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- The Project Gutenberg EBook of Relativity: The Special and General Theory
- by Albert Einstein
- (#1 in our series by Albert Einstein)
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- Title: Relativity: The Special and General Theory
- Author: Albert Einstein
- Release Date: February, 2004 [EBook #5001]
- [Yes, we are more than one year ahead of schedule]
- [This file was first posted on April 1, 2002]
- Edition: 10
- Language: English
- *** START OF THE PROJECT GUTENBERG EBOOK, RELATIVITY ***
- ALBERT EINSTEIN REFERENCE ARCHIVE
- RELATIVITY: THE SPECIAL AND GENERAL THEORY
- BY ALBERT EINSTEIN
- Written: 1916 (this revised edition: 1924)
- Source: Relativity: The Special and General Theory (1920)
- Publisher: Methuen & Co Ltd
- First Published: December, 1916
- Translated: Robert W. Lawson (Authorised translation)
- Transcription/Markup: Brian Basgen <brian@marxists.org>
- Transcription to text: Gregory B. Newby <gbnewby@petascale.org>
- Thanks to: Einstein Reference Archive (marxists.org)
- The Einstein Reference Archive is online at:
- http://www.marxists.org/reference/archive/einstein/index.htm
- Transcriber note: This file is a plain text rendition of HTML.
- Because many equations cannot be presented effectively in plain text,
- images are supplied for many equations and for all figures and tables.
- CONTENTS
- Preface
- Part I: The Special Theory of Relativity
- 01. Physical Meaning of Geometrical Propositions
- 02. The System of Co-ordinates
- 03. Space and Time in Classical Mechanics
- 04. The Galileian System of Co-ordinates
- 05. The Principle of Relativity (in the Restricted Sense)
- 06. The Theorem of the Addition of Velocities employed in
- Classical Mechanics
- 07. The Apparent Incompatability of the Law of Propagation of
- Light with the Principle of Relativity
- 08. On the Idea of Time in Physics
- 09. The Relativity of Simultaneity
- 10. On the Relativity of the Conception of Distance
- 11. The Lorentz Transformation
- 12. The Behaviour of Measuring-Rods and Clocks in Motion
- 13. Theorem of the Addition of Velocities. The Experiment of Fizeau
- 14. The Hueristic Value of the Theory of Relativity
- 15. General Results of the Theory
- 16. Expereince and the Special Theory of Relativity
- 17. Minkowski's Four-dimensial Space
- Part II: The General Theory of Relativity
- 18. Special and General Principle of Relativity
- 19. The Gravitational Field
- 20. The Equality of Inertial and Gravitational Mass as an Argument
- for the General Postulate of Relativity
- 21. In What Respects are the Foundations of Classical Mechanics
- and of the Special Theory of Relativity Unsatisfactory?
- 22. A Few Inferences from the General Principle of Relativity
- 23. Behaviour of Clocks and Measuring-Rods on a Rotating Body of
- Reference
- 24. Euclidean and non-Euclidean Continuum
- 25. Gaussian Co-ordinates
- 26. The Space-Time Continuum of the Speical Theory of Relativity
- Considered as a Euclidean Continuum
- 27. The Space-Time Continuum of the General Theory of Relativity
- is Not a Eculidean Continuum
- 28. Exact Formulation of the General Principle of Relativity
- 29. The Solution of the Problem of Gravitation on the Basis of the
- General Principle of Relativity
- Part III: Considerations on the Universe as a Whole
- 30. Cosmological Difficulties of Netwon's Theory
- 31. The Possibility of a "Finite" and yet "Unbounded" Universe
- 32. The Structure of Space According to the General Theory of
- Relativity
- Appendices:
- 01. Simple Derivation of the Lorentz Transformation (sup. ch. 11)
- 02. Minkowski's Four-Dimensional Space ("World") (sup. ch 17)
- 03. The Experimental Confirmation of the General Theory of Relativity
- 04. The Structure of Space According to the General Theory of
- Relativity (sup. ch 32)
- 05. Relativity and the Problem of Space
- Note: The fifth Appendix was added by Einstein at the time of the
- fifteenth re-printing of this book; and as a result is still under
- copyright restrictions so cannot be added without the permission of
- the publisher.
- PREFACE
- (December, 1916)
- The present book is intended, as far as possible, to give an exact
- insight into the theory of Relativity to those readers who, from a
- general scientific and philosophical point of view, are interested in
- the theory, but who are not conversant with the mathematical apparatus
- of theoretical physics. The work presumes a standard of education
- corresponding to that of a university matriculation examination, and,
- despite the shortness of the book, a fair amount of patience and force
- of will on the part of the reader. The author has spared himself no
- pains in his endeavour to present the main ideas in the simplest and
- most intelligible form, and on the whole, in the sequence and
- connection in which they actually originated. In the interest of
- clearness, it appeared to me inevitable that I should repeat myself
- frequently, without paying the slightest attention to the elegance of
- the presentation. I adhered scrupulously to the precept of that
- brilliant theoretical physicist L. Boltzmann, according to whom
- matters of elegance ought to be left to the tailor and to the cobbler.
