such a manner that its motion, as observed from the
470embankment, is uniform and in a straight line. If we were to observe
471the flying raven from the moving railway carriage. we should find that
472the motion of the raven would be one of different velocity and
473direction, but that it would still be uniform and in a straight line.
474Expressed in an abstract manner we may say : If a mass m is moving
475uniformly in a straight line with respect to a co-ordinate system K,
476then it will also be moving uniformly and in a straight line relative
477to a second co-ordinate system K1 provided that the latter is
478executing a uniform translatory motion with respect to K. In
479accordance with the discussion contained in the preceding section, it
480follows that:
481
482If K is a Galileian co-ordinate system. then every other co-ordinate
483system K' is a Galileian one, when, in relation to K, it is in a
484condition of uniform motion of translation. Relative to K1 the
485mechanical laws of Galilei-Newton hold good exactly as they do with
486respect to K.
487
488We advance a step farther in our generalisation when we express the
489tenet thus: If, relative to K, K1 is a uniformly moving co-ordinate
490system devoid of rotation, then natural phenomena run their course
491with respect to K1 according to exactly the same general laws as with
492respect to K. This statement is called the principle of relativity (in
493the restricted sense).
494
495As long as one was convinced that all natural phenomena were capable
496of representation with the help of classical mechanics, there was no
497need to doubt the validity of this principle of relativity. But in
498view of the more recent development of electrodynamics and optics it
499became more and more evident that classical mechanics affords an
500insufficient foundation for the physical description of all natural
501phenomena. At this juncture the question of the validity of the
502principle of relativity became ripe for discussion, and it did not
503appear impossible that the answer to this question might be in the
504negative.
505
506Nevertheless, there are two general facts which at the outset speak
507very much in favour of the validity of the principle of relativity.
508Even though classical mechanics does not supply us with a sufficiently
509broad basis for the theoretical presentation of all physical
510phenomena, still we must grant it a considerable measure of " truth,"
511since it supplies us with the actual motions of the heavenly bodies
512with a delicacy of detail little short of wonderful. The principle of
513relativity must therefore apply with great accuracy in the domain of
514mechanics. But that a principle of such broad generality should hold
515with such exactness in one domain of phenomena, and yet should be
516invalid for another, is a priori not very probable.
517
518We now proceed to the second argument, to which, moreover, we shall
519return later. If the principle of relativity (in the restricted sense)
520does not hold, then the Galileian co-ordinate systems K, K1, K2, etc.,
521which are moving uniformly relative to each other, will not be
522equivalent for the description of natural phenomena. In this case we
523should be constrained to believe that natural laws are capable of
524being formulated in a particularly simple manner, and of course only
525on condition that, from amongst all possible Galileian co-ordinate
526systems, we should have chosen one (K[0]) of a particular state of
527motion as our body of reference. We should then be justified (because
528of its merits for the description of natural phenomena) in calling
529this system " absolutely at rest," and all other Galileian systems K "
530in motion." If, for instance, our embankment were the system K[0] then
531our railway carriage would be a system K, relative to which less
532simple laws would hold than with respect to K[0]. This diminished
533simplicity would be due to the fact that the carriage K would be in
534motion (i.e."really")with respect to K[0]. In the general laws of
535nature which have been formulated with reference to K, the magnitude
536and direction of the velocity of the carriage would necessarily play a
537part. We should expect, for instance, that the note emitted by an
538organpipe placed with its axis parallel to the direction of travel
539would be different from that emitted if the axis of the pipe were
540placed perpendicular to this direction.
541
542Now in virtue of its motion in an orbit round the sun, our earth is
543comparable with a railway carriage travelling with a velocity of about
54430 kilometres per second. If the principle of relativity were not
545valid we should therefore expect that the direction of motion of the
546earth at any moment would enter into the laws of nature, and also that
547physical systems in their behaviour would be dependent on the
548orientation in space with respect to the earth. For owing to the
549alteration in direction of the velocity of revolution of the earth in
550the course of a year, the earth cannot be at rest relative to the
551hypothetical system K[0] throughout the whole year. However, the most
552careful observations have never revealed such anisotropic properties
553in terrestrial physical space, i.e. a physical non-equivalence of
554different directions. This is very powerful argument in favour of the
555principle of relativity.
556
557
558
559THE THEOREM OF THE
560ADDITION OF VELOCITIES
561EMPLOYED IN CLASSICAL MECHANICS
562
563
564Let us suppose our old friend the railway carriage to be travelling
565along the rails with a constant velocity v, and that a man traverses
566the length of the carriage in the direction of travel with a velocity
567w. How quickly or, in other words, with what velocity W does the man
568advance relative to the embankment during the process ? The only
569possible answer seems to result from the following consideration: If
570the man were to stand still for a second, he would advance relative to
571the embankment through a distance v equal numerically to the velocity
572of the carriage. As a consequence of his walking, however, he
573traverses an additional distance w relative to the carriage, and hence
574also relative to the embankment, in this second, the distance w being
575numerically equal to the velocity with which he is walking. Thus in
576total be covers the distance W=v+w relative to the embankment in the
577second considered. We shall see later that this result, which
578expresses the theorem of the addition of velocities employed in
579classical mechanics, cannot be maintained ; in other words, the law
580that we have just written down does not hold in reality. For the time
581being, however, we shall assume its correctness.
582
583
584
585THE APPARENT INCOMPATIBILITY OF THE
586LAW OF PROPAGATION OF LIGHT WITH THE
587PRINCIPLE OF RELATIVITY
588
589
590There is hardly a simpler law in physics than that according to which
591light is propagated in empty space. Every child at school knows, or
592believes he knows, that this propagation takes place in straight lines
593with a velocity c= 300,000 km./sec. At all events we know with great
594exactness that this velocity is the same for all colours, because if
595this were not the case, the minimum of emission would not be observed
596simultaneously for different colours during the eclipse of a fixed
597star by its dark neighbour. By means of similar considerations based
598on observa- tions of double stars, the Dutch astronomer De Sitter was
599also able to show that the velocity of propagation of light cannot
600depend on the velocity of motion of the body emitting the light. The
601assumption that this velocity of propagation is dependent on the
602direction "in space" is in itself improbable.
603
604In short, let us assume that the simple law of the constancy of the
605velocity of light c (in vacuum) is justifiably believed by the child
606at school. Who would imagine that this simple law has plunged the
607conscientiously thoughtful physicist into the greatest intellectual
608difficulties? Let us consider how these difficulties arise.
609
610Of course we must refer the process of the propagation of light (and
611indeed every other process) to a rigid reference-body (co-ordinate
612system). As such a system let us again choose our embankment. We shall
613imagine the air above it to have been removed. If a ray of light be
614sent along the embankment, we see from the above that the tip of the
615ray will be transmitted with the velocity c relative to the
616embankment. Now let us suppose that our railway carriage is again
617travelling along the railway lines with the velocity v, and that its
618direction is the same as that of the ray of light, but its velocity of
619course much less. Let us inquire about the velocity of propagation of
620the ray of light relative to the carriage. It is obvious that we can
621here apply the consideration of the previous section, since the ray of
622light plays the part of the man walking along relatively to the
623carriage. The velocity w of the man relative to the embankment is here
624replaced by the velocity of light relative to the embankment. w is the
625required velocity of light with respect to the carriage, and we have
626
627 w = c-v.
628
629The velocity of propagation ot a ray of light relative to the carriage
630thus comes cut smaller than c.
631
632But this result comes into conflict with the principle of relativity
633set forth in Section V. For, like every other general law of
634nature, the law of the transmission of light in vacuo [in vacuum]
635must, according to the principle of relativity, be the same for the
636railway carriage as reference-body as when the rails are the body of
637reference. But, from our above consideration, this would appear to be
638impossible. If every ray of light is propagated relative to the
639embankment with the velocity c, then for this reason it would appear
640that another law of propagation of light must necessarily hold with
641respect to the carriage -- a result contradictory to the principle of
642relativity.
643
644In view of this dilemma there appears to be nothing else for it than
645to abandon either the principle of relativity or the simple law of the
646propagation of light in vacuo. Those of you who have carefully
647followed the preceding discussion are almost sure to expect that we
648should retain the principle of relativity, which appeals so
649convincingly to the intellect because it is so natural and simple. The
650law of the propagation of light in vacuo would then have to be
651replaced by a more complicated law conformable to the principle of
652relativity. The development of theoretical physics shows, however,
653that we cannot pursue this course. The epoch-making theoretical
654investigations of H. A. Lorentz on the electrodynamical and optical
655phenomena connected with moving bodies show that experience in this
656domain leads conclusively to a theory of electromagnetic phenomena, of
657which the law of the constancy of the velocity of light in vacuo is a
658necessary consequence. Prominent theoretical physicists were theref
659ore more inclined to reject the principle of relativity, in spite of
660the fact that no empirical data had been found which were
661contradictory to this principle.
662
663At this juncture the theory of relativity entered the arena. As a
664result of an analysis of the physical conceptions of time and space,
665it became evident that in realily there is not the least
666incompatibilitiy between the principle of relativity and the law of
667propagation of light, and that by systematically holding fast to both
668these laws a logically rigid theory could be arrived at. This theory
669has been called the special theory of relativity to distinguish it
670from the extended theory, with which we shall deal later. In the
671following pages we shall present the fundamental ideas of the special
672theory of relativity.
673
674
675
676ON THE IDEA OF TIME IN PHYSICS
677
678
679Lightning has struck the rails on our railway embankment at two places
680A and B far distant from each other. I make the additional assertion
681that these two lightning flashes occurred simultaneously. If I ask you
682whether there is sense in this statement, you will answer my question
683with a decided "Yes." But if I now approach you with the request to
684explain to me the sense of the statement more precisely, you find
685after some consideration that the answer to this question is not so
686easy as it appears at first sight.
687
688After some time perhaps the following answer would occur to you: "The
689significance of the statement is clear in itself and needs no further
690explanation; of course it would require some consideration if I were
691to be commissioned to determine by observations whether in the actual
692case the two events took place simultaneously or not." I cannot be
693satisfied with this answer for the following reason. Supposing that as
694a result of ingenious considerations an able meteorologist were to
695discover that the lightning must always strike the places A and B
696simultaneously, then we should be faced with the task of testing
697whether or not this theoretical result is in accordance with the
698reality. We encounter the same difficulty with all physical statements
699in which the conception " simultaneous " plays a part. The concept
700does not exist for the physicist until he has the possibility of
701discovering whether or not it is fulfilled in an actual case. We thus
702require a definition of simultaneity such that this definition
703supplies us with the method by means of which, in the present case, he
704can decide by experiment whether or not both the lightning strokes
705occurred simultaneously. As long as this requirement is not satisfied,
706I allow myself to be deceived as a physicist (and of course the same
707applies if I am not a physicist), when I imagine that I am able to
708attach a meaning to the statement of simultaneity. (I would ask the
709reader not to proceed farther until he is fully convinced on this
710point.)
711
712After thinking the matter over for some time you then offer the
713following suggestion with which to test simultaneity. By measuring
714along the rails, the connecting line AB should be measured up and an
715observer placed at the mid-point M of the distance AB. This observer
716should be supplied with an arrangement (e.g. two mirrors inclined at
71790^0) which allows him visually to observe both places A and B at the
718same time. If the observer perceives the two flashes of lightning at
719the same time, then they are simultaneous.
720
721I am very pleased with this suggestion, but for all that I cannot
722regard the matter as quite settled, because I feel constrained to
723raise the following objection:
724
725"Your definition would certainly be right, if only I knew that the
726light by means of which the observer at M perceives the lightning
727flashes travels along the length A arrow M with the same velocity as
728along the length B arrow M. But an examination of this supposition
729would only be possible if we already had at our disposal the means of
730measuring time. It would thus appear as though we were moving here in
731a logical circle."
732
733After further consideration you cast a somewhat disdainful glance at
734me -- and rightly so -- and you declare:
735
736"I maintain my previous definition nevertheless, because in reality it
737assumes absolutely nothing about light. There is only one demand to be
738made of the definition of simultaneity, namely, that in every real
739case it must supply us with an empirical decision as to whether or not
740the conception that has to be defined is fulfilled. That my definition
741satisfies this demand is indisputable. That light requires the same
742time to traverse the path A arrow M as for the path B arrow M is in
743reality neither a supposition nor a hypothesis about the physical
744nature of light, but a stipulation which I can make of my own freewill
745in order to arrive at a definition of simultaneity."
746
747It is clear that this definition can be used to give an exact meaning
748not only to two events, but to as many events as we care to choose,
749and independently of the positions of the scenes of the events with
750respect to the body of reference * (here the railway embankment).
751We are thus led also to a definition of " time " in physics. For this
752purpose we suppose that clocks of identical construction are placed at
753the points A, B and C of the railway line (co-ordinate system) and
754that they are set in such a manner that the positions of their
755pointers are simultaneously (in the above sense) the same. Under these
756conditions we understand by the " time " of an event the reading
757(position of the hands) of that one of these clocks which is in the
758immediate vicinity (in space) of the event. In this manner a
759time-value is associated with every event which is essentially capable
760of observation.
761
762This stipulation contains a further physical hypothesis, the validity
763of which will hardly be doubted without empirical evidence to the
764contrary. It has been assumed that all these clocks go at the same
765rate if they are of identical construction. Stated more exactly: When
766two clocks arranged at rest in different places of a reference-body
767are set in such a manner that a particular position of the pointers of
768the one clock is simultaneous (in the above sense) with the same
769position, of the pointers of the other clock, then identical "
770settings " are always simultaneous (in the sense of the above
771definition).
772
773
774 Notes
775
776*) We suppose further, that, when three events A, B and C occur in
777different places in such a manner that A is simultaneous with B and B
778is simultaneous with C (simultaneous in the sense of the above
779definition), then the criterion for the simultaneity of the pair of
780events A, C is also satisfied. This assumption is a physical
781hypothesis about the the of propagation of light: it must certainly be
782fulfilled if we are to maintain the law of the constancy of the
783velocity of light in vacuo.
784
785
786
787THE RELATIVITY OF SIMULATNEITY
788
789
790Up to now our considerations have been referred to a particular body
791of reference, which we have styled a " railway embankment." We suppose
792a very long train travelling along the rails with the constant
793velocity v and in the direction indicated in Fig 1. People travelling
794in this train will with a vantage view the train as a rigid
795reference-body (co-ordinate system); they regard all events in
796
797 Fig. 01: file fig01.gif
798
799
800reference to the train. Then every event which takes place along the
801line also takes place at a particular point of the train. Also the
802definition of simultaneity can be given relative to the train in
803exactly the same way as with respect to the embankment. As a natural
804consequence, however, the following question arises :
805
806Are two events (e.g. the two strokes of lightning A and B) which are
807simultaneous with reference to the railway embankment also
808simultaneous relatively to the train? We shall show directly that the
809answer must be in the negative.
810
811When we say that the lightning strokes A and B are simultaneous with
812respect to be embankment, we mean: the rays of light emitted at the
813places A and B, where the lightning occurs, meet each other at the
814mid-point M of the length A arrow B of the embankment. But the events
815A and B also correspond to positions A and B on the train. Let M1 be
816the mid-point of the distance A arrow B on the travelling train. Just
817when the flashes (as judged from the embankment) of lightning occur,
818this point M1 naturally coincides with the point M but it moves
819towards the right in the diagram with the velocity v of the train. If
820an observer sitting in the position M1 in the train did not possess
821this velocity, then he would remain permanently at M, and the light
822rays emitted by the flashes of lightning A and B would reach him
823simultaneously, i.e. they would meet just where he is situated. Now in
824reality (considered with reference to the railway embankment) he is
825hastening towards the beam of light coming from B, whilst he is riding
826on ahead of the beam of light coming from A. Hence the observer will
827see the beam of light emitted from B earlier than he will see that
828emitted from A. Observers who take the railway train as their
829reference-body must therefore come to the conclusion that the
830lightning flash B took place earlier than the lightning flash A. We
831thus arrive at the important result:
832
833Events which are simultaneous with reference to the embankment are not
834simultaneous with respect to the train, and vice versa (relativity of
835simultaneity). Every reference-body (co-ordinate system) has its own
836particular time ; unless we are told the reference-body to which the
837statement of time refers, there is no meaning in a statement of the
838time of an event.
839
840Now before the advent of the theory of relativity it had always
841tacitly been assumed in physics that the statement of time had an
842absolute significance, i.e. that it is independent of the state of
843motion of the body of reference. But we have just seen that this
844assumption is incompatible with the most natural definition of
845simultaneity; if we discard this assumption, then the conflict between
846the law of the propagation of light in vacuo and the principle of
847relativity (developed in Section 7) disappears.
848
849We were led to that conflict by the considerations of Section 6,
850which are now no longer tenable. In that section we concluded that the
851man in the carriage, who traverses the distance w per second relative
852to the carriage, traverses the same distance also with respect to the
853embankment in each second of time. But, according to the foregoing
854considerations, the time required by a particular occurrence with
855respect to the carriage must not be considered equal to the duration
856of the same occurrence as judged from the embankment (as
857reference-body). Hence it cannot be contended that the man in walking
858travels the distance w relative to the railway line in a time which is
859equal to one second as judged from the embankment.
860
861Moreover, the considerations of Section 6 are based on yet a second
862assumption, which, in the light of a strict consideration, appears to
863be arbitrary, although it was always tacitly made even before the
864introduction of the theory of relativity.
865
866
867
868ON THE RELATIVITY OF THE CONCEPTION OF DISTANCE
869
870
871Let us consider two particular points on the train * travelling
872along the embankment with the velocity v, and inquire as to their
873distance apart. We already know that it is necessary to have a body of
874reference for the measurement of a distance, with respect to which
875body the distance can be measured up. It is the simplest plan to use
876the train itself as reference-body (co-ordinate system). An observer
877in the train measures the interval by marking off his measuring-rod in
878a straight line (e.g. along the floor of the carriage) as many times
879as is necessary to take him from the one marked point to the other.
880Then the number which tells us how often the rod has to be laid down
881is the required distance.