- I make no pretence of having withheld from the reader difficulties
- which are inherent to the subject. On the other hand, I have purposely
- treated the empirical physical foundations of the theory in a
- "step-motherly" fashion, so that readers unfamiliar with physics may
- not feel like the wanderer who was unable to see the forest for the
- trees. May the book bring some one a few happy hours of suggestive
- thought!
- December, 1916
- A. EINSTEIN
- PART I
- THE SPECIAL THEORY OF RELATIVITY
- PHYSICAL MEANING OF GEOMETRICAL PROPOSITIONS
- In your schooldays most of you who read this book made acquaintance
- with the noble building of Euclid's geometry, and you remember --
- perhaps with more respect than love -- the magnificent structure, on
- the lofty staircase of which you were chased about for uncounted hours
- by conscientious teachers. By reason of our past experience, you would
- certainly regard everyone with disdain who should pronounce even the
- most out-of-the-way proposition of this science to be untrue. But
- perhaps this feeling of proud certainty would leave you immediately if
- some one were to ask you: "What, then, do you mean by the assertion
- that these propositions are true?" Let us proceed to give this
- question a little consideration.
- Geometry sets out form certain conceptions such as "plane," "point,"
- and "straight line," with which we are able to associate more or less
- definite ideas, and from certain simple propositions (axioms) which,
- in virtue of these ideas, we are inclined to accept as "true." Then,
- on the basis of a logical process, the justification of which we feel
- ourselves compelled to admit, all remaining propositions are shown to
- follow from those axioms, i.e. they are proven. A proposition is then
- correct ("true") when it has been derived in the recognised manner
- from the axioms. The question of "truth" of the individual geometrical
- propositions is thus reduced to one of the "truth" of the axioms. Now
- it has long been known that the last question is not only unanswerable
- by the methods of geometry, but that it is in itself entirely without
- meaning. We cannot ask whether it is true that only one straight line
- goes through two points. We can only say that Euclidean geometry deals
- with things called "straight lines," to each of which is ascribed the
- property of being uniquely determined by two points situated on it.
- The concept "true" does not tally with the assertions of pure
- geometry, because by the word "true" we are eventually in the habit of
- designating always the correspondence with a "real" object; geometry,
- however, is not concerned with the relation of the ideas involved in
- it to objects of experience, but only with the logical connection of
- these ideas among themselves.
- It is not difficult to understand why, in spite of this, we feel
- constrained to call the propositions of geometry "true." Geometrical
- ideas correspond to more or less exact objects in nature, and these
- last are undoubtedly the exclusive cause of the genesis of those
- ideas. Geometry ought to refrain from such a course, in order to give
- to its structure the largest possible logical unity. The practice, for
- example, of seeing in a "distance" two marked positions on a
- practically rigid body is something which is lodged deeply in our
- habit of thought. We are accustomed further to regard three points as
- being situated on a straight line, if their apparent positions can be
- made to coincide for observation with one eye, under suitable choice
- of our place of observation.
- If, in pursuance of our habit of thought, we now supplement the
- propositions of Euclidean geometry by the single proposition that two
- points on a practically rigid body always correspond to the same
- distance (line-interval), independently of any changes in position to
- which we may subject the body, the propositions of Euclidean geometry
- then resolve themselves into propositions on the possible relative
- position of practically rigid bodies.* Geometry which has been
- supplemented in this way is then to be treated as a branch of physics.
- We can now legitimately ask as to the "truth" of geometrical
- propositions interpreted in this way, since we are justified in asking
- whether these propositions are satisfied for those real things we have
- associated with the geometrical ideas. In less exact terms we can
- express this by saying that by the "truth" of a geometrical
- proposition in this sense we understand its validity for a
- construction with rule and compasses.
- Of course the conviction of the "truth" of geometrical propositions in
- this sense is founded exclusively on rather incomplete experience. For
- the present we shall assume the "truth" of the geometrical
- propositions, then at a later stage (in the general theory of
- relativity) we shall see that this "truth" is limited, and we shall
- consider the extent of its limitation.
- Notes
- *) It follows that a natural object is associated also with a
- straight line. Three points A, B and C on a rigid body thus lie in a
- straight line when the points A and C being given, B is chosen such
- that the sum of the distances AB and BC is as short as possible. This
- incomplete suggestion will suffice for the present purpose.
- THE SYSTEM OF CO-ORDINATES
- On the basis of the physical interpretation of distance which has been
- indicated, we are also in a position to establish the distance between
- two points on a rigid body by means of measurements. For this purpose
- we require a " distance " (rod S) which is to be used once and for
- all, and which we employ as a standard measure. If, now, A and B are
- two points on a rigid body, we can construct the line joining them
- according to the rules of geometry ; then, starting from A, we can
- mark off the distance S time after time until we reach B. The number
- of these operations required is the numerical measure of the distance
- AB. This is the basis of all measurement of length. *
- Every description of the scene of an event or of the position of an
- object in space is based on the specification of the point on a rigid
- body (body of reference) with which that event or object coincides.