882
883It is a different matter when the distance has to be judged from the
884railway line. Here the following method suggests itself. If we call
885A^1 and B^1 the two points on the train whose distance apart is
886required, then both of these points are moving with the velocity v
887along the embankment. In the first place we require to determine the
888points A and B of the embankment which are just being passed by the
889two points A^1 and B^1 at a particular time t -- judged from the
890embankment. These points A and B of the embankment can be determined
891by applying the definition of time given in Section 8. The distance
892between these points A and B is then measured by repeated application
893of thee measuring-rod along the embankment.
894
895A priori it is by no means certain that this last measurement will
896supply us with the same result as the first. Thus the length of the
897train as measured from the embankment may be different from that
898obtained by measuring in the train itself. This circumstance leads us
899to a second objection which must be raised against the apparently
900obvious consideration of Section 6. Namely, if the man in the
901carriage covers the distance w in a unit of time -- measured from the
902train, -- then this distance -- as measured from the embankment -- is
903not necessarily also equal to w.
904
905
906 Notes
907
908*) e.g. the middle of the first and of the hundredth carriage.
909
910
911
912THE LORENTZ TRANSFORMATION
913
914
915The results of the last three sections show that the apparent
916incompatibility of the law of propagation of light with the principle
917of relativity (Section 7) has been derived by means of a
918consideration which borrowed two unjustifiable hypotheses from
919classical mechanics; these are as follows:
920
921(1) The time-interval (time) between two events is independent of the
922condition of motion of the body of reference.
923
924(2) The space-interval (distance) between two points of a rigid body
925is independent of the condition of motion of the body of reference.
926
927If we drop these hypotheses, then the dilemma of Section 7
928disappears, because the theorem of the addition of velocities derived
929in Section 6 becomes invalid. The possibility presents itself that
930the law of the propagation of light in vacuo may be compatible with
931the principle of relativity, and the question arises: How have we to
932modify the considerations of Section 6 in order to remove the
933apparent disagreement between these two fundamental results of
934experience? This question leads to a general one. In the discussion of
935Section 6 we have to do with places and times relative both to the
936train and to the embankment. How are we to find the place and time of
937an event in relation to the train, when we know the place and time of
938the event with respect to the railway embankment ? Is there a
939thinkable answer to this question of such a nature that the law of
940transmission of light in vacuo does not contradict the principle of
941relativity ? In other words : Can we conceive of a relation between
942place and time of the individual events relative to both
943reference-bodies, such that every ray of light possesses the velocity
944of transmission c relative to the embankment and relative to the train
945? This question leads to a quite definite positive answer, and to a
946perfectly definite transformation law for the space-time magnitudes of
947an event when changing over from one body of reference to another.
948
949Before we deal with this, we shall introduce the following incidental
950consideration. Up to the present we have only considered events taking
951place along the embankment, which had mathematically to assume the
952function of a straight line. In the manner indicated in Section 2
953we can imagine this reference-body supplemented laterally and in a
954vertical direction by means of a framework of rods, so that an event
955which takes place anywhere can be localised with reference to this
956framework. Fig. 2 Similarly, we can imagine the train travelling with
957the velocity v to be continued across the whole of space, so that
958every event, no matter how far off it may be, could also be localised
959with respect to the second framework. Without committing any
960fundamental error, we can disregard the fact that in reality these
961frameworks would continually interfere with each other, owing to the
962impenetrability of solid bodies. In every such framework we imagine
963three surfaces perpendicular to each other marked out, and designated
964as " co-ordinate planes " (" co-ordinate system "). A co-ordinate
965system K then corresponds to the embankment, and a co-ordinate system
966K' to the train. An event, wherever it may have taken place, would be
967fixed in space with respect to K by the three perpendiculars x, y, z
968on the co-ordinate planes, and with regard to time by a time value t.
969Relative to K1, the same event would be fixed in respect of space and
970time by corresponding values x1, y1, z1, t1, which of course are not
971identical with x, y, z, t. It has already been set forth in detail how
972these magnitudes are to be regarded as results of physical
973measurements.
974
975Obviously our problem can be exactly formulated in the following
976manner. What are the values x1, y1, z1, t1, of an event with respect
977to K1, when the magnitudes x, y, z, t, of the same event with respect
978to K are given ? The relations must be so chosen that the law of the
979transmission of light in vacuo is satisfied for one and the same ray
980of light (and of course for every ray) with respect to K and K1. For
981the relative orientation in space of the co-ordinate systems indicated
982in the diagram ([7]Fig. 2), this problem is solved by means of the
983equations :
984
985 eq. 1: file eq01.gif
986
987 y1 = y
988 z1 = z
989
990 eq. 2: file eq02.gif
991
992This system of equations is known as the " Lorentz transformation." *
993
994If in place of the law of transmission of light we had taken as our
995basis the tacit assumptions of the older mechanics as to the absolute
996character of times and lengths, then instead of the above we should
997have obtained the following equations:
998
999 x1 = x - vt
1000 y1 = y
1001 z1 = z
1002 t1 = t
1003
1004This system of equations is often termed the " Galilei
1005transformation." The Galilei transformation can be obtained from the
1006Lorentz transformation by substituting an infinitely large value for
1007the velocity of light c in the latter transformation.
1008
1009Aided by the following illustration, we can readily see that, in
1010accordance with the Lorentz transformation, the law of the
1011transmission of light in vacuo is satisfied both for the
1012reference-body K and for the reference-body K1. A light-signal is sent
1013along the positive x-axis, and this light-stimulus advances in
1014accordance with the equation
1015
1016 x = ct,
1017
1018i.e. with the velocity c. According to the equations of the Lorentz
1019transformation, this simple relation between x and t involves a
1020relation between x1 and t1. In point of fact, if we substitute for x
1021the value ct in the first and fourth equations of the Lorentz
1022transformation, we obtain:
1023
1024 eq. 3: file eq03.gif
1025
1026
1027 eq. 4: file eq04.gif
1028
1029from which, by division, the expression
1030
1031 x1 = ct1
1032
1033immediately follows. If referred to the system K1, the propagation of
1034light takes place according to this equation. We thus see that the
1035velocity of transmission relative to the reference-body K1 is also
1036equal to c. The same result is obtained for rays of light advancing in
1037any other direction whatsoever. Of cause this is not surprising, since
1038the equations of the Lorentz transformation were derived conformably
1039to this point of view.
1040
1041
1042 Notes
1043
1044*) A simple derivation of the Lorentz transformation is given in
1045Appendix I.
1046
1047
1048
1049THE BEHAVIOUR OF MEASURING-RODS AND CLOCKS IN MOTION
1050
1051
1052Place a metre-rod in the x1-axis of K1 in such a manner that one end
1053(the beginning) coincides with the point x1=0 whilst the other end
1054(the end of the rod) coincides with the point x1=I. What is the length
1055of the metre-rod relatively to the system K? In order to learn this,
1056we need only ask where the beginning of the rod and the end of the rod
1057lie with respect to K at a particular time t of the system K. By means
1058of the first equation of the Lorentz transformation the values of
1059these two points at the time t = 0 can be shown to be
1060
1061 eq. 05a: file eq05a.gif
1062
1063
1064 eq. 05b: file eq05b.gif
1065
1066
1067the distance between the points being eq. 06 .
1068
1069But the metre-rod is moving with the velocity v relative to K. It
1070therefore follows that the length of a rigid metre-rod moving in the
1071direction of its length with a velocity v is eq. 06 of a metre.
1072
1073The rigid rod is thus shorter when in motion than when at rest, and
1074the more quickly it is moving, the shorter is the rod. For the
1075velocity v=c we should have eq. 06a ,
1076
1077and for stiII greater velocities the square-root becomes imaginary.
1078From this we conclude that in the theory of relativity the velocity c
1079plays the part of a limiting velocity, which can neither be reached
1080nor exceeded by any real body.
1081
1082Of course this feature of the velocity c as a limiting velocity also
1083clearly follows from the equations of the Lorentz transformation, for
1084these became meaningless if we choose values of v greater than c.
1085
1086If, on the contrary, we had considered a metre-rod at rest in the
1087x-axis with respect to K, then we should have found that the length of
1088the rod as judged from K1 would have been eq. 06 ;
1089
1090this is quite in accordance with the principle of relativity which
1091forms the basis of our considerations.
1092
1093A Priori it is quite clear that we must be able to learn something
1094about the physical behaviour of measuring-rods and clocks from the
1095equations of transformation, for the magnitudes z, y, x, t, are
1096nothing more nor less than the results of measurements obtainable by
1097means of measuring-rods and clocks. If we had based our considerations
1098on the Galileian transformation we should not have obtained a
1099contraction of the rod as a consequence of its motion.
1100
1101Let us now consider a seconds-clock which is permanently situated at
1102the origin (x1=0) of K1. t1=0 and t1=I are two successive ticks of
1103this clock. The first and fourth equations of the Lorentz
1104transformation give for these two ticks :
1105
1106 t = 0
1107
1108and
1109
1110 eq. 07: file eq07.gif
1111
1112As judged from K, the clock is moving with the velocity v; as judged
1113from this reference-body, the time which elapses between two strokes
1114of the clock is not one second, but
1115
1116 eq. 08: file eq08.gif
1117
1118seconds, i.e. a somewhat larger time. As a consequence of its motion
1119the clock goes more slowly than when at rest. Here also the velocity c
1120plays the part of an unattainable limiting velocity.
1121
1122
1123
1124THEOREM OF THE ADDITION OF VELOCITIES.
1125THE EXPERIMENT OF FIZEAU
1126
1127
1128Now in practice we can move clocks and measuring-rods only with
1129velocities that are small compared with the velocity of light; hence
1130we shall hardly be able to compare the results of the previous section
1131directly with the reality. But, on the other hand, these results must
1132strike you as being very singular, and for that reason I shall now
1133draw another conclusion from the theory, one which can easily be
1134derived from the foregoing considerations, and which has been most
1135elegantly confirmed by experiment.
1136
1137In Section 6 we derived the theorem of the addition of velocities
1138in one direction in the form which also results from the hypotheses of
1139classical mechanics- This theorem can also be deduced readily horn the
1140Galilei transformation (Section 11). In place of the man walking
1141inside the carriage, we introduce a point moving relatively to the
1142co-ordinate system K1 in accordance with the equation
1143
1144 x1 = wt1
1145
1146By means of the first and fourth equations of the Galilei
1147transformation we can express x1 and t1 in terms of x and t, and we
1148then obtain
1149
1150 x = (v + w)t
1151
1152This equation expresses nothing else than the law of motion of the
1153point with reference to the system K (of the man with reference to the
1154embankment). We denote this velocity by the symbol W, and we then
1155obtain, as in Section 6,
1156
1157 W=v+w A)
1158
1159But we can carry out this consideration just as well on the basis of
1160the theory of relativity. In the equation
1161
1162 x1 = wt1 B)
1163
1164we must then express x1and t1 in terms of x and t, making use of the
1165first and fourth equations of the Lorentz transformation. Instead of
1166the equation (A) we then obtain the equation
1167
1168 eq. 09: file eq09.gif
1169
1170
1171which corresponds to the theorem of addition for velocities in one
1172direction according to the theory of relativity. The question now
1173arises as to which of these two theorems is the better in accord with
1174experience. On this point we axe enlightened by a most important
1175experiment which the brilliant physicist Fizeau performed more than
1176half a century ago, and which has been repeated since then by some of
1177the best experimental physicists, so that there can be no doubt about
1178its result. The experiment is concerned with the following question.
1179Light travels in a motionless liquid with a particular velocity w. How
1180quickly does it travel in the direction of the arrow in the tube T
1181(see the accompanying diagram, Fig. 3) when the liquid above
1182mentioned is flowing through the tube with a velocity v ?
1183
1184In accordance with the principle of relativity we shall certainly have
1185to take for granted that the propagation of light always takes place
1186with the same velocity w with respect to the liquid, whether the
1187latter is in motion with reference to other bodies or not. The
1188velocity of light relative to the liquid and the velocity of the
1189latter relative to the tube are thus known, and we require the
1190velocity of light relative to the tube.
1191
1192It is clear that we have the problem of Section 6 again before us. The
1193tube plays the part of the railway embankment or of the co-ordinate
1194system K, the liquid plays the part of the carriage or of the
1195co-ordinate system K1, and finally, the light plays the part of the
1196
1197 Figure 03: file fig03.gif
1198
1199
1200man walking along the carriage, or of the moving point in the present
1201section. If we denote the velocity of the light relative to the tube
1202by W, then this is given by the equation (A) or (B), according as the
1203Galilei transformation or the Lorentz transformation corresponds to
1204the facts. Experiment * decides in favour of equation (B) derived
1205from the theory of relativity, and the agreement is, indeed, very
1206exact. According to recent and most excellent measurements by Zeeman,
1207the influence of the velocity of flow v on the propagation of light is
1208represented by formula (B) to within one per cent.
1209
1210Nevertheless we must now draw attention to the fact that a theory of
1211this phenomenon was given by H. A. Lorentz long before the statement
1212of the theory of relativity. This theory was of a purely
1213electrodynamical nature, and was obtained by the use of particular
1214hypotheses as to the electromagnetic structure of matter. This
1215circumstance, however, does not in the least diminish the
1216conclusiveness of the experiment as a crucial test in favour of the
1217theory of relativity, for the electrodynamics of Maxwell-Lorentz, on
1218which the original theory was based, in no way opposes the theory of
1219relativity. Rather has the latter been developed trom electrodynamics
1220as an astoundingly simple combination and generalisation of the
1221hypotheses, formerly independent of each other, on which
1222electrodynamics was built.
1223
1224
1225 Notes
1226
1227*) Fizeau found eq. 10 , where eq. 11
1228
1229is the index of refraction of the liquid. On the other hand, owing to
1230the smallness of eq. 12 as compared with I,
1231
1232we can replace (B) in the first place by eq. 13 , or to the same order
1233of approximation by
1234
1235eq. 14 , which agrees with Fizeau's result.
1236
1237
1238
1239THE HEURISTIC VALUE OF THE THEORY OF RELATIVITY
1240
1241
1242Our train of thought in the foregoing pages can be epitomised in the
1243following manner. Experience has led to the conviction that, on the
1244one hand, the principle of relativity holds true and that on the other
1245hand the velocity of transmission of light in vacuo has to be
1246considered equal to a constant c. By uniting these two postulates we
1247obtained the law of transformation for the rectangular co-ordinates x,
1248y, z and the time t of the events which constitute the processes of
1249nature. In this connection we did not obtain the Galilei
1250transformation, but, differing from classical mechanics, the Lorentz
1251transformation.
1252
1253The law of transmission of light, the acceptance of which is justified
1254by our actual knowledge, played an important part in this process of
1255thought. Once in possession of the Lorentz transformation, however, we
1256can combine this with the principle of relativity, and sum up the
1257theory thus:
1258
1259Every general law of nature must be so constituted that it is
1260transformed into a law of exactly the same form when, instead of the
1261space-time variables x, y, z, t of the original coordinate system K,
1262we introduce new space-time variables x1, y1, z1, t1 of a co-ordinate
1263system K1. In this connection the relation between the ordinary and
1264the accented magnitudes is given by the Lorentz transformation. Or in
1265brief : General laws of nature are co-variant with respect to Lorentz
1266transformations.
1267
1268This is a definite mathematical condition that the theory of
1269relativity demands of a natural law, and in virtue of this, the theory
1270becomes a valuable heuristic aid in the search for general laws of
1271nature. If a general law of nature were to be found which did not
1272satisfy this condition, then at least one of the two fundamental
1273assumptions of the theory would have been disproved. Let us now
1274examine what general results the latter theory has hitherto evinced.
1275
1276
1277
1278GENERAL RESULTS OF THE THEORY
1279
1280
1281It is clear from our previous considerations that the (special) theory
1282of relativity has grown out of electrodynamics and optics. In these
1283fields it has not appreciably altered the predictions of theory, but
1284it has considerably simplified the theoretical structure, i.e. the
1285derivation of laws, and -- what is incomparably more important -- it
1286has considerably reduced the number of independent hypothese forming
1287the basis of theory. The special theory of relativity has rendered the
1288Maxwell-Lorentz theory so plausible, that the latter would have been
1289generally accepted by physicists even if experiment had decided less
1290unequivocally in its favour.
1291
1292Classical mechanics required to be modified before it could come into
1293line with the demands of the special theory of relativity. For the
1294main part, however, this modification affects only the laws for rapid
1295motions, in which the velocities of matter v are not very small as
1296compared with the velocity of light. We have experience of such rapid
1297motions only in the case of electrons and ions; for other motions the
1298variations from the laws of classical mechanics are too small to make
1299themselves evident in practice. We shall not consider the motion of
1300stars until we come to speak of the general theory of relativity. In
1301accordance with the theory of relativity the kinetic energy of a
1302material point of mass m is no longer given by the well-known
1303expression
1304
1305 eq. 15: file eq15.gif
1306
1307but by the expression
1308
1309 eq. 16: file eq16.gif
1310
1311
1312This expression approaches infinity as the velocity v approaches the
1313velocity of light c. The velocity must therefore always remain less
1314than c, however great may be the energies used to produce the
1315acceleration. If we develop the expression for the kinetic energy in
1316the form of a series, we obtain
1317
1318 eq. 17: file eq17.gif
1319
1320
1321When eq. 18 is small compared with unity, the third of these terms is
1322always small in comparison with the second,
1323
1324which last is alone considered in classical mechanics. The first term
1325mc^2 does not contain the velocity, and requires no consideration if
1326we are only dealing with the question as to how the energy of a
1327point-mass; depends on the velocity. We shall speak of its essential
1328significance later.
1329
1330The most important result of a general character to which the special
1331theory of relativity has led is concerned with the conception of mass.
1332Before the advent of relativity, physics recognised two conservation
1333laws of fundamental importance, namely, the law of the canservation of
1334energy and the law of the conservation of mass these two fundamental
1335laws appeared to be quite independent of each other. By means of the
1336theory of relativity they have been united into one law. We shall now
1337briefly consider how this unification came about, and what meaning is
1338to be attached to it.