- This applies not only to scientific description, but also to everyday
- life. If I analyse the place specification " Times Square, New York,"
- **A I arrive at the following result. The earth is the rigid body
- to which the specification of place refers; " Times Square, New York,"
- is a well-defined point, to which a name has been assigned, and with
- which the event coincides in space.**B
- This primitive method of place specification deals only with places on
- the surface of rigid bodies, and is dependent on the existence of
- points on this surface which are distinguishable from each other. But
- we can free ourselves from both of these limitations without altering
- the nature of our specification of position. If, for instance, a cloud
- is hovering over Times Square, then we can determine its position
- relative to the surface of the earth by erecting a pole
- perpendicularly on the Square, so that it reaches the cloud. The
- length of the pole measured with the standard measuring-rod, combined
- with the specification of the position of the foot of the pole,
- supplies us with a complete place specification. On the basis of this
- illustration, we are able to see the manner in which a refinement of
- the conception of position has been developed.
- (a) We imagine the rigid body, to which the place specification is
- referred, supplemented in such a manner that the object whose position
- we require is reached by. the completed rigid body.
- (b) In locating the position of the object, we make use of a number
- (here the length of the pole measured with the measuring-rod) instead
- of designated points of reference.
- (c) We speak of the height of the cloud even when the pole which
- reaches the cloud has not been erected. By means of optical
- observations of the cloud from different positions on the ground, and
- taking into account the properties of the propagation of light, we
- determine the length of the pole we should have required in order to
- reach the cloud.
- From this consideration we see that it will be advantageous if, in the
- description of position, it should be possible by means of numerical
- measures to make ourselves independent of the existence of marked
- positions (possessing names) on the rigid body of reference. In the
- physics of measurement this is attained by the application of the
- Cartesian system of co-ordinates.
- This consists of three plane surfaces perpendicular to each other and
- rigidly attached to a rigid body. Referred to a system of
- co-ordinates, the scene of any event will be determined (for the main
- part) by the specification of the lengths of the three perpendiculars
- or co-ordinates (x, y, z) which can be dropped from the scene of the
- event to those three plane surfaces. The lengths of these three
- perpendiculars can be determined by a series of manipulations with
- rigid measuring-rods performed according to the rules and methods laid
- down by Euclidean geometry.
- In practice, the rigid surfaces which constitute the system of
- co-ordinates are generally not available ; furthermore, the magnitudes
- of the co-ordinates are not actually determined by constructions with
- rigid rods, but by indirect means. If the results of physics and
- astronomy are to maintain their clearness, the physical meaning of
- specifications of position must always be sought in accordance with
- the above considerations. ***
- We thus obtain the following result: Every description of events in
- space involves the use of a rigid body to which such events have to be
- referred. The resulting relationship takes for granted that the laws
- of Euclidean geometry hold for "distances;" the "distance" being
- represented physically by means of the convention of two marks on a
- rigid body.
- Notes
- * Here we have assumed that there is nothing left over i.e. that
- the measurement gives a whole number. This difficulty is got over by
- the use of divided measuring-rods, the introduction of which does not
- demand any fundamentally new method.
- **A Einstein used "Potsdamer Platz, Berlin" in the original text.
- In the authorised translation this was supplemented with "Tranfalgar
- Square, London". We have changed this to "Times Square, New York", as
- this is the most well known/identifiable location to English speakers
- in the present day. [Note by the janitor.]
- **B It is not necessary here to investigate further the significance
- of the expression "coincidence in space." This conception is
- sufficiently obvious to ensure that differences of opinion are
- scarcely likely to arise as to its applicability in practice.
- *** A refinement and modification of these views does not become
- necessary until we come to deal with the general theory of relativity,
- treated in the second part of this book.
- SPACE AND TIME IN CLASSICAL MECHANICS
- The purpose of mechanics is to describe how bodies change their
- position in space with "time." I should load my conscience with grave
- sins against the sacred spirit of lucidity were I to formulate the
- aims of mechanics in this way, without serious reflection and detailed
- explanations. Let us proceed to disclose these sins.
- It is not clear what is to be understood here by "position" and
- "space." I stand at the window of a railway carriage which is
- travelling uniformly, and drop a stone on the embankment, without
- throwing it. Then, disregarding the influence of the air resistance, I
- see the stone descend in a straight line. A pedestrian who observes
- the misdeed from the footpath notices that the stone falls to earth in
- a parabolic curve. I now ask: Do the "positions" traversed by the
- stone lie "in reality" on a straight line or on a parabola? Moreover,
- what is meant here by motion "in space" ? From the considerations of
- the previous section the answer is self-evident. In the first place we
- entirely shun the vague word "space," of which, we must honestly
- acknowledge, we cannot form the slightest conception, and we replace
- it by "motion relative to a practically rigid body of reference." The
- positions relative to the body of reference (railway carriage or
- embankment) have already been defined in detail in the preceding
- section. If instead of " body of reference " we insert " system of
- co-ordinates," which is a useful idea for mathematical description, we
- are in a position to say : The stone traverses a straight line
- relative to a system of co-ordinates rigidly attached to the carriage,
- but relative to a system of co-ordinates rigidly attached to the
- ground (embankment) it describes a parabola. With the aid of this
- example it is clearly seen that there is no such thing as an
- independently existing trajectory (lit. "path-curve"*), but only
- a trajectory relative to a particular body of reference.