1339
1340The principle of relativity requires that the law of the concervation
1341of energy should hold not only with reference to a co-ordinate system
1342K, but also with respect to every co-ordinate system K1 which is in a
1343state of uniform motion of translation relative to K, or, briefly,
1344relative to every " Galileian " system of co-ordinates. In contrast to
1345classical mechanics; the Lorentz transformation is the deciding factor
1346in the transition from one such system to another.
1347
1348By means of comparatively simple considerations we are led to draw the
1349following conclusion from these premises, in conjunction with the
1350fundamental equations of the electrodynamics of Maxwell: A body moving
1351with the velocity v, which absorbs * an amount of energy E[0] in
1352the form of radiation without suffering an alteration in velocity in
1353the process, has, as a consequence, its energy increased by an amount
1354
1355 eq. 19: file eq19.gif
1356
1357In consideration of the expression given above for the kinetic energy
1358of the body, the required energy of the body comes out to be
1359
1360 eq. 20: file eq20.gif
1361
1362
1363Thus the body has the same energy as a body of mass
1364
1365 eq.21: file eq21.gif
1366
1367moving with the velocity v. Hence we can say: If a body takes up an
1368amount of energy E[0], then its inertial mass increases by an amount
1369
1370 eq. 22: file eq22.gif
1371
1372
1373the inertial mass of a body is not a constant but varies according to
1374the change in the energy of the body. The inertial mass of a system of
1375bodies can even be regarded as a measure of its energy. The law of the
1376conservation of the mass of a system becomes identical with the law of
1377the conservation of energy, and is only valid provided that the system
1378neither takes up nor sends out energy. Writing the expression for the
1379energy in the form
1380
1381 eq. 23: file eq23.gif
1382
1383we see that the term mc^2, which has hitherto attracted our attention,
1384is nothing else than the energy possessed by the body ** before it
1385absorbed the energy E[0].
1386
1387A direct comparison of this relation with experiment is not possible
1388at the present time (1920; see *** Note, p. 48), owing to the fact that
1389the changes in energy E[0] to which we can Subject a system are not
1390large enough to make themselves perceptible as a change in the
1391inertial mass of the system.
1392
1393 eq. 22: file eq22.gif
1394
1395
1396is too small in comparison with the mass m, which was present before
1397the alteration of the energy. It is owing to this circumstance that
1398classical mechanics was able to establish successfully the
1399conservation of mass as a law of independent validity.
1400
1401Let me add a final remark of a fundamental nature. The success of the
1402Faraday-Maxwell interpretation of electromagnetic action at a distance
1403resulted in physicists becoming convinced that there are no such
1404things as instantaneous actions at a distance (not involving an
1405intermediary medium) of the type of Newton's law of gravitation.
1406According to the theory of relativity, action at a distance with the
1407velocity of light always takes the place of instantaneous action at a
1408distance or of action at a distance with an infinite velocity of
1409transmission. This is connected with the fact that the velocity c
1410plays a fundamental role in this theory. In Part II we shall see in
1411what way this result becomes modified in the general theory of
1412relativity.
1413
1414
1415 Notes
1416
1417*) E[0] is the energy taken up, as judged from a co-ordinate system
1418moving with the body.
1419
1420**) As judged from a co-ordinate system moving with the body.
1421
1422***[Note] The equation E = mc^2 has been thoroughly proved time and
1423again since this time.
1424
1425
1426
1427EXPERIENCE AND THE SPECIAL THEORY OF RELATIVITY
1428
1429
1430To what extent is the special theory of relativity supported by
1431experience? This question is not easily answered for the reason
1432already mentioned in connection with the fundamental experiment of
1433Fizeau. The special theory of relativity has crystallised out from the
1434Maxwell-Lorentz theory of electromagnetic phenomena. Thus all facts of
1435experience which support the electromagnetic theory also support the
1436theory of relativity. As being of particular importance, I mention
1437here the fact that the theory of relativity enables us to predict the
1438effects produced on the light reaching us from the fixed stars. These
1439results are obtained in an exceedingly simple manner, and the effects
1440indicated, which are due to the relative motion of the earth with
1441reference to those fixed stars are found to be in accord with
1442experience. We refer to the yearly movement of the apparent position
1443of the fixed stars resulting from the motion of the earth round the
1444sun (aberration), and to the influence of the radial components of the
1445relative motions of the fixed stars with respect to the earth on the
1446colour of the light reaching us from them. The latter effect manifests
1447itself in a slight displacement of the spectral lines of the light
1448transmitted to us from a fixed star, as compared with the position of
1449the same spectral lines when they are produced by a terrestrial source
1450of light (Doppler principle). The experimental arguments in favour of
1451the Maxwell-Lorentz theory, which are at the same time arguments in
1452favour of the theory of relativity, are too numerous to be set forth
1453here. In reality they limit the theoretical possibilities to such an
1454extent, that no other theory than that of Maxwell and Lorentz has been
1455able to hold its own when tested by experience.
1456
1457But there are two classes of experimental facts hitherto obtained
1458which can be represented in the Maxwell-Lorentz theory only by the
1459introduction of an auxiliary hypothesis, which in itself -- i.e.
1460without making use of the theory of relativity -- appears extraneous.
1461
1462It is known that cathode rays and the so-called b-rays emitted by
1463radioactive substances consist of negatively electrified particles
1464(electrons) of very small inertia and large velocity. By examining the
1465deflection of these rays under the influence of electric and magnetic
1466fields, we can study the law of motion of these particles very
1467exactly.
1468
1469In the theoretical treatment of these electrons, we are faced with the
1470difficulty that electrodynamic theory of itself is unable to give an
1471account of their nature. For since electrical masses of one sign repel
1472each other, the negative electrical masses constituting the electron
1473would necessarily be scattered under the influence of their mutual
1474repulsions, unless there are forces of another kind operating between
1475them, the nature of which has hitherto remained obscure to us.* If
1476we now assume that the relative distances between the electrical
1477masses constituting the electron remain unchanged during the motion of
1478the electron (rigid connection in the sense of classical mechanics),
1479we arrive at a law of motion of the electron which does not agree with
1480experience. Guided by purely formal points of view, H. A. Lorentz was
1481the first to introduce the hypothesis that the form of the electron
1482experiences a contraction in the direction of motion in consequence of
1483that motion. the contracted length being proportional to the
1484expression
1485
1486 eq. 05: file eq05.gif
1487
1488This, hypothesis, which is not justifiable by any electrodynamical
1489facts, supplies us then with that particular law of motion which has
1490been confirmed with great precision in recent years.
1491
1492The theory of relativity leads to the same law of motion, without
1493requiring any special hypothesis whatsoever as to the structure and
1494the behaviour of the electron. We arrived at a similar conclusion in
1495Section 13 in connection with the experiment of Fizeau, the result
1496of which is foretold by the theory of relativity without the necessity
1497of drawing on hypotheses as to the physical nature of the liquid.
1498
1499The second class of facts to which we have alluded has reference to
1500the question whether or not the motion of the earth in space can be
1501made perceptible in terrestrial experiments. We have already remarked
1502in Section 5 that all attempts of this nature led to a negative
1503result. Before the theory of relativity was put forward, it was
1504difficult to become reconciled to this negative result, for reasons
1505now to be discussed. The inherited prejudices about time and space did
1506not allow any doubt to arise as to the prime importance of the
1507Galileian transformation for changing over from one body of reference
1508to another. Now assuming that the Maxwell-Lorentz equations hold for a
1509reference-body K, we then find that they do not hold for a
1510reference-body K1 moving uniformly with respect to K, if we assume
1511that the relations of the Galileian transformstion exist between the
1512co-ordinates of K and K1. It thus appears that, of all Galileian
1513co-ordinate systems, one (K) corresponding to a particular state of
1514motion is physically unique. This result was interpreted physically by
1515regarding K as at rest with respect to a hypothetical æther of space.
1516On the other hand, all coordinate systems K1 moving relatively to K
1517were to be regarded as in motion with respect to the æther. To this
1518motion of K1 against the æther ("æther-drift " relative to K1) were
1519attributed the more complicated laws which were supposed to hold
1520relative to K1. Strictly speaking, such an æther-drift ought also to
1521be assumed relative to the earth, and for a long time the efforts of
1522physicists were devoted to attempts to detect the existence of an
1523æther-drift at the earth's surface.
1524
1525In one of the most notable of these attempts Michelson devised a
1526method which appears as though it must be decisive. Imagine two
1527mirrors so arranged on a rigid body that the reflecting surfaces face
1528each other. A ray of light requires a perfectly definite time T to
1529pass from one mirror to the other and back again, if the whole system
1530be at rest with respect to the æther. It is found by calculation,
1531however, that a slightly different time T1 is required for this
1532process, if the body, together with the mirrors, be moving relatively
1533to the æther. And yet another point: it is shown by calculation that
1534for a given velocity v with reference to the æther, this time T1 is
1535different when the body is moving perpendicularly to the planes of the
1536mirrors from that resulting when the motion is parallel to these
1537planes. Although the estimated difference between these two times is
1538exceedingly small, Michelson and Morley performed an experiment
1539involving interference in which this difference should have been
1540clearly detectable. But the experiment gave a negative result -- a
1541fact very perplexing to physicists. Lorentz and FitzGerald rescued the
1542theory from this difficulty by assuming that the motion of the body
1543relative to the æther produces a contraction of the body in the
1544direction of motion, the amount of contraction being just sufficient
1545to compensate for the differeace in time mentioned above. Comparison
1546with the discussion in Section 11 shows that also from the
1547standpoint of the theory of relativity this solution of the difficulty
1548was the right one. But on the basis of the theory of relativity the
1549method of interpretation is incomparably more satisfactory. According
1550to this theory there is no such thing as a " specially favoured "
1551(unique) co-ordinate system to occasion the introduction of the
1552æther-idea, and hence there can be no æther-drift, nor any experiment
1553with which to demonstrate it. Here the contraction of moving bodies
1554follows from the two fundamental principles of the theory, without the
1555introduction of particular hypotheses ; and as the prime factor
1556involved in this contraction we find, not the motion in itself, to
1557which we cannot attach any meaning, but the motion with respect to the
1558body of reference chosen in the particular case in point. Thus for a
1559co-ordinate system moving with the earth the mirror system of
1560Michelson and Morley is not shortened, but it is shortened for a
1561co-ordinate system which is at rest relatively to the sun.
1562
1563
1564 Notes
1565
1566*) The general theory of relativity renders it likely that the
1567electrical masses of an electron are held together by gravitational
1568forces.
1569
1570
1571
1572MINKOWSKI'S FOUR-DIMENSIONAL SPACE
1573
1574
1575The non-mathematician is seized by a mysterious shuddering when he
1576hears of "four-dimensional" things, by a feeling not unlike that
1577awakened by thoughts of the occult. And yet there is no more
1578common-place statement than that the world in which we live is a
1579four-dimensional space-time continuum.
1580
1581Space is a three-dimensional continuum. By this we mean that it is
1582possible to describe the position of a point (at rest) by means of
1583three numbers (co-ordinales) x, y, z, and that there is an indefinite
1584number of points in the neighbourhood of this one, the position of
1585which can be described by co-ordinates such as x[1], y[1], z[1], which
1586may be as near as we choose to the respective values of the
1587co-ordinates x, y, z, of the first point. In virtue of the latter
1588property we speak of a " continuum," and owing to the fact that there
1589are three co-ordinates we speak of it as being " three-dimensional."
1590
1591Similarly, the world of physical phenomena which was briefly called "
1592world " by Minkowski is naturally four dimensional in the space-time
1593sense. For it is composed of individual events, each of which is
1594described by four numbers, namely, three space co-ordinates x, y, z,
1595and a time co-ordinate, the time value t. The" world" is in this sense
1596also a continuum; for to every event there are as many "neighbouring"
1597events (realised or at least thinkable) as we care to choose, the
1598co-ordinates x[1], y[1], z[1], t[1] of which differ by an indefinitely
1599small amount from those of the event x, y, z, t originally considered.
1600That we have not been accustomed to regard the world in this sense as
1601a four-dimensional continuum is due to the fact that in physics,
1602before the advent of the theory of relativity, time played a different
1603and more independent role, as compared with the space coordinates. It
1604is for this reason that we have been in the habit of treating time as
1605an independent continuum. As a matter of fact, according to classical
1606mechanics, time is absolute, i.e. it is independent of the position
1607and the condition of motion of the system of co-ordinates. We see this
1608expressed in the last equation of the Galileian transformation (t1 =
1609t)
1610
1611The four-dimensional mode of consideration of the "world" is natural
1612on the theory of relativity, since according to this theory time is
1613robbed of its independence. This is shown by the fourth equation of
1614the Lorentz transformation:
1615
1616 eq. 24: file eq24.gif
1617
1618
1619Moreover, according to this equation the time difference Dt1 of two
1620events with respect to K1 does not in general vanish, even when the
1621time difference Dt1 of the same events with reference to K vanishes.
1622Pure " space-distance " of two events with respect to K results in "
1623time-distance " of the same events with respect to K. But the
1624discovery of Minkowski, which was of importance for the formal
1625development of the theory of relativity, does not lie here. It is to
1626be found rather in the fact of his recognition that the
1627four-dimensional space-time continuum of the theory of relativity, in
1628its most essential formal properties, shows a pronounced relationship
1629to the three-dimensional continuum of Euclidean geometrical
1630space.* In order to give due prominence to this relationship,
1631however, we must replace the usual time co-ordinate t by an imaginary
1632magnitude eq. 25 proportional to it. Under these conditions, the
1633natural laws satisfying the demands of the (special) theory of
1634relativity assume mathematical forms, in which the time co-ordinate
1635plays exactly the same role as the three space co-ordinates. Formally,
1636these four co-ordinates correspond exactly to the three space
1637co-ordinates in Euclidean geometry. It must be clear even to the
1638non-mathematician that, as a consequence of this purely formal
1639addition to our knowledge, the theory perforce gained clearness in no
1640mean measure.
1641
1642These inadequate remarks can give the reader only a vague notion of
1643the important idea contributed by Minkowski. Without it the general
1644theory of relativity, of which the fundamental ideas are developed in
1645the following pages, would perhaps have got no farther than its long
1646clothes. Minkowski's work is doubtless difficult of access to anyone
1647inexperienced in mathematics, but since it is not necessary to have a
1648very exact grasp of this work in order to understand the fundamental
1649ideas of either the special or the general theory of relativity, I
1650shall leave it here at present, and revert to it only towards the end
1651of Part 2.
1652
1653
1654 Notes
1655
1656*) Cf. the somewhat more detailed discussion in Appendix II.
1657
1658
1659
1660
1661PART II
1662
1663THE GENERAL THEORY OF RELATIVITY
1664
1665
1666SPECIAL AND GENERAL PRINCIPLE OF RELATIVITY
1667
1668
1669The basal principle, which was the pivot of all our previous
1670considerations, was the special principle of relativity, i.e. the
1671principle of the physical relativity of all uniform motion. Let as
1672once more analyse its meaning carefully.
1673
1674It was at all times clear that, from the point of view of the idea it
1675conveys to us, every motion must be considered only as a relative
1676motion. Returning to the illustration we have frequently used of the
1677embankment and the railway carriage, we can express the fact of the
1678motion here taking place in the following two forms, both of which are
1679equally justifiable :
1680
1681(a) The carriage is in motion relative to the embankment,
1682(b) The embankment is in motion relative to the carriage.
1683
1684In (a) the embankment, in (b) the carriage, serves as the body of
1685reference in our statement of the motion taking place. If it is simply
1686a question of detecting or of describing the motion involved, it is in
1687principle immaterial to what reference-body we refer the motion. As
1688already mentioned, this is self-evident, but it must not be confused
1689with the much more comprehensive statement called "the principle of
1690relativity," which we have taken as the basis of our investigations.
1691
1692The principle we have made use of not only maintains that we may
1693equally well choose the carriage or the embankment as our
1694reference-body for the description of any event (for this, too, is
1695self-evident). Our principle rather asserts what follows : If we
1696formulate the general laws of nature as they are obtained from
1697experience, by making use of
1698
1699(a) the embankment as reference-body,
1700(b) the railway carriage as reference-body,
1701
1702then these general laws of nature (e.g. the laws of mechanics or the
1703law of the propagation of light in vacuo) have exactly the same form
1704in both cases. This can also be expressed as follows : For the
1705physical description of natural processes, neither of the reference
1706bodies K, K1 is unique (lit. " specially marked out ") as compared
1707with the other. Unlike the first, this latter statement need not of
1708necessity hold a priori; it is not contained in the conceptions of "
1709motion" and " reference-body " and derivable from them; only
1710experience can decide as to its correctness or incorrectness.
1711
1712Up to the present, however, we have by no means maintained the
1713equivalence of all bodies of reference K in connection with the
1714formulation of natural laws. Our course was more on the following
1715Iines. In the first place, we started out from the assumption that
1716there exists a reference-body K, whose condition of motion is such
1717that the Galileian law holds with respect to it : A particle left to
1718itself and sufficiently far removed from all other particles moves
1719uniformly in a straight line. With reference to K (Galileian
1720reference-body) the laws of nature were to be as simple as possible.
1721But in addition to K, all bodies of reference K1 should be given
1722preference in this sense, and they should be exactly equivalent to K
1723for the formulation of natural laws, provided that they are in a state
1724of uniform rectilinear and non-rotary motion with respect to K ; all
1725these bodies of reference are to be regarded as Galileian
1726reference-bodies. The validity of the principle of relativity was
1727assumed only for these reference-bodies, but not for others (e.g.
1728those possessing motion of a different kind). In this sense we speak
1729of the special principle of relativity, or special theory of
1730relativity.
1731
1732In contrast to this we wish to understand by the "general principle of
1733relativity" the following statement : All bodies of reference K, K1,
1734etc., are equivalent for the description of natural phenomena
1735(formulation of the general laws of nature), whatever may be their
1736state of motion. But before proceeding farther, it ought to be pointed
1737out that this formulation must be replaced later by a more abstract
1738one, for reasons which will become evident at a later stage.