- In order to have a complete description of the motion, we must specify
- how the body alters its position with time ; i.e. for every point on
- the trajectory it must be stated at what time the body is situated
- there. These data must be supplemented by such a definition of time
- that, in virtue of this definition, these time-values can be regarded
- essentially as magnitudes (results of measurements) capable of
- observation. If we take our stand on the ground of classical
- mechanics, we can satisfy this requirement for our illustration in the
- following manner. We imagine two clocks of identical construction ;
- the man at the railway-carriage window is holding one of them, and the
- man on the footpath the other. Each of the observers determines the
- position on his own reference-body occupied by the stone at each tick
- of the clock he is holding in his hand. In this connection we have not
- taken account of the inaccuracy involved by the finiteness of the
- velocity of propagation of light. With this and with a second
- difficulty prevailing here we shall have to deal in detail later.
- Notes
- *) That is, a curve along which the body moves.
- THE GALILEIAN SYSTEM OF CO-ORDINATES
- As is well known, the fundamental law of the mechanics of
- Galilei-Newton, which is known as the law of inertia, can be stated
- thus: A body removed sufficiently far from other bodies continues in a
- state of rest or of uniform motion in a straight line. This law not
- only says something about the motion of the bodies, but it also
- indicates the reference-bodies or systems of coordinates, permissible
- in mechanics, which can be used in mechanical description. The visible
- fixed stars are bodies for which the law of inertia certainly holds to
- a high degree of approximation. Now if we use a system of co-ordinates
- which is rigidly attached to the earth, then, relative to this system,
- every fixed star describes a circle of immense radius in the course of
- an astronomical day, a result which is opposed to the statement of the
- law of inertia. So that if we adhere to this law we must refer these
- motions only to systems of coordinates relative to which the fixed
- stars do not move in a circle. A system of co-ordinates of which the
- state of motion is such that the law of inertia holds relative to it
- is called a " Galileian system of co-ordinates." The laws of the
- mechanics of Galflei-Newton can be regarded as valid only for a
- Galileian system of co-ordinates.
- THE PRINCIPLE OF RELATIVITY
- (IN THE RESTRICTED SENSE)
- In order to attain the greatest possible clearness, let us return to
- our example of the railway carriage supposed to be travelling
- uniformly. We call its motion a uniform translation ("uniform" because
- it is of constant velocity and direction, " translation " because
- although the carriage changes its position relative to the embankment
- yet it does not rotate in so doing). Let us imagine a raven flying
- through the air in such a manner that its motion, as observed from the
- embankment, is uniform and in a straight line. If we were to observe
- the flying raven from the moving railway carriage. we should find that
- the motion of the raven would be one of different velocity and
- direction, but that it would still be uniform and in a straight line.
- Expressed in an abstract manner we may say : If a mass m is moving
- uniformly in a straight line with respect to a co-ordinate system K,
- then it will also be moving uniformly and in a straight line relative
- to a second co-ordinate system K1 provided that the latter is
- executing a uniform translatory motion with respect to K. In
- accordance with the discussion contained in the preceding section, it
- follows that:
- If K is a Galileian co-ordinate system. then every other co-ordinate
- system K' is a Galileian one, when, in relation to K, it is in a
- condition of uniform motion of translation. Relative to K1 the
- mechanical laws of Galilei-Newton hold good exactly as they do with
- respect to K.
- We advance a step farther in our generalisation when we express the
- tenet thus: If, relative to K, K1 is a uniformly moving co-ordinate
- system devoid of rotation, then natural phenomena run their course
- with respect to K1 according to exactly the same general laws as with
- respect to K. This statement is called the principle of relativity (in
- the restricted sense).
- As long as one was convinced that all natural phenomena were capable
- of representation with the help of classical mechanics, there was no
- need to doubt the validity of this principle of relativity. But in
- view of the more recent development of electrodynamics and optics it
- became more and more evident that classical mechanics affords an
- insufficient foundation for the physical description of all natural
- phenomena. At this juncture the question of the validity of the
- principle of relativity became ripe for discussion, and it did not
- appear impossible that the answer to this question might be in the
- negative.
- Nevertheless, there are two general facts which at the outset speak
- very much in favour of the validity of the principle of relativity.
- Even though classical mechanics does not supply us with a sufficiently
- broad basis for the theoretical presentation of all physical
- phenomena, still we must grant it a considerable measure of " truth,"
- since it supplies us with the actual motions of the heavenly bodies
- with a delicacy of detail little short of wonderful. The principle of
- relativity must therefore apply with great accuracy in the domain of
- mechanics. But that a principle of such broad generality should hold
- with such exactness in one domain of phenomena, and yet should be
- invalid for another, is a priori not very probable.