1739
1740Since the introduction of the special principle of relativity has been
1741justified, every intellect which strives after generalisation must
1742feel the temptation to venture the step towards the general principle
1743of relativity. But a simple and apparently quite reliable
1744consideration seems to suggest that, for the present at any rate,
1745there is little hope of success in such an attempt; Let us imagine
1746ourselves transferred to our old friend the railway carriage, which is
1747travelling at a uniform rate. As long as it is moving unifromly, the
1748occupant of the carriage is not sensible of its motion, and it is for
1749this reason that he can without reluctance interpret the facts of the
1750case as indicating that the carriage is at rest, but the embankment in
1751motion. Moreover, according to the special principle of relativity,
1752this interpretation is quite justified also from a physical point of
1753view.
1754
1755If the motion of the carriage is now changed into a non-uniform
1756motion, as for instance by a powerful application of the brakes, then
1757the occupant of the carriage experiences a correspondingly powerful
1758jerk forwards. The retarded motion is manifested in the mechanical
1759behaviour of bodies relative to the person in the railway carriage.
1760The mechanical behaviour is different from that of the case previously
1761considered, and for this reason it would appear to be impossible that
1762the same mechanical laws hold relatively to the non-uniformly moving
1763carriage, as hold with reference to the carriage when at rest or in
1764uniform motion. At all events it is clear that the Galileian law does
1765not hold with respect to the non-uniformly moving carriage. Because of
1766this, we feel compelled at the present juncture to grant a kind of
1767absolute physical reality to non-uniform motion, in opposition to the
1768general principle of relatvity. But in what follows we shall soon see
1769that this conclusion cannot be maintained.
1770
1771
1772
1773THE GRAVITATIONAL FIELD
1774
1775
1776"If we pick up a stone and then let it go, why does it fall to the
1777ground ?" The usual answer to this question is: "Because it is
1778attracted by the earth." Modern physics formulates the answer rather
1779differently for the following reason. As a result of the more careful
1780study of electromagnetic phenomena, we have come to regard action at a
1781distance as a process impossible without the intervention of some
1782intermediary medium. If, for instance, a magnet attracts a piece of
1783iron, we cannot be content to regard this as meaning that the magnet
1784acts directly on the iron through the intermediate empty space, but we
1785are constrained to imagine -- after the manner of Faraday -- that the
1786magnet always calls into being something physically real in the space
1787around it, that something being what we call a "magnetic field." In
1788its turn this magnetic field operates on the piece of iron, so that
1789the latter strives to move towards the magnet. We shall not discuss
1790here the justification for this incidental conception, which is indeed
1791a somewhat arbitrary one. We shall only mention that with its aid
1792electromagnetic phenomena can be theoretically represented much more
1793satisfactorily than without it, and this applies particularly to the
1794transmission of electromagnetic waves. The effects of gravitation also
1795are regarded in an analogous manner.
1796
1797The action of the earth on the stone takes place indirectly. The earth
1798produces in its surrounding a gravitational field, which acts on the
1799stone and produces its motion of fall. As we know from experience, the
1800intensity of the action on a body dimishes according to a quite
1801definite law, as we proceed farther and farther away from the earth.
1802From our point of view this means : The law governing the properties
1803of the gravitational field in space must be a perfectly definite one,
1804in order correctly to represent the diminution of gravitational action
1805with the distance from operative bodies. It is something like this:
1806The body (e.g. the earth) produces a field in its immediate
1807neighbourhood directly; the intensity and direction of the field at
1808points farther removed from the body are thence determined by the law
1809which governs the properties in space of the gravitational fields
1810themselves.
1811
1812In contrast to electric and magnetic fields, the gravitational field
1813exhibits a most remarkable property, which is of fundamental
1814importance for what follows. Bodies which are moving under the sole
1815influence of a gravitational field receive an acceleration, which does
1816not in the least depend either on the material or on the physical
1817state of the body. For instance, a piece of lead and a piece of wood
1818fall in exactly the same manner in a gravitational field (in vacuo),
1819when they start off from rest or with the same initial velocity. This
1820law, which holds most accurately, can be expressed in a different form
1821in the light of the following consideration.
1822
1823According to Newton's law of motion, we have
1824
1825(Force) = (inertial mass) x (acceleration),
1826
1827where the "inertial mass" is a characteristic constant of the
1828accelerated body. If now gravitation is the cause of the acceleration,
1829we then have
1830
1831(Force) = (gravitational mass) x (intensity of the gravitational
1832field),
1833
1834where the "gravitational mass" is likewise a characteristic constant
1835for the body. From these two relations follows:
1836
1837 eq. 26: file eq26.gif
1838
1839
1840If now, as we find from experience, the acceleration is to be
1841independent of the nature and the condition of the body and always the
1842same for a given gravitational field, then the ratio of the
1843gravitational to the inertial mass must likewise be the same for all
1844bodies. By a suitable choice of units we can thus make this ratio
1845equal to unity. We then have the following law: The gravitational mass
1846of a body is equal to its inertial law.
1847
1848It is true that this important law had hitherto been recorded in
1849mechanics, but it had not been interpreted. A satisfactory
1850interpretation can be obtained only if we recognise the following fact
1851: The same quality of a body manifests itself according to
1852circumstances as " inertia " or as " weight " (lit. " heaviness '). In
1853the following section we shall show to what extent this is actually
1854the case, and how this question is connected with the general
1855postulate of relativity.
1856
1857
1858
1859
1860THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS
1861AS AN ARGUMENT FOR THE GENERAL POSTULE OF RELATIVITY
1862
1863
1864We imagine a large portion of empty space, so far removed from stars
1865and other appreciable masses, that we have before us approximately the
1866conditions required by the fundamental law of Galilei. It is then
1867possible to choose a Galileian reference-body for this part of space
1868(world), relative to which points at rest remain at rest and points in
1869motion continue permanently in uniform rectilinear motion. As
1870reference-body let us imagine a spacious chest resembling a room with
1871an observer inside who is equipped with apparatus. Gravitation
1872naturally does not exist for this observer. He must fasten himself
1873with strings to the floor, otherwise the slightest impact against the
1874floor will cause him to rise slowly towards the ceiling of the room.
1875
1876To the middle of the lid of the chest is fixed externally a hook with
1877rope attached, and now a " being " (what kind of a being is immaterial
1878to us) begins pulling at this with a constant force. The chest
1879together with the observer then begin to move "upwards" with a
1880uniformly accelerated motion. In course of time their velocity will
1881reach unheard-of values -- provided that we are viewing all this from
1882another reference-body which is not being pulled with a rope.
1883
1884But how does the man in the chest regard the Process ? The
1885acceleration of the chest will be transmitted to him by the reaction
1886of the floor of the chest. He must therefore take up this pressure by
1887means of his legs if he does not wish to be laid out full length on
1888the floor. He is then standing in the chest in exactly the same way as
1889anyone stands in a room of a home on our earth. If he releases a body
1890which he previously had in his land, the accelertion of the chest will
1891no longer be transmitted to this body, and for this reason the body
1892will approach the floor of the chest with an accelerated relative
1893motion. The observer will further convince himself that the
1894acceleration of the body towards the floor of the chest is always of
1895the same magnitude, whatever kind of body he may happen to use for the
1896experiment.
1897
1898Relying on his knowledge of the gravitational field (as it was
1899discussed in the preceding section), the man in the chest will thus
1900come to the conclusion that he and the chest are in a gravitational
1901field which is constant with regard to time. Of course he will be
1902puzzled for a moment as to why the chest does not fall in this
1903gravitational field. just then, however, he discovers the hook in the
1904middle of the lid of the chest and the rope which is attached to it,
1905and he consequently comes to the conclusion that the chest is
1906suspended at rest in the gravitational field.
1907
1908Ought we to smile at the man and say that he errs in his conclusion ?
1909I do not believe we ought to if we wish to remain consistent ; we must
1910rather admit that his mode of grasping the situation violates neither
1911reason nor known mechanical laws. Even though it is being accelerated
1912with respect to the "Galileian space" first considered, we can
1913nevertheless regard the chest as being at rest. We have thus good
1914grounds for extending the principle of relativity to include bodies of
1915reference which are accelerated with respect to each other, and as a
1916result we have gained a powerful argument for a generalised postulate
1917of relativity.
1918
1919We must note carefully that the possibility of this mode of
1920interpretation rests on the fundamental property of the gravitational
1921field of giving all bodies the same acceleration, or, what comes to
1922the same thing, on the law of the equality of inertial and
1923gravitational mass. If this natural law did not exist, the man in the
1924accelerated chest would not be able to interpret the behaviour of the
1925bodies around him on the supposition of a gravitational field, and he
1926would not be justified on the grounds of experience in supposing his
1927reference-body to be " at rest."
1928
1929Suppose that the man in the chest fixes a rope to the inner side of
1930the lid, and that he attaches a body to the free end of the rope. The
1931result of this will be to strech the rope so that it will hang "
1932vertically " downwards. If we ask for an opinion of the cause of
1933tension in the rope, the man in the chest will say: "The suspended
1934body experiences a downward force in the gravitational field, and this
1935is neutralised by the tension of the rope ; what determines the
1936magnitude of the tension of the rope is the gravitational mass of the
1937suspended body." On the other hand, an observer who is poised freely
1938in space will interpret the condition of things thus : " The rope must
1939perforce take part in the accelerated motion of the chest, and it
1940transmits this motion to the body attached to it. The tension of the
1941rope is just large enough to effect the acceleration of the body. That
1942which determines the magnitude of the tension of the rope is the
1943inertial mass of the body." Guided by this example, we see that our
1944extension of the principle of relativity implies the necessity of the
1945law of the equality of inertial and gravitational mass. Thus we have
1946obtained a physical interpretation of this law.
1947
1948From our consideration of the accelerated chest we see that a general
1949theory of relativity must yield important results on the laws of
1950gravitation. In point of fact, the systematic pursuit of the general
1951idea of relativity has supplied the laws satisfied by the
1952gravitational field. Before proceeding farther, however, I must warn
1953the reader against a misconception suggested by these considerations.
1954A gravitational field exists for the man in the chest, despite the
1955fact that there was no such field for the co-ordinate system first
1956chosen. Now we might easily suppose that the existence of a
1957gravitational field is always only an apparent one. We might also
1958think that, regardless of the kind of gravitational field which may be
1959present, we could always choose another reference-body such that no
1960gravitational field exists with reference to it. This is by no means
1961true for all gravitational fields, but only for those of quite special
1962form. It is, for instance, impossible to choose a body of reference
1963such that, as judged from it, the gravitational field of the earth (in
1964its entirety) vanishes.
1965
1966We can now appreciate why that argument is not convincing, which we
1967brought forward against the general principle of relativity at theend
1968of Section 18. It is certainly true that the observer in the
1969railway carriage experiences a jerk forwards as a result of the
1970application of the brake, and that he recognises, in this the
1971non-uniformity of motion (retardation) of the carriage. But he is
1972compelled by nobody to refer this jerk to a " real " acceleration
1973(retardation) of the carriage. He might also interpret his experience
1974thus: " My body of reference (the carriage) remains permanently at
1975rest. With reference to it, however, there exists (during the period
1976of application of the brakes) a gravitational field which is directed
1977forwards and which is variable with respect to time. Under the
1978influence of this field, the embankment together with the earth moves
1979non-uniformly in such a manner that their original velocity in the
1980backwards direction is continuously reduced."
1981
1982
1983
1984IN WHAT RESPECTS ARE THE FOUNDATIONS OF CLASSICAL MECHANICS AND OF THE
1985SPECIAL THEORY OF RELATIVITY UNSATISFACTORY?
1986
1987
1988We have already stated several times that classical mechanics starts
1989out from the following law: Material particles sufficiently far
1990removed from other material particles continue to move uniformly in a
1991straight line or continue in a state of rest. We have also repeatedly
1992emphasised that this fundamental law can only be valid for bodies of
1993reference K which possess certain unique states of motion, and which
1994are in uniform translational motion relative to each other. Relative
1995to other reference-bodies K the law is not valid. Both in classical
1996mechanics and in the special theory of relativity we therefore
1997differentiate between reference-bodies K relative to which the
1998recognised " laws of nature " can be said to hold, and
1999reference-bodies K relative to which these laws do not hold.
2000
2001But no person whose mode of thought is logical can rest satisfied with
2002this condition of things. He asks : " How does it come that certain
2003reference-bodies (or their states of motion) are given priority over
2004other reference-bodies (or their states of motion) ? What is the
2005reason for this Preference? In order to show clearly what I mean by
2006this question, I shall make use of a comparison.
2007
2008I am standing in front of a gas range. Standing alongside of each
2009other on the range are two pans so much alike that one may be mistaken
2010for the other. Both are half full of water. I notice that steam is
2011being emitted continuously from the one pan, but not from the other. I
2012am surprised at this, even if I have never seen either a gas range or
2013a pan before. But if I now notice a luminous something of bluish
2014colour under the first pan but not under the other, I cease to be
2015astonished, even if I have never before seen a gas flame. For I can
2016only say that this bluish something will cause the emission of the
2017steam, or at least possibly it may do so. If, however, I notice the
2018bluish something in neither case, and if I observe that the one
2019continuously emits steam whilst the other does not, then I shall
2020remain astonished and dissatisfied until I have discovered some
2021circumstance to which I can attribute the different behaviour of the
2022two pans.
2023
2024Analogously, I seek in vain for a real something in classical
2025mechanics (or in the special theory of relativity) to which I can
2026attribute the different behaviour of bodies considered with respect to
2027the reference systems K and K1.* Newton saw this objection and
2028attempted to invalidate it, but without success. But E. Mach recognsed
2029it most clearly of all, and because of this objection he claimed that
2030mechanics must be placed on a new basis. It can only be got rid of by
2031means of a physics which is conformable to the general principle of
2032relativity, since the equations of such a theory hold for every body
2033of reference, whatever may be its state of motion.
2034
2035
2036 Notes
2037
2038*) The objection is of importance more especially when the state of
2039motion of the reference-body is of such a nature that it does not
2040require any external agency for its maintenance, e.g. in the case when
2041the reference-body is rotating uniformly.
2042
2043
2044
2045A FEW INFERENCES FROM THE GENERAL PRINCIPLE OF RELATIVITY
2046
2047
2048The considerations of Section 20 show that the general principle of
2049relativity puts us in a position to derive properties of the
2050gravitational field in a purely theoretical manner. Let us suppose,
2051for instance, that we know the space-time " course " for any natural
2052process whatsoever, as regards the manner in which it takes place in
2053the Galileian domain relative to a Galileian body of reference K. By
2054means of purely theoretical operations (i.e. simply by calculation) we
2055are then able to find how this known natural process appears, as seen
2056from a reference-body K1 which is accelerated relatively to K. But
2057since a gravitational field exists with respect to this new body of
2058reference K1, our consideration also teaches us how the gravitational
2059field influences the process studied.
2060
2061For example, we learn that a body which is in a state of uniform
2062rectilinear motion with respect to K (in accordance with the law of
2063Galilei) is executing an accelerated and in general curvilinear motion
2064with respect to the accelerated reference-body K1 (chest). This
2065acceleration or curvature corresponds to the influence on the moving
2066body of the gravitational field prevailing relatively to K. It is
2067known that a gravitational field influences the movement of bodies in
2068this way, so that our consideration supplies us with nothing
2069essentially new.
2070
2071However, we obtain a new result of fundamental importance when we
2072carry out the analogous consideration for a ray of light. With respect
2073to the Galileian reference-body K, such a ray of light is transmitted
2074rectilinearly with the velocity c. It can easily be shown that the
2075path of the same ray of light is no longer a straight line when we
2076consider it with reference to the accelerated chest (reference-body
2077K1). From this we conclude, that, in general, rays of light are
2078propagated curvilinearly in gravitational fields. In two respects this
2079result is of great importance.
2080
2081In the first place, it can be compared with the reality. Although a
2082detailed examination of the question shows that the curvature of light
2083rays required by the general theory of relativity is only exceedingly
2084small for the gravitational fields at our disposal in practice, its
2085estimated magnitude for light rays passing the sun at grazing
2086incidence is nevertheless 1.7 seconds of arc. This ought to manifest
2087itself in the following way. As seen from the earth, certain fixed
2088stars appear to be in the neighbourhood of the sun, and are thus
2089capable of observation during a total eclipse of the sun. At such
2090times, these stars ought to appear to be displaced outwards from the
2091sun by an amount indicated above, as compared with their apparent
2092position in the sky when the sun is situated at another part of the
2093heavens. The examination of the correctness or otherwise of this
2094deduction is a problem of the greatest importance, the early solution
2095of which is to be expected of astronomers.[2]*
2096
2097In the second place our result shows that, according to the general
2098theory of relativity, the law of the constancy of the velocity of
2099light in vacuo, which constitutes one of the two fundamental
2100assumptions in the special theory of relativity and to which we have
2101already frequently referred, cannot claim any unlimited validity. A
2102curvature of rays of light can only take place when the velocity of
2103propagation of light varies with position. Now we might think that as
2104a consequence of this, the special theory of relativity and with it
2105the whole theory of relativity would be laid in the dust. But in
2106reality this is not the case. We can only conclude that the special
2107theory of relativity cannot claim an unlinlited domain of validity ;
2108its results hold only so long as we are able to disregard the
2109influences of gravitational fields on the phenomena (e.g. of light).
2110
2111Since it has often been contended by opponents of the theory of
2112relativity that the special theory of relativity is overthrown by the
2113general theory of relativity, it is perhaps advisable to make the
2114facts of the case clearer by means of an appropriate comparison.
2115Before the development of electrodynamics the laws of electrostatics
2116were looked upon as the laws of electricity. At the present time we
2117know that electric fields can be derived correctly from electrostatic
2118considerations only for the case, which is never strictly realised, in
2119which the electrical masses are quite at rest relatively to each
2120other, and to the co-ordinate system. Should we be justified in saying
2121that for this reason electrostatics is overthrown by the
2122field-equations of Maxwell in electrodynamics ? Not in the least.