- We now proceed to the second argument, to which, moreover, we shall
- return later. If the principle of relativity (in the restricted sense)
- does not hold, then the Galileian co-ordinate systems K, K1, K2, etc.,
- which are moving uniformly relative to each other, will not be
- equivalent for the description of natural phenomena. In this case we
- should be constrained to believe that natural laws are capable of
- being formulated in a particularly simple manner, and of course only
- on condition that, from amongst all possible Galileian co-ordinate
- systems, we should have chosen one (K[0]) of a particular state of
- motion as our body of reference. We should then be justified (because
- of its merits for the description of natural phenomena) in calling
- this system " absolutely at rest," and all other Galileian systems K "
- in motion." If, for instance, our embankment were the system K[0] then
- our railway carriage would be a system K, relative to which less
- simple laws would hold than with respect to K[0]. This diminished
- simplicity would be due to the fact that the carriage K would be in
- motion (i.e."really")with respect to K[0]. In the general laws of
- nature which have been formulated with reference to K, the magnitude
- and direction of the velocity of the carriage would necessarily play a
- part. We should expect, for instance, that the note emitted by an
- organpipe placed with its axis parallel to the direction of travel
- would be different from that emitted if the axis of the pipe were
- placed perpendicular to this direction.
- Now in virtue of its motion in an orbit round the sun, our earth is
- comparable with a railway carriage travelling with a velocity of about
- 30 kilometres per second. If the principle of relativity were not
- valid we should therefore expect that the direction of motion of the
- earth at any moment would enter into the laws of nature, and also that
- physical systems in their behaviour would be dependent on the
- orientation in space with respect to the earth. For owing to the
- alteration in direction of the velocity of revolution of the earth in
- the course of a year, the earth cannot be at rest relative to the
- hypothetical system K[0] throughout the whole year. However, the most
- careful observations have never revealed such anisotropic properties
- in terrestrial physical space, i.e. a physical non-equivalence of
- different directions. This is very powerful argument in favour of the
- principle of relativity.
- THE THEOREM OF THE
- ADDITION OF VELOCITIES
- EMPLOYED IN CLASSICAL MECHANICS
- Let us suppose our old friend the railway carriage to be travelling
- along the rails with a constant velocity v, and that a man traverses
- the length of the carriage in the direction of travel with a velocity
- w. How quickly or, in other words, with what velocity W does the man
- advance relative to the embankment during the process ? The only
- possible answer seems to result from the following consideration: If
- the man were to stand still for a second, he would advance relative to
- the embankment through a distance v equal numerically to the velocity
- of the carriage. As a consequence of his walking, however, he
- traverses an additional distance w relative to the carriage, and hence
- also relative to the embankment, in this second, the distance w being
- numerically equal to the velocity with which he is walking. Thus in
- total be covers the distance W=v+w relative to the embankment in the
- second considered. We shall see later that this result, which
- expresses the theorem of the addition of velocities employed in
- classical mechanics, cannot be maintained ; in other words, the law
- that we have just written down does not hold in reality. For the time
- being, however, we shall assume its correctness.
- THE APPARENT INCOMPATIBILITY OF THE
- LAW OF PROPAGATION OF LIGHT WITH THE
- PRINCIPLE OF RELATIVITY
- There is hardly a simpler law in physics than that according to which
- light is propagated in empty space. Every child at school knows, or
- believes he knows, that this propagation takes place in straight lines
- with a velocity c= 300,000 km./sec. At all events we know with great
- exactness that this velocity is the same for all colours, because if
- this were not the case, the minimum of emission would not be observed
- simultaneously for different colours during the eclipse of a fixed
- star by its dark neighbour. By means of similar considerations based
- on observa- tions of double stars, the Dutch astronomer De Sitter was
- also able to show that the velocity of propagation of light cannot
- depend on the velocity of motion of the body emitting the light. The
- assumption that this velocity of propagation is dependent on the
- direction "in space" is in itself improbable.
- In short, let us assume that the simple law of the constancy of the
- velocity of light c (in vacuum) is justifiably believed by the child
- at school. Who would imagine that this simple law has plunged the
- conscientiously thoughtful physicist into the greatest intellectual
- difficulties? Let us consider how these difficulties arise.
- Of course we must refer the process of the propagation of light (and
- indeed every other process) to a rigid reference-body (co-ordinate
- system). As such a system let us again choose our embankment. We shall
- imagine the air above it to have been removed. If a ray of light be
- sent along the embankment, we see from the above that the tip of the
- ray will be transmitted with the velocity c relative to the
- embankment. Now let us suppose that our railway carriage is again
- travelling along the railway lines with the velocity v, and that its
- direction is the same as that of the ray of light, but its velocity of
- course much less. Let us inquire about the velocity of propagation of
- the ray of light relative to the carriage. It is obvious that we can
- here apply the consideration of the previous section, since the ray of
- light plays the part of the man walking along relatively to the
- carriage. The velocity w of the man relative to the embankment is here
- replaced by the velocity of light relative to the embankment. w is the
- required velocity of light with respect to the carriage, and we have
- w = c-v.
- The velocity of propagation ot a ray of light relative to the carriage
- thus comes cut smaller than c.