2123Electrostatics is contained in electrodynamics as a limiting case ;
2124the laws of the latter lead directly to those of the former for the
2125case in which the fields are invariable with regard to time. No fairer
2126destiny could be allotted to any physical theory, than that it should
2127of itself point out the way to the introduction of a more
2128comprehensive theory, in which it lives on as a limiting case.
2129
2130In the example of the transmission of light just dealt with, we have
2131seen that the general theory of relativity enables us to derive
2132theoretically the influence of a gravitational field on the course of
2133natural processes, the Iaws of which are already known when a
2134gravitational field is absent. But the most attractive problem, to the
2135solution of which the general theory of relativity supplies the key,
2136concerns the investigation of the laws satisfied by the gravitational
2137field itself. Let us consider this for a moment.
2138
2139We are acquainted with space-time domains which behave (approximately)
2140in a " Galileian " fashion under suitable choice of reference-body,
2141i.e. domains in which gravitational fields are absent. If we now refer
2142such a domain to a reference-body K1 possessing any kind of motion,
2143then relative to K1 there exists a gravitational field which is
2144variable with respect to space and time.[3]** The character of this
2145field will of course depend on the motion chosen for K1. According to
2146the general theory of relativity, the general law of the gravitational
2147field must be satisfied for all gravitational fields obtainable in
2148this way. Even though by no means all gravitationial fields can be
2149produced in this way, yet we may entertain the hope that the general
2150law of gravitation will be derivable from such gravitational fields of
2151a special kind. This hope has been realised in the most beautiful
2152manner. But between the clear vision of this goal and its actual
2153realisation it was necessary to surmount a serious difficulty, and as
2154this lies deep at the root of things, I dare not withhold it from the
2155reader. We require to extend our ideas of the space-time continuum
2156still farther.
2157
2158
2159 Notes
2160
2161*) By means of the star photographs of two expeditions equipped by
2162a Joint Committee of the Royal and Royal Astronomical Societies, the
2163existence of the deflection of light demanded by theory was first
2164confirmed during the solar eclipse of 29th May, 1919. (Cf. Appendix
2165III.)
2166
2167**) This follows from a generalisation of the discussion in
2168Section 20
2169
2170
2171
2172BEHAVIOUR OF CLOCKS AND MEASURING-RODS ON A ROTATING BODY OF REFERENCE
2173
2174
2175Hitherto I have purposely refrained from speaking about the physical
2176interpretation of space- and time-data in the case of the general
2177theory of relativity. As a consequence, I am guilty of a certain
2178slovenliness of treatment, which, as we know from the special theory
2179of relativity, is far from being unimportant and pardonable. It is now
2180high time that we remedy this defect; but I would mention at the
2181outset, that this matter lays no small claims on the patience and on
2182the power of abstraction of the reader.
2183
2184We start off again from quite special cases, which we have frequently
2185used before. Let us consider a space time domain in which no
2186gravitational field exists relative to a reference-body K whose state
2187of motion has been suitably chosen. K is then a Galileian
2188reference-body as regards the domain considered, and the results of
2189the special theory of relativity hold relative to K. Let us supposse
2190the same domain referred to a second body of reference K1, which is
2191rotating uniformly with respect to K. In order to fix our ideas, we
2192shall imagine K1 to be in the form of a plane circular disc, which
2193rotates uniformly in its own plane about its centre. An observer who
2194is sitting eccentrically on the disc K1 is sensible of a force which
2195acts outwards in a radial direction, and which would be interpreted as
2196an effect of inertia (centrifugal force) by an observer who was at
2197rest with respect to the original reference-body K. But the observer
2198on the disc may regard his disc as a reference-body which is " at rest
2199" ; on the basis of the general principle of relativity he is
2200justified in doing this. The force acting on himself, and in fact on
2201all other bodies which are at rest relative to the disc, he regards as
2202the effect of a gravitational field. Nevertheless, the
2203space-distribution of this gravitational field is of a kind that would
2204not be possible on Newton's theory of gravitation.* But since the
2205observer believes in the general theory of relativity, this does not
2206disturb him; he is quite in the right when he believes that a general
2207law of gravitation can be formulated- a law which not only explains
2208the motion of the stars correctly, but also the field of force
2209experienced by himself.
2210
2211The observer performs experiments on his circular disc with clocks and
2212measuring-rods. In doing so, it is his intention to arrive at exact
2213definitions for the signification of time- and space-data with
2214reference to the circular disc K1, these definitions being based on
2215his observations. What will be his experience in this enterprise ?
2216
2217To start with, he places one of two identically constructed clocks at
2218the centre of the circular disc, and the other on the edge of the
2219disc, so that they are at rest relative to it. We now ask ourselves
2220whether both clocks go at the same rate from the standpoint of the
2221non-rotating Galileian reference-body K. As judged from this body, the
2222clock at the centre of the disc has no velocity, whereas the clock at
2223the edge of the disc is in motion relative to K in consequence of the
2224rotation. According to a result obtained in Section 12, it follows
2225that the latter clock goes at a rate permanently slower than that of
2226the clock at the centre of the circular disc, i.e. as observed from K.
2227It is obvious that the same effect would be noted by an observer whom
2228we will imagine sitting alongside his clock at the centre of the
2229circular disc. Thus on our circular disc, or, to make the case more
2230general, in every gravitational field, a clock will go more quickly or
2231less quickly, according to the position in which the clock is situated
2232(at rest). For this reason it is not possible to obtain a reasonable
2233definition of time with the aid of clocks which are arranged at rest
2234with respect to the body of reference. A similar difficulty presents
2235itself when we attempt to apply our earlier definition of simultaneity
2236in such a case, but I do not wish to go any farther into this
2237question.
2238
2239Moreover, at this stage the definition of the space co-ordinates also
2240presents insurmountable difficulties. If the observer applies his
2241standard measuring-rod (a rod which is short as compared with the
2242radius of the disc) tangentially to the edge of the disc, then, as
2243judged from the Galileian system, the length of this rod will be less
2244than I, since, according to Section 12, moving bodies suffer a
2245shortening in the direction of the motion. On the other hand, the
2246measaring-rod will not experience a shortening in length, as judged
2247from K, if it is applied to the disc in the direction of the radius.
2248If, then, the observer first measures the circumference of the disc
2249with his measuring-rod and then the diameter of the disc, on dividing
2250the one by the other, he will not obtain as quotient the familiar
2251number p = 3.14 . . ., but a larger number,[4]** whereas of course,
2252for a disc which is at rest with respect to K, this operation would
2253yield p exactly. This proves that the propositions of Euclidean
2254geometry cannot hold exactly on the rotating disc, nor in general in a
2255gravitational field, at least if we attribute the length I to the rod
2256in all positions and in every orientation. Hence the idea of a
2257straight line also loses its meaning. We are therefore not in a
2258position to define exactly the co-ordinates x, y, z relative to the
2259disc by means of the method used in discussing the special theory, and
2260as long as the co- ordinates and times of events have not been
2261defined, we cannot assign an exact meaning to the natural laws in
2262which these occur.
2263
2264Thus all our previous conclusions based on general relativity would
2265appear to be called in question. In reality we must make a subtle
2266detour in order to be able to apply the postulate of general
2267relativity exactly. I shall prepare the reader for this in the
2268following paragraphs.
2269
2270
2271 Notes
2272
2273*) The field disappears at the centre of the disc and increases
2274proportionally to the distance from the centre as we proceed outwards.
2275
2276**) Throughout this consideration we have to use the Galileian
2277(non-rotating) system K as reference-body, since we may only assume
2278the validity of the results of the special theory of relativity
2279relative to K (relative to K1 a gravitational field prevails).
2280
2281
2282
2283EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM
2284
2285
2286The surface of a marble table is spread out in front of me. I can get
2287from any one point on this table to any other point by passing
2288continuously from one point to a " neighbouring " one, and repeating
2289this process a (large) number of times, or, in other words, by going
2290from point to point without executing "jumps." I am sure the reader
2291will appreciate with sufficient clearness what I mean here by "
2292neighbouring " and by " jumps " (if he is not too pedantic). We
2293express this property of the surface by describing the latter as a
2294continuum.
2295
2296Let us now imagine that a large number of little rods of equal length
2297have been made, their lengths being small compared with the dimensions
2298of the marble slab. When I say they are of equal length, I mean that
2299one can be laid on any other without the ends overlapping. We next lay
2300four of these little rods on the marble slab so that they constitute a
2301quadrilateral figure (a square), the diagonals of which are equally
2302long. To ensure the equality of the diagonals, we make use of a little
2303testing-rod. To this square we add similar ones, each of which has one
2304rod in common with the first. We proceed in like manner with each of
2305these squares until finally the whole marble slab is laid out with
2306squares. The arrangement is such, that each side of a square belongs
2307to two squares and each corner to four squares.
2308
2309It is a veritable wonder that we can carry out this business without
2310getting into the greatest difficulties. We only need to think of the
2311following. If at any moment three squares meet at a corner, then two
2312sides of the fourth square are already laid, and, as a consequence,
2313the arrangement of the remaining two sides of the square is already
2314completely determined. But I am now no longer able to adjust the
2315quadrilateral so that its diagonals may be equal. If they are equal of
2316their own accord, then this is an especial favour of the marble slab
2317and of the little rods, about which I can only be thankfully
2318surprised. We must experience many such surprises if the construction
2319is to be successful.
2320
2321If everything has really gone smoothly, then I say that the points of
2322the marble slab constitute a Euclidean continuum with respect to the
2323little rod, which has been used as a " distance " (line-interval). By
2324choosing one corner of a square as " origin" I can characterise every
2325other corner of a square with reference to this origin by means of two
2326numbers. I only need state how many rods I must pass over when,
2327starting from the origin, I proceed towards the " right " and then "
2328upwards," in order to arrive at the corner of the square under
2329consideration. These two numbers are then the " Cartesian co-ordinates
2330" of this corner with reference to the " Cartesian co-ordinate system"
2331which is determined by the arrangement of little rods.
2332
2333By making use of the following modification of this abstract
2334experiment, we recognise that there must also be cases in which the
2335experiment would be unsuccessful. We shall suppose that the rods "
2336expand " by in amount proportional to the increase of temperature. We
2337heat the central part of the marble slab, but not the periphery, in
2338which case two of our little rods can still be brought into
2339coincidence at every position on the table. But our construction of
2340squares must necessarily come into disorder during the heating,
2341because the little rods on the central region of the table expand,
2342whereas those on the outer part do not.
2343
2344With reference to our little rods -- defined as unit lengths -- the
2345marble slab is no longer a Euclidean continuum, and we are also no
2346longer in the position of defining Cartesian co-ordinates directly
2347with their aid, since the above construction can no longer be carried
2348out. But since there are other things which are not influenced in a
2349similar manner to the little rods (or perhaps not at all) by the
2350temperature of the table, it is possible quite naturally to maintain
2351the point of view that the marble slab is a " Euclidean continuum."
2352This can be done in a satisfactory manner by making a more subtle
2353stipulation about the measurement or the comparison of lengths.
2354
2355But if rods of every kind (i.e. of every material) were to behave in
2356the same way as regards the influence of temperature when they are on
2357the variably heated marble slab, and if we had no other means of
2358detecting the effect of temperature than the geometrical behaviour of
2359our rods in experiments analogous to the one described above, then our
2360best plan would be to assign the distance one to two points on the
2361slab, provided that the ends of one of our rods could be made to
2362coincide with these two points ; for how else should we define the
2363distance without our proceeding being in the highest measure grossly
2364arbitrary ? The method of Cartesian coordinates must then be
2365discarded, and replaced by another which does not assume the validity
2366of Euclidean geometry for rigid bodies.* The reader will notice
2367that the situation depicted here corresponds to the one brought about
2368by the general postitlate of relativity (Section 23).
2369
2370
2371 Notes
2372
2373*) Mathematicians have been confronted with our problem in the
2374following form. If we are given a surface (e.g. an ellipsoid) in
2375Euclidean three-dimensional space, then there exists for this surface
2376a two-dimensional geometry, just as much as for a plane surface. Gauss
2377undertook the task of treating this two-dimensional geometry from
2378first principles, without making use of the fact that the surface
2379belongs to a Euclidean continuum of three dimensions. If we imagine
2380constructions to be made with rigid rods in the surface (similar to
2381that above with the marble slab), we should find that different laws
2382hold for these from those resulting on the basis of Euclidean plane
2383geometry. The surface is not a Euclidean continuum with respect to the
2384rods, and we cannot define Cartesian co-ordinates in the surface.
2385Gauss indicated the principles according to which we can treat the
2386geometrical relationships in the surface, and thus pointed out the way
2387to the method of Riemman of treating multi-dimensional, non-Euclidean
2388continuum. Thus it is that mathematicians long ago solved the formal
2389problems to which we are led by the general postulate of relativity.
2390
2391
2392
2393GAUSSIAN CO-ORDINATES
2394
2395
2396According to Gauss, this combined analytical and geometrical mode of
2397handling the problem can be arrived at in the following way. We
2398imagine a system of arbitrary curves (see Fig. 4) drawn on the surface
2399of the table. These we designate as u-curves, and we indicate each of
2400them by means of a number. The Curves u= 1, u= 2 and u= 3 are drawn in
2401the diagram. Between the curves u= 1 and u= 2 we must imagine an
2402infinitely large number to be drawn, all of which correspond to real
2403numbers lying between 1 and 2. fig. 04 We have then a system of
2404u-curves, and this "infinitely dense" system covers the whole surface
2405of the table. These u-curves must not intersect each other, and
2406through each point of the surface one and only one curve must pass.
2407Thus a perfectly definite value of u belongs to every point on the
2408surface of the marble slab. In like manner we imagine a system of
2409v-curves drawn on the surface. These satisfy the same conditions as
2410the u-curves, they are provided with numbers in a corresponding
2411manner, and they may likewise be of arbitrary shape. It follows that a
2412value of u and a value of v belong to every point on the surface of
2413the table. We call these two numbers the co-ordinates of the surface
2414of the table (Gaussian co-ordinates). For example, the point P in the
2415diagram has the Gaussian co-ordinates u= 3, v= 1. Two neighbouring
2416points P and P1 on the surface then correspond to the co-ordinates
2417
2418 P: u,v
2419
2420 P1: u + du, v + dv,
2421
2422where du and dv signify very small numbers. In a similar manner we may
2423indicate the distance (line-interval) between P and P1, as measured
2424with a little rod, by means of the very small number ds. Then
2425according to Gauss we have
2426
2427 ds2 = g[11]du2 + 2g[12]dudv = g[22]dv2
2428
2429where g[11], g[12], g[22], are magnitudes which depend in a perfectly
2430definite way on u and v. The magnitudes g[11], g[12] and g[22],
2431determine the behaviour of the rods relative to the u-curves and
2432v-curves, and thus also relative to the surface of the table. For the
2433case in which the points of the surface considered form a Euclidean
2434continuum with reference to the measuring-rods, but only in this case,
2435it is possible to draw the u-curves and v-curves and to attach numbers
2436to them, in such a manner, that we simply have :
2437
2438 ds2 = du2 + dv2
2439
2440Under these conditions, the u-curves and v-curves are straight lines
2441in the sense of Euclidean geometry, and they are perpendicular to each
2442other. Here the Gaussian coordinates are samply Cartesian ones. It is
2443clear that Gauss co-ordinates are nothing more than an association of
2444two sets of numbers with the points of the surface considered, of such
2445a nature that numerical values differing very slightly from each other
2446are associated with neighbouring points " in space."
2447
2448So far, these considerations hold for a continuum of two dimensions.
2449But the Gaussian method can be applied also to a continuum of three,
2450four or more dimensions. If, for instance, a continuum of four
2451dimensions be supposed available, we may represent it in the following
2452way. With every point of the continuum, we associate arbitrarily four
2453numbers, x[1], x[2], x[3], x[4], which are known as " co-ordinates."
2454Adjacent points correspond to adjacent values of the coordinates. If a
2455distance ds is associated with the adjacent points P and P1, this
2456distance being measurable and well defined from a physical point of
2457view, then the following formula holds:
2458
2459ds2 = g[11]dx[1]^2 + 2g[12]dx[1]dx[2] . . . . g[44]dx[4]^2,
2460
2461where the magnitudes g[11], etc., have values which vary with the
2462position in the continuum. Only when the continuum is a Euclidean one
2463is it possible to associate the co-ordinates x[1] . . x[4]. with the
2464points of the continuum so that we have simply
2465
2466ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.
2467
2468In this case relations hold in the four-dimensional continuum which
2469are analogous to those holding in our three-dimensional measurements.
2470
2471However, the Gauss treatment for ds2 which we have given above is not
2472always possible. It is only possible when sufficiently small regions
2473of the continuum under consideration may be regarded as Euclidean
2474continua. For example, this obviously holds in the case of the marble
2475slab of the table and local variation of temperature. The temperature
2476is practically constant for a small part of the slab, and thus the
2477geometrical behaviour of the rods is almost as it ought to be
2478according to the rules of Euclidean geometry. Hence the imperfections
2479of the construction of squares in the previous section do not show
2480themselves clearly until this construction is extended over a
2481considerable portion of the surface of the table.
2482
2483We can sum this up as follows: Gauss invented a method for the
2484mathematical treatment of continua in general, in which "
2485size-relations " (" distances " between neighbouring points) are
2486defined. To every point of a continuum are assigned as many numbers
2487(Gaussian coordinates) as the continuum has dimensions. This is done
2488in such a way, that only one meaning can be attached to the
2489assignment, and that numbers (Gaussian coordinates) which differ by an
2490indefinitely small amount are assigned to adjacent points. The
2491Gaussian coordinate system is a logical generalisation of the
2492Cartesian co-ordinate system. It is also applicable to non-Euclidean
2493continua, but only when, with respect to the defined "size" or
2494"distance," small parts of the continuum under consideration behave
2495more nearly like a Euclidean system, the smaller the part of the
2496continuum under our notice.
2497
2498
2499
2500THE SPACE-TIME CONTINUUM OF THE SPEICAL THEORY OF RELATIVITY CONSIDERED AS A
2501EUCLIDEAN CONTINUUM
2502
2503
2504We are now in a position to formulate more exactly the idea of
2505Minkowski, which was only vaguely indicated in Section 17. In
2506accordance with the special theory of relativity, certain co-ordinate
2507systems are given preference for the description of the
2508four-dimensional, space-time continuum. We called these " Galileian
2509co-ordinate systems." For these systems, the four co-ordinates x, y,
2510z, t, which determine an event or -- in other words, a point of the
2511four-dimensional continuum -- are defined physically in a simple
2512manner, as set forth in detail in the first part of this book. For the
2513transition from one Galileian system to another, which is moving
2514uniformly with reference to the first, the equations of the Lorentz
2515transformation are valid. These last form the basis for the derivation
2516of deductions from the special theory of relativity, and in themselves
2517they are nothing more than the expression of the universal validity of
2518the law of transmission of light for all Galileian systems of
2519reference.
2520
2521Minkowski found that the Lorentz transformations satisfy the following
2522simple conditions. Let us consider two neighbouring events, the
2523relative position of which in the four-dimensional continuum is given
2524with respect to a Galileian reference-body K by the space co-ordinate
2525differences dx, dy, dz and the time-difference dt. With reference to a
2526second Galileian system we shall suppose that the corresponding
2527differences for these two events are dx1, dy1, dz1, dt1. Then these
2528magnitudes always fulfil the condition*
2529
2530 dx2 + dy2 + dz2 - c^2dt2 = dx1 2 + dy1 2 + dz1 2 - c^2dt1 2.
2531
2532The validity of the Lorentz transformation follows from this
2533condition. We can express this as follows: The magnitude
2534
2535 ds2 = dx2 + dy2 + dz2 - c^2dt2,
2536
2537which belongs to two adjacent points of the four-dimensional
2538space-time continuum, has the same value for all selected (Galileian)
2539reference-bodies. If we replace x, y, z, sq. rt. -I . ct , by x[1],
2540x[2], x[3], x[4], we also obtaill the result that
2541
2542 ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.
2543
2544is independent of the choice of the body of reference. We call the
2545magnitude ds the " distance " apart of the two events or
2546four-dimensional points.
2547
2548Thus, if we choose as time-variable the imaginary variable sq. rt. -I
2549. ct instead of the real quantity t, we can regard the space-time
2550contintium -- accordance with the special theory of relativity -- as a
2551", Euclidean " four-dimensional continuum, a result which follows from
2552the considerations of the preceding section.
2553
2554
2555 Notes
2556
2557*) Cf. Appendixes I and 2. The relations which are derived
2558there for the co-ordlnates themselves are valid also for co-ordinate
2559differences, and thus also for co-ordinate differentials (indefinitely
2560small differences).
2561
2562
2563
2564THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF REALTIIVTY IS NOT A
2565ECULIDEAN CONTINUUM
2566
2567
2568In the first part of this book we were able to make use of space-time
2569co-ordinates which allowed of a simple and direct physical
2570interpretation, and which, according to Section 26, can be regarded
2571as four-dimensional Cartesian co-ordinates. This was possible on the
2572basis of the law of the constancy of the velocity of tight. But
2573according to Section 21 the general theory of relativity cannot
2574retain this law. On the contrary, we arrived at the result that
2575according to this latter theory the velocity of light must always
2576depend on the co-ordinates when a gravitational field is present. In
2577connection with a specific illustration in Section 23, we found
2578that the presence of a gravitational field invalidates the definition
2579of the coordinates and the ifine, which led us to our objective in the
2580special theory of relativity.
2581
2582In view of the resuIts of these considerations we are led to the
2583conviction that, according to the general principle of relativity, the
2584space-time continuum cannot be regarded as a Euclidean one, but that
2585here we have the general case, corresponding to the marble slab with
2586local variations of temperature, and with which we made acquaintance
2587as an example of a two-dimensional continuum. Just as it was there
2588impossible to construct a Cartesian co-ordinate system from equal
2589rods, so here it is impossible to build up a system (reference-body)
2590from rigid bodies and clocks, which shall be of such a nature that
2591measuring-rods and clocks, arranged rigidly with respect to one
2592another, shaIll indicate position and time directly. Such was the
2593essence of the difficulty with which we were confronted in Section
259423.
2595
2596But the considerations of Sections 25 and 26 show us the way to
2597surmount this difficulty. We refer the fourdimensional space-time
2598continuum in an arbitrary manner to Gauss co-ordinates. We assign to
2599every point of the continuum (event) four numbers, x[1], x[2], x[3],
2600x[4] (co-ordinates), which have not the least direct physical
2601significance, but only serve the purpose of numbering the points of
2602the continuum in a definite but arbitrary manner. This arrangement
2603does not even need to be of such a kind that we must regard x[1],
2604x[2], x[3], as "space" co-ordinates and x[4], as a " time "
2605co-ordinate.
2606
2607The reader may think that such a description of the world would be
2608quite inadequate. What does it mean to assign to an event the
2609particular co-ordinates x[1], x[2], x[3], x[4], if in themselves these
2610co-ordinates have no significance ? More careful consideration shows,
2611however, that this anxiety is unfounded. Let us consider, for
2612instance, a material point with any kind of motion. If this point had
2613only a momentary existence without duration, then it would to
2614described in space-time by a single system of values x[1], x[2], x[3],
2615x[4]. Thus its permanent existence must be characterised by an
2616infinitely large number of such systems of values, the co-ordinate
2617values of which are so close together as to give continuity;
2618corresponding to the material point, we thus have a (uni-dimensional)
2619line in the four-dimensional continuum. In the same way, any such
2620lines in our continuum correspond to many points in motion. The only
2621statements having regard to these points which can claim a physical
2622existence are in reality the statements about their encounters. In our
2623mathematical treatment, such an encounter is expressed in the fact
2624that the two lines which represent the motions of the points in
2625question have a particular system of co-ordinate values, x[1], x[2],
2626x[3], x[4], in common. After mature consideration the reader will
2627doubtless admit that in reality such encounters constitute the only
2628actual evidence of a time-space nature with which we meet in physical
2629statements.
2630
2631When we were describing the motion of a material point relative to a
2632body of reference, we stated nothing more than the encounters of this
2633point with particular points of the reference-body. We can also
2634determine the corresponding values of the time by the observation of
2635encounters of the body with clocks, in conjunction with the
2636observation of the encounter of the hands of clocks with particular
2637points on the dials. It is just the same in the case of
2638space-measurements by means of measuring-rods, as a litttle
2639consideration will show.
2640
2641The following statements hold generally : Every physical description
2642resolves itself into a number of statements, each of which refers to
2643the space-time coincidence of two events A and B. In terms of Gaussian
2644co-ordinates, every such statement is expressed by the agreement of
2645their four co-ordinates x[1], x[2], x[3], x[4]. Thus in reality, the
2646description of the time-space continuum by means of Gauss co-ordinates
2647completely replaces the description with the aid of a body of
2648reference, without suffering from the defects of the latter mode of
2649description; it is not tied down to the Euclidean character of the
2650continuum which has to be represented.
2651
2652
2653
2654EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY
2655
2656
2657We are now in a position to replace the pro. visional formulation of
2658the general principle of relativity given in Section 18 by an exact
2659formulation. The form there used, "All bodies of reference K, K1,
2660etc., are equivalent for the description of natural phenomena
2661(formulation of the general laws of nature), whatever may be their
2662state of motion," cannot be maintained, because the use of rigid
2663reference-bodies, in the sense of the method followed in the special
2664theory of relativity, is in general not possible in space-time
2665description. The Gauss co-ordinate system has to take the place of the
2666body of reference. The following statement corresponds to the
2667fundamental idea of the general principle of relativity: "All Gaussian
2668co-ordinate systems are essentially equivalent for the formulation of
2669the general laws of nature."
2670
2671We can state this general principle of relativity in still another
2672form, which renders it yet more clearly intelligible than it is when
2673in the form of the natural extension of the special principle of
2674relativity. According to the special theory of relativity, the
2675equations which express the general laws of nature pass over into
2676equations of the same form when, by making use of the Lorentz
2677transformation, we replace the space-time variables x, y, z, t, of a
2678(Galileian) reference-body K by the space-time variables x1, y1, z1,
2679t1, of a new reference-body K1. According to the general theory of
2680relativity, on the other hand, by application of arbitrary
2681substitutions of the Gauss variables x[1], x[2], x[3], x[4], the
2682equations must pass over into equations of the same form; for every
2683transformation (not only the Lorentz transformation) corresponds to
2684the transition of one Gauss co-ordinate system into another.
2685
2686If we desire to adhere to our "old-time" three-dimensional view of
2687things, then we can characterise the development which is being
2688undergone by the fundamental idea of the general theory of relativity
2689as follows : The special theory of relativity has reference to
2690Galileian domains, i.e. to those in which no gravitational field
2691exists. In this connection a Galileian reference-body serves as body
2692of reference, i.e. a rigid body the state of motion of which is so
2693chosen that the Galileian law of the uniform rectilinear motion of
2694"isolated" material points holds relatively to it.
2695
2696Certain considerations suggest that we should refer the same Galileian
2697domains to non-Galileian reference-bodies also. A gravitational field
2698of a special kind is then present with respect to these bodies (cf.
2699Sections 20 and 23).
2700
2701In gravitational fields there are no such things as rigid bodies with
2702Euclidean properties; thus the fictitious rigid body of reference is
2703of no avail in the general theory of relativity. The motion of clocks
2704is also influenced by gravitational fields, and in such a way that a
2705physical definition of time which is made directly with the aid of
2706clocks has by no means the same degree of plausibility as in the
2707special theory of relativity.
2708
2709For this reason non-rigid reference-bodies are used, which are as a
2710whole not only moving in any way whatsoever, but which also suffer
2711alterations in form ad lib. during their motion. Clocks, for which the
2712law of motion is of any kind, however irregular, serve for the
2713definition of time. We have to imagine each of these clocks fixed at a
2714point on the non-rigid reference-body. These clocks satisfy only the
2715one condition, that the "readings" which are observed simultaneously
2716on adjacent clocks (in space) differ from each other by an
2717indefinitely small amount. This non-rigid reference-body, which might
2718appropriately be termed a "reference-mollusc", is in the main
2719equivalent to a Gaussian four-dimensional co-ordinate system chosen
2720arbitrarily. That which gives the "mollusc" a certain
2721comprehensibility as compared with the Gauss co-ordinate system is the
2722(really unjustified) formal retention of the separate existence of the
2723space co-ordinates as opposed to the time co-ordinate. Every point on
2724the mollusc is treated as a space-point, and every material point
2725which is at rest relatively to it as at rest, so long as the mollusc
2726is considered as reference-body. The general principle of relativity
2727requires that all these molluscs can be used as reference-bodies with
2728equal right and equal success in the formulation of the general laws
2729of nature; the laws themselves must be quite independent of the choice
2730of mollusc.
2731
2732The great power possessed by the general principle of relativity lies
2733in the comprehensive limitation which is imposed on the laws of nature
2734in consequence of what we have seen above.
2735
2736
2737
2738THE SOLUTION OF THE PROBLEM OF GRAVITATION ON THE BASIS OF THE GENERAL
2739PRINCIPLE OF RELATIVITY
2740
2741
2742If the reader has followed all our previous considerations, he will
2743have no further difficulty in understanding the methods leading to the
2744solution of the problem of gravitation.
2745
2746We start off on a consideration of a Galileian domain, i.e. a domain
2747in which there is no gravitational field relative to the Galileian
2748reference-body K. The behaviour of measuring-rods and clocks with
2749reference to K is known from the special theory of relativity,
2750likewise the behaviour of "isolated" material points; the latter move
2751uniformly and in straight lines.
2752
2753Now let us refer this domain to a random Gauss coordinate system or to
2754a "mollusc" as reference-body K1. Then with respect to K1 there is a
2755gravitational field G (of a particular kind). We learn the behaviour
2756of measuring-rods and clocks and also of freely-moving material points
2757with reference to K1 simply by mathematical transformation. We
2758interpret this behaviour as the behaviour of measuring-rods, docks and
2759material points tinder the influence of the gravitational field G.
2760Hereupon we introduce a hypothesis: that the influence of the
2761gravitational field on measuringrods, clocks and freely-moving
2762material points continues to take place according to the same laws,
2763even in the case where the prevailing gravitational field is not
2764derivable from the Galfleian special care, simply by means of a
2765transformation of co-ordinates.
2766
2767The next step is to investigate the space-time behaviour of the
2768gravitational field G, which was derived from the Galileian special
2769case simply by transformation of the coordinates. This behaviour is
2770formulated in a law, which is always valid, no matter how the
2771reference-body (mollusc) used in the description may be chosen.
2772
2773This law is not yet the general law of the gravitational field, since
2774the gravitational field under consideration is of a special kind. In
2775order to find out the general law-of-field of gravitation we still
2776require to obtain a generalisation of the law as found above. This can
2777be obtained without caprice, however, by taking into consideration the
2778following demands:
2779
2780(a) The required generalisation must likewise satisfy the general
2781postulate of relativity.
2782
2783(b) If there is any matter in the domain under consideration, only its
2784inertial mass, and thus according to Section 15 only its energy is
2785of importance for its etfect in exciting a field.
2786
2787(c) Gravitational field and matter together must satisfy the law of
2788the conservation of energy (and of impulse).
2789
2790Finally, the general principle of relativity permits us to determine
2791the influence of the gravitational field on the course of all those
2792processes which take place according to known laws when a
2793gravitational field is absent i.e. which have already been fitted into
2794the frame of the special theory of relativity. In this connection we
2795proceed in principle according to the method which has already been
2796explained for measuring-rods, clocks and freely moving material
2797points.
2798
2799The theory of gravitation derived in this way from the general
2800postulate of relativity excels not only in its beauty ; nor in
2801removing the defect attaching to classical mechanics which was brought
2802to light in Section 21; nor in interpreting the empirical law of
2803the equality of inertial and gravitational mass ; but it has also
2804already explained a result of observation in astronomy, against which
2805classical mechanics is powerless.
2806
2807If we confine the application of the theory to the case where the
2808gravitational fields can be regarded as being weak, and in which all
2809masses move with respect to the coordinate system with velocities
2810which are small compared with the velocity of light, we then obtain as
2811a first approximation the Newtonian theory. Thus the latter theory is
2812obtained here without any particular assumption, whereas Newton had to
2813introduce the hypothesis that the force of attraction between mutually
2814attracting material points is inversely proportional to the square of
2815the distance between them. If we increase the accuracy of the
2816calculation, deviations from the theory of Newton make their
2817appearance, practically all of which must nevertheless escape the test
2818of observation owing to their smallness.
2819
2820We must draw attention here to one of these deviations. According to
2821Newton's theory, a planet moves round the sun in an ellipse, which
2822would permanently maintain its position with respect to the fixed
2823stars, if we could disregard the motion of the fixed stars themselves
2824and the action of the other planets under consideration. Thus, if we
2825correct the observed motion of the planets for these two influences,
2826and if Newton's theory be strictly correct, we ought to obtain for the
2827orbit of the planet an ellipse, which is fixed with reference to the
2828fixed stars. This deduction, which can be tested with great accuracy,
2829has been confirmed for all the planets save one, with the precision
2830that is capable of being obtained by the delicacy of observation
2831attainable at the present time. The sole exception is Mercury, the
2832planet which lies nearest the sun. Since the time of Leverrier, it has
2833been known that the ellipse corresponding to the orbit of Mercury,
2834after it has been corrected for the influences mentioned above, is not
2835stationary with respect to the fixed stars, but that it rotates
2836exceedingly slowly in the plane of the orbit and in the sense of the
2837orbital motion. The value obtained for this rotary movement of the
2838orbital ellipse was 43 seconds of arc per century, an amount ensured
2839to be correct to within a few seconds of arc. This effect can be
2840explained by means of classical mechanics only on the assumption of
2841hypotheses which have little probability, and which were devised
2842solely for this purponse.
2843
2844On the basis of the general theory of relativity, it is found that the
2845ellipse of every planet round the sun must necessarily rotate in the
2846manner indicated above ; that for all the planets, with the exception
2847of Mercury, this rotation is too small to be detected with the
2848delicacy of observation possible at the present time ; but that in the
2849case of Mercury it must amount to 43 seconds of arc per century, a
2850result which is strictly in agreement with observation.
2851
2852Apart from this one, it has hitherto been possible to make only two
2853deductions from the theory which admit of being tested by observation,
2854to wit, the curvature of light rays by the gravitational field of the
2855sun,*x and a displacement of the spectral lines of light reaching
2856us from large stars, as compared with the corresponding lines for
2857light produced in an analogous manner terrestrially (i.e. by the same
2858kind of atom).** These two deductions from the theory have both
2859been confirmed.
2860
2861
2862 Notes
2863
2864*) First observed by Eddington and others in 1919. (Cf. Appendix
2865III, pp. 126-129).
2866
2867**) Established by Adams in 1924. (Cf. p. 132)
2868
2869
2870
2871
2872PART III
2873
2874CONSIDERATIONS ON THE UNIVERSE AS A WHOLE
2875
2876
2877COSMOLOGICAL DIFFICULTIES OF NEWTON'S THEORY
2878
2879
2880Part from the difficulty discussed in Section 21, there is a second
2881fundamental difficulty attending classical celestial mechanics, which,
2882to the best of my knowledge, was first discussed in detail by the
2883astronomer Seeliger. If we ponder over the question as to how the
2884universe, considered as a whole, is to be regarded, the first answer
2885that suggests itself to us is surely this: As regards space (and time)
2886the universe is infinite. There are stars everywhere, so that the
2887density of matter, although very variable in detail, is nevertheless
2888on the average everywhere the same. In other words: However far we
2889might travel through space, we should find everywhere an attenuated
2890swarm of fixed stars of approrimately the same kind and density.
2891
2892This view is not in harmony with the theory of Newton. The latter
2893theory rather requires that the universe should have a kind of centre
2894in which the density of the stars is a maximum, and that as we proceed
2895outwards from this centre the group-density of the stars should
2896diminish, until finally, at great distances, it is succeeded by an
2897infinite region of emptiness. The stellar universe ought to be a
2898finite island in the infinite ocean of space.*
2899
2900This conception is in itself not very satisfactory. It is still less
2901satisfactory because it leads to the result that the light emitted by
2902the stars and also individual stars of the stellar system are
2903perpetually passing out into infinite space, never to return, and
2904without ever again coming into interaction with other objects of
2905nature. Such a finite material universe would be destined to become
2906gradually but systematically impoverished.