- But this result comes into conflict with the principle of relativity
- set forth in Section V. For, like every other general law of
- nature, the law of the transmission of light in vacuo [in vacuum]
- must, according to the principle of relativity, be the same for the
- railway carriage as reference-body as when the rails are the body of
- reference. But, from our above consideration, this would appear to be
- impossible. If every ray of light is propagated relative to the
- embankment with the velocity c, then for this reason it would appear
- that another law of propagation of light must necessarily hold with
- respect to the carriage -- a result contradictory to the principle of
- relativity.
- In view of this dilemma there appears to be nothing else for it than
- to abandon either the principle of relativity or the simple law of the
- propagation of light in vacuo. Those of you who have carefully
- followed the preceding discussion are almost sure to expect that we
- should retain the principle of relativity, which appeals so
- convincingly to the intellect because it is so natural and simple. The
- law of the propagation of light in vacuo would then have to be
- replaced by a more complicated law conformable to the principle of
- relativity. The development of theoretical physics shows, however,
- that we cannot pursue this course. The epoch-making theoretical
- investigations of H. A. Lorentz on the electrodynamical and optical
- phenomena connected with moving bodies show that experience in this
- domain leads conclusively to a theory of electromagnetic phenomena, of
- which the law of the constancy of the velocity of light in vacuo is a
- necessary consequence. Prominent theoretical physicists were theref
- ore more inclined to reject the principle of relativity, in spite of
- the fact that no empirical data had been found which were
- contradictory to this principle.
- At this juncture the theory of relativity entered the arena. As a
- result of an analysis of the physical conceptions of time and space,
- it became evident that in realily there is not the least
- incompatibilitiy between the principle of relativity and the law of
- propagation of light, and that by systematically holding fast to both
- these laws a logically rigid theory could be arrived at. This theory
- has been called the special theory of relativity to distinguish it
- from the extended theory, with which we shall deal later. In the
- following pages we shall present the fundamental ideas of the special
- theory of relativity.
- ON THE IDEA OF TIME IN PHYSICS
- Lightning has struck the rails on our railway embankment at two places
- A and B far distant from each other. I make the additional assertion
- that these two lightning flashes occurred simultaneously. If I ask you
- whether there is sense in this statement, you will answer my question
- with a decided "Yes." But if I now approach you with the request to
- explain to me the sense of the statement more precisely, you find
- after some consideration that the answer to this question is not so
- easy as it appears at first sight.
- After some time perhaps the following answer would occur to you: "The
- significance of the statement is clear in itself and needs no further
- explanation; of course it would require some consideration if I were
- to be commissioned to determine by observations whether in the actual
- case the two events took place simultaneously or not." I cannot be
- satisfied with this answer for the following reason. Supposing that as
- a result of ingenious considerations an able meteorologist were to
- discover that the lightning must always strike the places A and B
- simultaneously, then we should be faced with the task of testing
- whether or not this theoretical result is in accordance with the
- reality. We encounter the same difficulty with all physical statements
- in which the conception " simultaneous " plays a part. The concept
- does not exist for the physicist until he has the possibility of
- discovering whether or not it is fulfilled in an actual case. We thus
- require a definition of simultaneity such that this definition
- supplies us with the method by means of which, in the present case, he
- can decide by experiment whether or not both the lightning strokes
- occurred simultaneously. As long as this requirement is not satisfied,
- I allow myself to be deceived as a physicist (and of course the same
- applies if I am not a physicist), when I imagine that I am able to
- attach a meaning to the statement of simultaneity. (I would ask the
- reader not to proceed farther until he is fully convinced on this
- point.)
- After thinking the matter over for some time you then offer the
- following suggestion with which to test simultaneity. By measuring
- along the rails, the connecting line AB should be measured up and an
- observer placed at the mid-point M of the distance AB. This observer
- should be supplied with an arrangement (e.g. two mirrors inclined at
- 90^0) which allows him visually to observe both places A and B at the
- same time. If the observer perceives the two flashes of lightning at
- the same time, then they are simultaneous.
- I am very pleased with this suggestion, but for all that I cannot
- regard the matter as quite settled, because I feel constrained to
- raise the following objection:
- "Your definition would certainly be right, if only I knew that the
- light by means of which the observer at M perceives the lightning
- flashes travels along the length A arrow M with the same velocity as
- along the length B arrow M. But an examination of this supposition
- would only be possible if we already had at our disposal the means of
- measuring time. It would thus appear as though we were moving here in
- a logical circle."
- After further consideration you cast a somewhat disdainful glance at
- me -- and rightly so -- and you declare:
- "I maintain my previous definition nevertheless, because in reality it
- assumes absolutely nothing about light. There is only one demand to be
- made of the definition of simultaneity, namely, that in every real
- case it must supply us with an empirical decision as to whether or not
- the conception that has to be defined is fulfilled. That my definition
- satisfies this demand is indisputable. That light requires the same
- time to traverse the path A arrow M as for the path B arrow M is in
- reality neither a supposition nor a hypothesis about the physical
- nature of light, but a stipulation which I can make of my own freewill
- in order to arrive at a definition of simultaneity."