2907
2908In order to escape this dilemma, Seeliger suggested a modification of
2909Newton's law, in which he assumes that for great distances the force
2910of attraction between two masses diminishes more rapidly than would
2911result from the inverse square law. In this way it is possible for the
2912mean density of matter to be constant everywhere, even to infinity,
2913without infinitely large gravitational fields being produced. We thus
2914free ourselves from the distasteful conception that the material
2915universe ought to possess something of the nature of a centre. Of
2916course we purchase our emancipation from the fundamental difficulties
2917mentioned, at the cost of a modification and complication of Newton's
2918law which has neither empirical nor theoretical foundation. We can
2919imagine innumerable laws which would serve the same purpose, without
2920our being able to state a reason why one of them is to be preferred to
2921the others ; for any one of these laws would be founded just as little
2922on more general theoretical principles as is the law of Newton.
2923
2924
2925 Notes
2926
2927*) Proof -- According to the theory of Newton, the number of "lines
2928of force" which come from infinity and terminate in a mass m is
2929proportional to the mass m. If, on the average, the Mass density p[0]
2930is constant throughout tithe universe, then a sphere of volume V will
2931enclose the average man p[0]V. Thus the number of lines of force
2932passing through the surface F of the sphere into its interior is
2933proportional to p[0] V. For unit area of the surface of the sphere the
2934number of lines of force which enters the sphere is thus proportional
2935to p[0] V/F or to p[0]R. Hence the intensity of the field at the
2936surface would ultimately become infinite with increasing radius R of
2937the sphere, which is impossible.
2938
2939
2940
2941THE POSSIBILITY OF A "FINITE" AND YET "UNBOUNDED" UNIVERSE
2942
2943
2944But speculations on the structure of the universe also move in quite
2945another direction. The development of non-Euclidean geometry led to
2946the recognition of the fact, that we can cast doubt on the
2947infiniteness of our space without coming into conflict with the laws
2948of thought or with experience (Riemann, Helmholtz). These questions
2949have already been treated in detail and with unsurpassable lucidity by
2950Helmholtz and Poincaré, whereas I can only touch on them briefly here.
2951
2952In the first place, we imagine an existence in two dimensional space.
2953Flat beings with flat implements, and in particular flat rigid
2954measuring-rods, are free to move in a plane. For them nothing exists
2955outside of this plane: that which they observe to happen to themselves
2956and to their flat " things " is the all-inclusive reality of their
2957plane. In particular, the constructions of plane Euclidean geometry
2958can be carried out by means of the rods e.g. the lattice construction,
2959considered in Section 24. In contrast to ours, the universe of
2960these beings is two-dimensional; but, like ours, it extends to
2961infinity. In their universe there is room for an infinite number of
2962identical squares made up of rods, i.e. its volume (surface) is
2963infinite. If these beings say their universe is " plane," there is
2964sense in the statement, because they mean that they can perform the
2965constructions of plane Euclidean geometry with their rods. In this
2966connection the individual rods always represent the same distance,
2967independently of their position.
2968
2969Let us consider now a second two-dimensional existence, but this time
2970on a spherical surface instead of on a plane. The flat beings with
2971their measuring-rods and other objects fit exactly on this surface and
2972they are unable to leave it. Their whole universe of observation
2973extends exclusively over the surface of the sphere. Are these beings
2974able to regard the geometry of their universe as being plane geometry
2975and their rods withal as the realisation of " distance " ? They cannot
2976do this. For if they attempt to realise a straight line, they will
2977obtain a curve, which we " three-dimensional beings " designate as a
2978great circle, i.e. a self-contained line of definite finite length,
2979which can be measured up by means of a measuring-rod. Similarly, this
2980universe has a finite area that can be compared with the area, of a
2981square constructed with rods. The great charm resulting from this
2982consideration lies in the recognition of the fact that the universe of
2983these beings is finite and yet has no limits.
2984
2985But the spherical-surface beings do not need to go on a world-tour in
2986order to perceive that they are not living in a Euclidean universe.
2987They can convince themselves of this on every part of their " world,"
2988provided they do not use too small a piece of it. Starting from a
2989point, they draw " straight lines " (arcs of circles as judged in
2990three dimensional space) of equal length in all directions. They will
2991call the line joining the free ends of these lines a " circle." For a
2992plane surface, the ratio of the circumference of a circle to its
2993diameter, both lengths being measured with the same rod, is, according
2994to Euclidean geometry of the plane, equal to a constant value p, which
2995is independent of the diameter of the circle. On their spherical
2996surface our flat beings would find for this ratio the value
2997
2998 eq. 27: file eq27.gif
2999
3000i.e. a smaller value than p, the difference being the more
3001considerable, the greater is the radius of the circle in comparison
3002with the radius R of the " world-sphere." By means of this relation
3003the spherical beings can determine the radius of their universe ("
3004world "), even when only a relatively small part of their worldsphere
3005is available for their measurements. But if this part is very small
3006indeed, they will no longer be able to demonstrate that they are on a
3007spherical " world " and not on a Euclidean plane, for a small part of
3008a spherical surface differs only slightly from a piece of a plane of
3009the same size.
3010
3011Thus if the spherical surface beings are living on a planet of which
3012the solar system occupies only a negligibly small part of the
3013spherical universe, they have no means of determining whether they are
3014living in a finite or in an infinite universe, because the " piece of
3015universe " to which they have access is in both cases practically
3016plane, or Euclidean. It follows directly from this discussion, that
3017for our sphere-beings the circumference of a circle first increases
3018with the radius until the " circumference of the universe " is
3019reached, and that it thenceforward gradually decreases to zero for
3020still further increasing values of the radius. During this process the
3021area of the circle continues to increase more and more, until finally
3022it becomes equal to the total area of the whole " world-sphere."
3023
3024Perhaps the reader will wonder why we have placed our " beings " on a
3025sphere rather than on another closed surface. But this choice has its
3026justification in the fact that, of all closed surfaces, the sphere is
3027unique in possessing the property that all points on it are
3028equivalent. I admit that the ratio of the circumference c of a circle
3029to its radius r depends on r, but for a given value of r it is the
3030same for all points of the " worldsphere "; in other words, the "
3031world-sphere " is a " surface of constant curvature."
3032
3033To this two-dimensional sphere-universe there is a three-dimensional
3034analogy, namely, the three-dimensional spherical space which was
3035discovered by Riemann. its points are likewise all equivalent. It
3036possesses a finite volume, which is determined by its "radius"
3037(2p2R3). Is it possible to imagine a spherical space? To imagine a
3038space means nothing else than that we imagine an epitome of our "
3039space " experience, i.e. of experience that we can have in the
3040movement of " rigid " bodies. In this sense we can imagine a spherical
3041space.
3042
3043Suppose we draw lines or stretch strings in all directions from a
3044point, and mark off from each of these the distance r with a
3045measuring-rod. All the free end-points of these lengths lie on a
3046spherical surface. We can specially measure up the area (F) of this
3047surface by means of a square made up of measuring-rods. If the
3048universe is Euclidean, then F = 4pR2 ; if it is spherical, then F is
3049always less than 4pR2. With increasing values of r, F increases from
3050zero up to a maximum value which is determined by the " world-radius,"
3051but for still further increasing values of r, the area gradually
3052diminishes to zero. At first, the straight lines which radiate from
3053the starting point diverge farther and farther from one another, but
3054later they approach each other, and finally they run together again at
3055a "counter-point" to the starting point. Under such conditions they
3056have traversed the whole spherical space. It is easily seen that the
3057three-dimensional spherical space is quite analogous to the
3058two-dimensional spherical surface. It is finite (i.e. of finite
3059volume), and has no bounds.
3060
3061It may be mentioned that there is yet another kind of curved space: "
3062elliptical space." It can be regarded as a curved space in which the
3063two " counter-points " are identical (indistinguishable from each
3064other). An elliptical universe can thus be considered to some extent
3065as a curved universe possessing central symmetry.
3066
3067It follows from what has been said, that closed spaces without limits
3068are conceivable. From amongst these, the spherical space (and the
3069elliptical) excels in its simplicity, since all points on it are
3070equivalent. As a result of this discussion, a most interesting
3071question arises for astronomers and physicists, and that is whether
3072the universe in which we live is infinite, or whether it is finite in
3073the manner of the spherical universe. Our experience is far from being
3074sufficient to enable us to answer this question. But the general
3075theory of relativity permits of our answering it with a moduate degree
3076of certainty, and in this connection the difficulty mentioned in
3077Section 30 finds its solution.
3078
3079
3080
3081THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY
3082
3083
3084According to the general theory of relativity, the geometrical
3085properties of space are not independent, but they are determined by
3086matter. Thus we can draw conclusions about the geometrical structure
3087of the universe only if we base our considerations on the state of the
3088matter as being something that is known. We know from experience that,
3089for a suitably chosen co-ordinate system, the velocities of the stars
3090are small as compared with the velocity of transmission of light. We
3091can thus as a rough approximation arrive at a conclusion as to the
3092nature of the universe as a whole, if we treat the matter as being at
3093rest.
3094
3095We already know from our previous discussion that the behaviour of
3096measuring-rods and clocks is influenced by gravitational fields, i.e.
3097by the distribution of matter. This in itself is sufficient to exclude
3098the possibility of the exact validity of Euclidean geometry in our
3099universe. But it is conceivable that our universe differs only
3100slightly from a Euclidean one, and this notion seems all the more
3101probable, since calculations show that the metrics of surrounding
3102space is influenced only to an exceedingly small extent by masses even
3103of the magnitude of our sun. We might imagine that, as regards
3104geometry, our universe behaves analogously to a surface which is
3105irregularly curved in its individual parts, but which nowhere departs
3106appreciably from a plane: something like the rippled surface of a
3107lake. Such a universe might fittingly be called a quasi-Euclidean
3108universe. As regards its space it would be infinite. But calculation
3109shows that in a quasi-Euclidean universe the average density of matter
3110would necessarily be nil. Thus such a universe could not be inhabited
3111by matter everywhere ; it would present to us that unsatisfactory
3112picture which we portrayed in Section 30.
3113
3114If we are to have in the universe an average density of matter which
3115differs from zero, however small may be that difference, then the
3116universe cannot be quasi-Euclidean. On the contrary, the results of
3117calculation indicate that if matter be distributed uniformly, the
3118universe would necessarily be spherical (or elliptical). Since in
3119reality the detailed distribution of matter is not uniform, the real
3120universe will deviate in individual parts from the spherical, i.e. the
3121universe will be quasi-spherical. But it will be necessarily finite.
3122In fact, the theory supplies us with a simple connection * between
3123the space-expanse of the universe and the average density of matter in
3124it.
3125
3126
3127 Notes
3128
3129*) For the radius R of the universe we obtain the equation
3130
3131 eq. 28: file eq28.gif
3132
3133The use of the C.G.S. system in this equation gives 2/k = 1^.08.10^27;
3134p is the average density of the matter and k is a constant connected
3135with the Newtonian constant of gravitation.
3136
3137
3138
3139APPENDIX I
3140
3141SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION
3142(SUPPLEMENTARY TO SECTION 11)
3143
3144
3145For the relative orientation of the co-ordinate systems indicated in
3146Fig. 2, the x-axes of both systems pernumently coincide. In the
3147present case we can divide the problem into parts by considering first
3148only events which are localised on the x-axis. Any such event is
3149represented with respect to the co-ordinate system K by the abscissa x
3150and the time t, and with respect to the system K1 by the abscissa x'
3151and the time t'. We require to find x' and t' when x and t are given.
3152
3153A light-signal, which is proceeding along the positive axis of x, is
3154transmitted according to the equation
3155
3156 x = ct
3157
3158or
3159
3160 x - ct = 0 . . . (1).
3161
3162Since the same light-signal has to be transmitted relative to K1 with
3163the velocity c, the propagation relative to the system K1 will be
3164represented by the analogous formula
3165
3166 x' - ct' = O . . . (2)
3167
3168Those space-time points (events) which satisfy (x) must also satisfy
3169(2). Obviously this will be the case when the relation
3170
3171 (x' - ct') = l (x - ct) . . . (3).
3172
3173is fulfilled in general, where l indicates a constant ; for, according
3174to (3), the disappearance of (x - ct) involves the disappearance of
3175(x' - ct').
3176
3177If we apply quite similar considerations to light rays which are being
3178transmitted along the negative x-axis, we obtain the condition
3179
3180 (x' + ct') = µ(x + ct) . . . (4).
3181
3182By adding (or subtracting) equations (3) and (4), and introducing for
3183convenience the constants a and b in place of the constants l and µ,
3184where
3185
3186 eq. 29: file eq29.gif
3187
3188and
3189
3190 eq. 30: file eq30.gif
3191
3192we obtain the equations
3193
3194 eq. 31: file eq31.gif
3195
3196We should thus have the solution of our problem, if the constants a
3197and b were known. These result from the following discussion.
3198
3199For the origin of K1 we have permanently x' = 0, and hence according
3200to the first of the equations (5)
3201
3202 eq. 32: file eq32.gif
3203
3204If we call v the velocity with which the origin of K1 is moving
3205relative to K, we then have
3206
3207 eq. 33: file eq33.gif
3208
3209The same value v can be obtained from equations (5), if we calculate
3210the velocity of another point of K1 relative to K, or the velocity
3211(directed towards the negative x-axis) of a point of K with respect to
3212K'. In short, we can designate v as the relative velocity of the two
3213systems.
3214
3215Furthermore, the principle of relativity teaches us that, as judged
3216from K, the length of a unit measuring-rod which is at rest with
3217reference to K1 must be exactly the same as the length, as judged from
3218K', of a unit measuring-rod which is at rest relative to K. In order
3219to see how the points of the x-axis appear as viewed from K, we only
3220require to take a " snapshot " of K1 from K; this means that we have
3221to insert a particular value of t (time of K), e.g. t = 0. For this
3222value of t we then obtain from the first of the equations (5)
3223
3224 x' = ax
3225
3226Two points of the x'-axis which are separated by the distance Dx' = I
3227when measured in the K1 system are thus separated in our instantaneous
3228photograph by the distance
3229
3230 eq. 34: file eq34.gif
3231
3232But if the snapshot be taken from K'(t' = 0), and if we eliminate t
3233from the equations (5), taking into account the expression (6), we
3234obtain
3235
3236 eq. 35: file eq35.gif
3237
3238From this we conclude that two points on the x-axis separated by the
3239distance I (relative to K) will be represented on our snapshot by the
3240distance
3241
3242 eq. 36: file eq36.gif
3243
3244But from what has been said, the two snapshots must be identical;
3245hence Dx in (7) must be equal to Dx' in (7a), so that we obtain
3246
3247 eq. 37: file eq37.gif
3248
3249The equations (6) and (7b) determine the constants a and b. By
3250inserting the values of these constants in (5), we obtain the first
3251and the fourth of the equations given in Section 11.
3252
3253 eq. 38: file eq38.gif
3254
3255Thus we have obtained the Lorentz transformation for events on the
3256x-axis. It satisfies the condition
3257
3258 x'2 - c^2t'2 = x2 - c^2t2 . . . (8a).
3259
3260The extension of this result, to include events which take place
3261outside the x-axis, is obtained by retaining equations (8) and
3262supplementing them by the relations
3263
3264 eq. 39: file eq39.gif
3265
3266In this way we satisfy the postulate of the constancy of the velocity
3267of light in vacuo for rays of light of arbitrary direction, both for
3268the system K and for the system K'. This may be shown in the following
3269manner.
3270
3271We suppose a light-signal sent out from the origin of K at the time t
3272= 0. It will be propagated according to the equation
3273
3274 eq. 40: file eq40.gif
3275
3276or, if we square this equation, according to the equation
3277
3278 x2 + y2 + z2 = c^2t2 = 0 . . . (10).
3279
3280It is required by the law of propagation of light, in conjunction with
3281the postulate of relativity, that the transmission of the signal in
3282question should take place -- as judged from K1 -- in accordance with
3283the corresponding formula
3284
3285 r' = ct'
3286
3287or,
3288
3289 x'2 + y'2 + z'2 - c^2t'2 = 0 . . . (10a).
3290
3291In order that equation (10a) may be a consequence of equation (10), we
3292must have
3293
3294 x'2 + y'2 + z'2 - c^2t'2 = s (x2 + y2 + z2 - c^2t2) (11).
3295
3296Since equation (8a) must hold for points on the x-axis, we thus have s
3297= I. It is easily seen that the Lorentz transformation really
3298satisfies equation (11) for s = I; for (11) is a consequence of (8a)
3299and (9), and hence also of (8) and (9). We have thus derived the
3300Lorentz transformation.
3301
3302The Lorentz transformation represented by (8) and (9) still requires
3303to be generalised. Obviously it is immaterial whether the axes of K1
3304be chosen so that they are spatially parallel to those of K. It is
3305also not essential that the velocity of translation of K1 with respect
3306to K should be in the direction of the x-axis. A simple consideration
3307shows that we are able to construct the Lorentz transformation in this
3308general sense from two kinds of transformations, viz. from Lorentz
3309transformations in the special sense and from purely spatial
3310transformations. which corresponds to the replacement of the
3311rectangular co-ordinate system by a new system with its axes pointing
3312in other directions.
3313
3314Mathematically, we can characterise the generalised Lorentz
3315transformation thus :
3316
3317It expresses x', y', x', t', in terms of linear homogeneous functions
3318of x, y, x, t, of such a kind that the relation
3319
3320 x'2 + y'2 + z'2 - c^2t'2 = x2 + y2 + z2 - c^2t2 (11a).
3321
3322is satisficd identically. That is to say: If we substitute their
3323expressions in x, y, x, t, in place of x', y', x', t', on the
3324left-hand side, then the left-hand side of (11a) agrees with the
3325right-hand side.
3326
3327
3328
3329APPENDIX II
3330
3331MINKOWSKI'S FOUR-DIMENSIONAL SPACE ("WORLD")
3332(SUPPLEMENTARY TO SECTION 17)
3333
3334
3335We can characterise the Lorentz transformation still more simply if we
3336introduce the imaginary eq. 25 in place of t, as time-variable. If, in
3337accordance with this, we insert
3338
3339 x[1] = x
3340 x[2] = y
3341 x[3] = z
3342 x[4] = eq. 25
3343
3344and similarly for the accented system K1, then the condition which is
3345identically satisfied by the transformation can be expressed thus :
3346
3347x[1]'2 + x[2]'2 + x[3]'2 + x[4]'2 = x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2
3348 (12).
3349
3350That is, by the afore-mentioned choice of " coordinates," (11a) [see
3351the end of Appendix II] is transformed into this equation.
3352
3353We see from (12) that the imaginary time co-ordinate x[4], enters into
3354the condition of transformation in exactly the same way as the space
3355co-ordinates x[1], x[2], x[3]. It is due to this fact that, according
3356to the theory of relativity, the " time "x[4], enters into natural
3357laws in the same form as the space co ordinates x[1], x[2], x[3].
3358
3359A four-dimensional continuum described by the "co-ordinates" x[1],
3360x[2], x[3], x[4], was called "world" by Minkowski, who also termed a
3361point-event a " world-point." From a "happening" in three-dimensional
3362space, physics becomes, as it were, an " existence " in the
3363four-dimensional " world."
3364
3365This four-dimensional " world " bears a close similarity to the
3366three-dimensional " space " of (Euclidean) analytical geometry. If we
3367introduce into the latter a new Cartesian co-ordinate system (x'[1],
3368x'[2], x'[3]) with the same origin, then x'[1], x'[2], x'[3], are
3369linear homogeneous functions of x[1], x[2], x[3] which identically
3370satisfy the equation
3371
3372 x'[1]^2 + x'[2]^2 + x'[3]^2 = x[1]^2 + x[2]^2 + x[3]^2
3373
3374The analogy with (12) is a complete one. We can regard Minkowski's "
3375world " in a formal manner as a four-dimensional Euclidean space (with
3376an imaginary time coordinate) ; the Lorentz transformation corresponds
3377to a " rotation " of the co-ordinate system in the fourdimensional "
3378world."
3379
3380
3381
3382APPENDIX III
3383
3384THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY
3385
3386
3387From a systematic theoretical point of view, we may imagine the
3388process of evolution of an empirical science to be a continuous
3389process of induction. Theories are evolved and are expressed in short
3390compass as statements of a large number of individual observations in
3391the form of empirical laws, from which the general laws can be
3392ascertained by comparison. Regarded in this way, the development of a
3393science bears some resemblance to the compilation of a classified
3394catalogue. It is, as it were, a purely empirical enterprise.
3395
3396But this point of view by no means embraces the whole of the actual
3397process ; for it slurs over the important part played by intuition and
3398deductive thought in the development of an exact science. As soon as a
3399science has emerged from its initial stages, theoretical advances are
3400no longer achieved merely by a process of arrangement. Guided by
3401empirical data, the investigator rather develops a system of thought
3402which, in general, is built up logically from a small number of
3403fundamental assumptions, the so-called axioms. We call such a system
3404of thought a theory. The theory finds the justification for its
3405existence in the fact that it correlates a large number of single
3406observations, and it is just here that the " truth " of the theory
3407lies.
3408
3409Corresponding to the same complex of empirical data, there may be
3410several theories, which differ from one another to a considerable
3411extent. But as regards the deductions from the theories which are
3412capable of being tested, the agreement between the theories may be so
3413complete that it becomes difficult to find any deductions in which the
3414two theories differ from each other. As an example, a case of general
3415interest is available in the province of biology, in the Darwinian
3416theory of the development of species by selection in the struggle for
3417existence, and in the theory of development which is based on the
3418hypothesis of the hereditary transmission of acquired characters.
3419
3420We have another instance of far-reaching agreement between the
3421deductions from two theories in Newtonian mechanics on the one hand,
3422and the general theory of relativity on the other. This agreement goes
3423so far, that up to the preseat we have been able to find only a few
3424deductions from the general theory of relativity which are capable of
3425investigation, and to which the physics of pre-relativity days does
3426not also lead, and this despite the profound difference in the
3427fundamental assumptions of the two theories. In what follows, we shall
3428again consider these important deductions, and we shall also discuss
3429the empirical evidence appertaining to them which has hitherto been
3430obtained.
3431
3432 (a) Motion of the Perihelion of Mercury
3433
3434According to Newtonian mechanics and Newton's law of gravitation, a
3435planet which is revolving round the sun would describe an ellipse
3436round the latter, or, more correctly, round the common centre of
3437gravity of the sun and the planet. In such a system, the sun, or the
3438common centre of gravity, lies in one of the foci of the orbital
3439ellipse in such a manner that, in the course of a planet-year, the
3440distance sun-planet grows from a minimum to a maximum, and then
3441decreases again to a minimum. If instead of Newton's law we insert a
3442somewhat different law of attraction into the calculation, we find
3443that, according to this new law, the motion would still take place in
3444such a manner that the distance sun-planet exhibits periodic
3445variations; but in this case the angle described by the line joining
3446sun and planet during such a period (from perihelion--closest
3447proximity to the sun--to perihelion) would differ from 360^0. The line
3448of the orbit would not then be a closed one but in the course of time
3449it would fill up an annular part of the orbital plane, viz. between
3450the circle of least and the circle of greatest distance of the planet
3451from the sun.
3452
3453According also to the general theory of relativity, which differs of
3454course from the theory of Newton, a small variation from the
3455Newton-Kepler motion of a planet in its orbit should take place, and
3456in such away, that the angle described by the radius sun-planet
3457between one perhelion and the next should exceed that corresponding to
3458one complete revolution by an amount given by
3459
3460 eq. 41: file eq41.gif
3461
3462(N.B. -- One complete revolution corresponds to the angle 2p in the
3463absolute angular measure customary in physics, and the above
3464expression giver the amount by which the radius sun-planet exceeds
3465this angle during the interval between one perihelion and the next.)
3466In this expression a represents the major semi-axis of the ellipse, e
3467its eccentricity, c the velocity of light, and T the period of
3468revolution of the planet. Our result may also be stated as follows :
3469According to the general theory of relativity, the major axis of the
3470ellipse rotates round the sun in the same sense as the orbital motion
3471of the planet. Theory requires that this rotation should amount to 43
3472seconds of arc per century for the planet Mercury, but for the other
3473Planets of our solar system its magnitude should be so small that it
3474would necessarily escape detection. *
3475
3476In point of fact, astronomers have found that the theory of Newton
3477does not suffice to calculate the observed motion of Mercury with an
3478exactness corresponding to that of the delicacy of observation
3479attainable at the present time. After taking account of all the
3480disturbing influences exerted on Mercury by the remaining planets, it
3481was found (Leverrier: 1859; and Newcomb: 1895) that an unexplained
3482perihelial movement of the orbit of Mercury remained over, the amount
3483of which does not differ sensibly from the above mentioned +43 seconds
3484of arc per century. The uncertainty of the empirical result amounts to
3485a few seconds only.
3486
3487 (b) Deflection of Light by a Gravitational Field
3488
3489In Section 22 it has been already mentioned that according to the
3490general theory of relativity, a ray of light will experience a
3491curvature of its path when passing through a gravitational field, this
3492curvature being similar to that experienced by the path of a body
3493which is projected through a gravitational field. As a result of this
3494theory, we should expect that a ray of light which is passing close to
3495a heavenly body would be deviated towards the latter. For a ray of
3496light which passes the sun at a distance of D sun-radii from its
3497centre, the angle of deflection (a) should amount to
3498
3499 eq. 42: file eq42.gif
3500
3501It may be added that, according to the theory, half of Figure 05 this
3502deflection is produced by the Newtonian field of attraction of the
3503sun, and the other half by the geometrical modification (" curvature
3504") of space caused by the sun.
3505
3506This result admits of an experimental test by means of the
3507photographic registration of stars during a total eclipse of the sun.
3508The only reason why we must wait for a total eclipse is because at
3509every other time the atmosphere is so strongly illuminated by the
3510light from the sun that the stars situated near the sun's disc are
3511invisible. The predicted effect can be seen clearly from the
3512accompanying diagram. If the sun (S) were not present, a star which is
3513practically infinitely distant would be seen in the direction D[1], as
3514observed front the earth. But as a consequence of the deflection of
3515light from the star by the sun, the star will be seen in the direction
3516D[2], i.e. at a somewhat greater distance from the centre of the sun
3517than corresponds to its real position.
3518
3519In practice, the question is tested in the following way. The stars in
3520the neighbourhood of the sun are photographed during a solar eclipse.
3521In addition, a second photograph of the same stars is taken when the
3522sun is situated at another position in the sky, i.e. a few months
3523earlier or later. As compared whh the standard photograph, the
3524positions of the stars on the eclipse-photograph ought to appear
3525displaced radially outwards (away from the centre of the sun) by an
3526amount corresponding to the angle a.
3527
3528We are indebted to the [British] Royal Society and to the Royal
3529Astronomical Society for the investigation of this important
3530deduction. Undaunted by the [first world] war and by difficulties of
3531both a material and a psychological nature aroused by the war, these
3532societies equipped two expeditions -- to Sobral (Brazil), and to the
3533island of Principe (West Africa) -- and sent several of Britain's most
3534celebrated astronomers (Eddington, Cottingham, Crommelin, Davidson),
3535in order to obtain photographs of the solar eclipse of 29th May, 1919.
3536The relative discrepancies to be expected between the stellar
3537photographs obtained during the eclipse and the comparison photographs
3538amounted to a few hundredths of a millimetre only. Thus great accuracy
3539was necessary in making the adjustments required for the taking of the
3540photographs, and in their subsequent measurement.
3541
3542The results of the measurements confirmed the theory in a thoroughly
3543satisfactory manner. The rectangular components of the observed and of
3544the calculated deviations of the stars (in seconds of arc) are set
3545forth in the following table of results :
3546
3547 Table 01: file table01.gif
3548
3549 (c) Displacement of Spectral Lines Towards the Red
3550
3551In Section 23 it has been shown that in a system K1 which is in
3552rotation with regard to a Galileian system K, clocks of identical
3553construction, and which are considered at rest with respect to the
3554rotating reference-body, go at rates which are dependent on the
3555positions of the clocks. We shall now examine this dependence
3556quantitatively. A clock, which is situated at a distance r from the
3557centre of the disc, has a velocity relative to K which is given by
3558
3559 V = wr
3560
3561where w represents the angular velocity of rotation of the disc K1
3562with respect to K. If v[0], represents the number of ticks of the
3563clock per unit time (" rate " of the clock) relative to K when the
3564clock is at rest, then the " rate " of the clock (v) when it is moving
3565relative to K with a velocity V, but at rest with respect to the disc,
3566will, in accordance with Section 12, be given by
3567
3568 eq. 43: file eq43.gif
3569
3570or with sufficient accuracy by
3571
3572 eq. 44: file eq44.gif
3573
3574This expression may also be stated in the following form:
3575
3576 eq. 45: file eq45.gif
3577
3578If we represent the difference of potential of the centrifugal force
3579between the position of the clock and the centre of the disc by f,
3580i.e. the work, considered negatively, which must be performed on the
3581unit of mass against the centrifugal force in order to transport it
3582from the position of the clock on the rotating disc to the centre of
3583the disc, then we have
3584
3585 eq. 46: file eq46.gif
3586
3587From this it follows that
3588
3589 eq. 47: file eq47.gif
3590
3591In the first place, we see from this expression that two clocks of
3592identical construction will go at different rates when situated at
3593different distances from the centre of the disc. This result is aiso
3594valid from the standpoint of an observer who is rotating with the
3595disc.
3596
3597Now, as judged from the disc, the latter is in a gravititional field
3598of potential f, hence the result we have obtained will hold quite
3599generally for gravitational fields. Furthermore, we can regard an atom
3600which is emitting spectral lines as a clock, so that the following
3601statement will hold:
3602
3603An atom absorbs or emits light of a frequency which is dependent on
3604the potential of the gravitational field in which it is situated.
3605
3606The frequency of an atom situated on the surface of a heavenly body
3607will be somewhat less than the frequency of an atom of the same
3608element which is situated in free space (or on the surface of a
3609smaller celestial body).
3610
3611Now f = - K (M/r), where K is Newton's constant of gravitation, and M
3612is the mass of the heavenly body. Thus a displacement towards the red
3613ought to take place for spectral lines produced at the surface of
3614stars as compared with the spectral lines of the same element produced
3615at the surface of the earth, the amount of this displacement being
3616
3617 eq. 48: file eq48.gif
3618
3619For the sun, the displacement towards the red predicted by theory
3620amounts to about two millionths of the wave-length. A trustworthy
3621calculation is not possible in the case of the stars, because in
3622general neither the mass M nor the radius r are known.
3623
3624It is an open question whether or not this effect exists, and at the
3625present time (1920) astronomers are working with great zeal towards
3626the solution. Owing to the smallness of the effect in the case of the
3627sun, it is difficult to form an opinion as to its existence. Whereas
3628Grebe and Bachem (Bonn), as a result of their own measurements and
3629those of Evershed and Schwarzschild on the cyanogen bands, have placed
3630the existence of the effect almost beyond doubt, while other
3631investigators, particularly St. John, have been led to the opposite
3632opinion in consequence of their measurements.
3633
3634Mean displacements of lines towards the less refrangible end of the
3635spectrum are certainly revealed by statistical investigations of the
3636fixed stars ; but up to the present the examination of the available
3637data does not allow of any definite decision being arrived at, as to
3638whether or not these displacements are to be referred in reality to
3639the effect of gravitation. The results of observation have been
3640collected together, and discussed in detail from the standpoint of the
3641question which has been engaging our attention here, in a paper by E.
3642Freundlich entitled "Zur Prüfung der allgemeinen
3643Relativit¨aut;ts-Theorie" (Die Naturwissenschaften, 1919, No. 35,
3644p. 520: Julius Springer, Berlin).
3645
3646At all events, a definite decision will be reached during the next few
3647years. If the displacement of spectral lines towards the red by the
3648gravitational potential does not exist, then the general theory of
3649relativity will be untenable. On the other hand, if the cause of the
3650displacement of spectral lines be definitely traced to the
3651gravitational potential, then the study of this displacement will
3652furnish us with important information as to the mass of the heavenly
3653bodies. [5][A]
3654
3655
3656 Notes
3657
3658*) Especially since the next planet Venus has an orbit that is
3659almost an exact circle, which makes it more difficult to locate the
3660perihelion with precision.
3661
3662The displacentent of spectral lines towards the red end of the
3663spectrum was definitely established by Adams in 1924, by observations
3664on the dense companion of Sirius, for which the effect is about thirty
3665times greater than for the Sun. R.W.L. -- translator
3666
3667
3668
3669APPENDIX IV
3670
3671THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY
3672(SUPPLEMENTARY TO SECTION 32)
3673
3674
3675Since the publication of the first edition of this little book, our
3676knowledge about the structure of space in the large (" cosmological
3677problem ") has had an important development, which ought to be
3678mentioned even in a popular presentation of the subject.
3679
3680My original considerations on the subject were based on two
3681hypotheses:
3682
3683(1) There exists an average density of matter in the whole of space
3684which is everywhere the same and different from zero.
3685
3686(2) The magnitude (" radius ") of space is independent of time.
3687
3688Both these hypotheses proved to be consistent, according to the
3689general theory of relativity, but only after a hypothetical term was
3690added to the field equations, a term which was not required by the
3691theory as such nor did it seem natural from a theoretical point of
3692view (" cosmological term of the field equations ").
3693
3694Hypothesis (2) appeared unavoidable to me at the time, since I thought
3695that one would get into bottomless speculations if one departed from
3696it.
3697
3698However, already in the 'twenties, the Russian mathematician Friedman
3699showed that a different hypothesis was natural from a purely
3700theoretical point of view. He realized that it was possible to
3701preserve hypothesis (1) without introducing the less natural
3702cosmological term into the field equations of gravitation, if one was
3703ready to drop hypothesis (2). Namely, the original field equations
3704admit a solution in which the " world radius " depends on time
3705(expanding space). In that sense one can say, according to Friedman,
3706that the theory demands an expansion of space.
3707
3708A few years later Hubble showed, by a special investigation of the
3709extra-galactic nebulae (" milky ways "), that the spectral lines
3710emitted showed a red shift which increased regularly with the distance
3711of the nebulae. This can be interpreted in regard to our present
3712knowledge only in the sense of Doppler's principle, as an expansive
3713motion of the system of stars in the large -- as required, according
3714to Friedman, by the field equations of gravitation. Hubble's discovery
3715can, therefore, be considered to some extent as a confirmation of the
3716theory.
3717
3718There does arise, however, a strange difficulty. The interpretation of
3719the galactic line-shift discovered by Hubble as an expansion (which
3720can hardly be doubted from a theoretical point of view), leads to an
3721origin of this expansion which lies " only " about 10^9 years ago,
3722while physical astronomy makes it appear likely that the development
3723of individual stars and systems of stars takes considerably longer. It
3724is in no way known how this incongruity is to be overcome.
3725
3726I further want to rernark that the theory of expanding space, together
3727with the empirical data of astronomy, permit no decision to be reached
3728about the finite or infinite character of (three-dimensional) space,
3729while the original " static " hypothesis of space yielded the closure
3730(finiteness) of space.
3731
3732
3733K = co-ordinate system
3734x, y = two-dimensional co-ordinates
3735x, y, z = three-dimensional co-ordinates
3736x, y, z, t = four-dimensional co-ordinates
3737
3738t = time
3739I = distance
3740v = velocity
3741
3742F = force
3743G = gravitational field
3744
3745
3746
3747
3748
3749*** END OF THE PROJECT GUTENBERG EBOOK, RELATIVITY ***
3750
3751This file should be named relat10.zip and contains numerous
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