- It is clear that this definition can be used to give an exact meaning
- not only to two events, but to as many events as we care to choose,
- and independently of the positions of the scenes of the events with
- respect to the body of reference * (here the railway embankment).
- We are thus led also to a definition of " time " in physics. For this
- purpose we suppose that clocks of identical construction are placed at
- the points A, B and C of the railway line (co-ordinate system) and
- that they are set in such a manner that the positions of their
- pointers are simultaneously (in the above sense) the same. Under these
- conditions we understand by the " time " of an event the reading
- (position of the hands) of that one of these clocks which is in the
- immediate vicinity (in space) of the event. In this manner a
- time-value is associated with every event which is essentially capable
- of observation.
- This stipulation contains a further physical hypothesis, the validity
- of which will hardly be doubted without empirical evidence to the
- contrary. It has been assumed that all these clocks go at the same
- rate if they are of identical construction. Stated more exactly: When
- two clocks arranged at rest in different places of a reference-body
- are set in such a manner that a particular position of the pointers of
- the one clock is simultaneous (in the above sense) with the same
- position, of the pointers of the other clock, then identical "
- settings " are always simultaneous (in the sense of the above
- definition).
- Notes
- *) We suppose further, that, when three events A, B and C occur in
- different places in such a manner that A is simultaneous with B and B
- is simultaneous with C (simultaneous in the sense of the above
- definition), then the criterion for the simultaneity of the pair of
- events A, C is also satisfied. This assumption is a physical
- hypothesis about the the of propagation of light: it must certainly be
- fulfilled if we are to maintain the law of the constancy of the
- velocity of light in vacuo.
- THE RELATIVITY OF SIMULATNEITY
- Up to now our considerations have been referred to a particular body
- of reference, which we have styled a " railway embankment." We suppose
- a very long train travelling along the rails with the constant
- velocity v and in the direction indicated in Fig 1. People travelling
- in this train will with a vantage view the train as a rigid
- reference-body (co-ordinate system); they regard all events in
- Fig. 01: file fig01.gif
- reference to the train. Then every event which takes place along the
- line also takes place at a particular point of the train. Also the
- definition of simultaneity can be given relative to the train in
- exactly the same way as with respect to the embankment. As a natural
- consequence, however, the following question arises :
- Are two events (e.g. the two strokes of lightning A and B) which are
- simultaneous with reference to the railway embankment also
- simultaneous relatively to the train? We shall show directly that the
- answer must be in the negative.
- When we say that the lightning strokes A and B are simultaneous with
- respect to be embankment, we mean: the rays of light emitted at the
- places A and B, where the lightning occurs, meet each other at the
- mid-point M of the length A arrow B of the embankment. But the events
- A and B also correspond to positions A and B on the train. Let M1 be
- the mid-point of the distance A arrow B on the travelling train. Just
- when the flashes (as judged from the embankment) of lightning occur,
- this point M1 naturally coincides with the point M but it moves
- towards the right in the diagram with the velocity v of the train. If
- an observer sitting in the position M1 in the train did not possess
- this velocity, then he would remain permanently at M, and the light
- rays emitted by the flashes of lightning A and B would reach him
- simultaneously, i.e. they would meet just where he is situated. Now in
- reality (considered with reference to the railway embankment) he is
- hastening towards the beam of light coming from B, whilst he is riding
- on ahead of the beam of light coming from A. Hence the observer will
- see the beam of light emitted from B earlier than he will see that
- emitted from A. Observers who take the railway train as their
- reference-body must therefore come to the conclusion that the
- lightning flash B took place earlier than the lightning flash A. We
- thus arrive at the important result:
- Events which are simultaneous with reference to the embankment are not
- simultaneous with respect to the train, and vice versa (relativity of
- simultaneity). Every reference-body (co-ordinate system) has its own
- particular time ; unless we are told the reference-body to which the
- statement of time refers, there is no meaning in a statement of the
- time of an event.
- Now before the advent of the theory of relativity it had always
- tacitly been assumed in physics that the statement of time had an
- absolute significance, i.e. that it is independent of the state of
- motion of the body of reference. But we have just seen that this
- assumption is incompatible with the most natural definition of
- simultaneity; if we discard this assumption, then the conflict between
- the law of the propagation of light in vacuo and the principle of
- relativity (developed in Section 7) disappears.
- We were led to that conflict by the considerations of Section 6,
- which are now no longer tenable. In that section we concluded that the
- man in the carriage, who traverses the distance w per second relative
- to the carriage, traverses the same distance also with respect to the
- embankment in each second of time. But, according to the foregoing
- considerations, the time required by a particular occurrence with
- respect to the carriage must not be considered equal to the duration
- of the same occurrence as judged from the embankment (as
- reference-body). Hence it cannot be contended that the man in walking
- travels the distance w relative to the railway line in a time which is
- equal to one second as judged from the embankment.
- Moreover, the considerations of Section 6 are based on yet a second
- assumption, which, in the light of a strict consideration, appears to
- be arbitrary, although it was always tacitly made even before the
- introduction of the theory of relativity.
- ON THE RELATIVITY OF THE CONCEPTION OF DISTANCE
- Let us consider two particular points on the train * travelling
- along the embankment with the velocity v, and inquire as to their
- distance apart. We already know that it is necessary to have a body of
- reference for the measurement of a distance, with respect to which
- body the distance can be measured up. It is the simplest plan to use
- the train itself as reference-body (co-ordinate system). An observer
- in the train measures the interval by marking off his measuring-rod in
- a straight line (e.g. along the floor of the carriage) as many times
- as is necessary to take him from the one marked point to the other.
- Then the number which tells us how often the rod has to be laid down
- is the required distance.
- It is a different matter when the distance has to be judged from the
- railway line. Here the following method suggests itself. If we call
- A^1 and B^1 the two points on the train whose distance apart is
- required, then both of these points are moving with the velocity v
- along the embankment. In the first place we require to determine the
- points A and B of the embankment which are just being passed by the
- two points A^1 and B^1 at a particular time t -- judged from the
- embankment. These points A and B of the embankment can be determined
- by applying the definition of time given in Section 8. The distance
- between these points A and B is then measured by repeated application
- of thee measuring-rod along the embankment.
- A priori it is by no means certain that this last measurement will
- supply us with the same result as the first. Thus the length of the
- train as measured from the embankment may be different from that
- obtained by measuring in the train itself. This circumstance leads us
- to a second objection which must be raised against the apparently
- obvious consideration of Section 6. Namely, if the man in the
- carriage covers the distance w in a unit of time -- measured from the
- train, -- then this distance -- as measured from the embankment -- is
- not necessarily also equal to w.
- Notes
- *) e.g. the middle of the first and of the hundredth carriage.
- THE LORENTZ TRANSFORMATION
- The results of the last three sections show that the apparent
- incompatibility of the law of propagation of light with the principle
- of relativity (Section 7) has been derived by means of a
- consideration which borrowed two unjustifiable hypotheses from
- classical mechanics; these are as follows:
- (1) The time-interval (time) between two events is independent of the
- condition of motion of the body of reference.
- (2) The space-interval (distance) between two points of a rigid body
- is independent of the condition of motion of the body of reference.
- If we drop these hypotheses, then the dilemma of Section 7
- disappears, because the theorem of the addition of velocities derived
- in Section 6 becomes invalid. The possibility presents itself that
- the law of the propagation of light in vacuo may be compatible with
- the principle of relativity, and the question arises: How have we to
- modify the considerations of Section 6 in order to remove the
- apparent disagreement between these two fundamental results of
- experience? This question leads to a general one. In the discussion of
- Section 6 we have to do with places and times relative both to the
- train and to the embankment. How are we to find the place and time of
- an event in relation to the train, when we know the place and time of
- the event with respect to the railway embankment ? Is there a
- thinkable answer to this question of such a nature that the law of
- transmission of light in vacuo does not contradict the principle of
- relativity ? In other words : Can we conceive of a relation between
- place and time of the individual events relative to both
- reference-bodies, such that every ray of light possesses the velocity
- of transmission c relative to the embankment and relative to the train
- ? This question leads to a quite definite positive answer, and to a
- perfectly definite transformation law for the space-time magnitudes of
- an event when changing over from one body of reference to another.
- Before we deal with this, we shall introduce the following incidental
- consideration. Up to the present we have only considered events taking
- place along the embankment, which had mathematically to assume the
- function of a straight line. In the manner indicated in Section 2
- we can imagine this reference-body supplemented laterally and in a
- vertical direction by means of a framework of rods, so that an event
- which takes place anywhere can be localised with reference to this
- framework. Fig. 2 Similarly, we can imagine the train travelling with
- the velocity v to be continued across the whole of space, so that
- every event, no matter how far off it may be, could also be localised
- with respect to the second framework. Without committing any
- fundamental error, we can disregard the fact that in reality these
- frameworks would continually interfere with each other, owing to the
- impenetrability of solid bodies. In every such framework we imagine
- three surfaces perpendicular to each other marked out, and designated
- as " co-ordinate planes " (" co-ordinate system "). A co-ordinate
- system K then corresponds to the embankment, and a co-ordinate system
- K' to the train. An event, wherever it may have taken place, would be
- fixed in space with respect to K by the three perpendiculars x, y, z
- on the co-ordinate planes, and with regard to time by a time value t.
- Relative to K1, the same event would be fixed in respect of space and
- time by corresponding values x1, y1, z1, t1, which of course are not
- identical with x, y, z, t. It has already been set forth in detail how
- these magnitudes are to be regarded as results of physical
- measurements.
- Obviously our problem can be exactly formulated in the following
- manner. What are the values x1, y1, z1, t1, of an event with respect
- to K1, when the magnitudes x, y, z, t, of the same event with respect
- to K are given ? The relations must be so chosen that the law of the
- transmission of light in vacuo is satisfied for one and the same ray
- of light (and of course for every ray) with respect to K and K1. For
- the relative orientation in space of the co-ordinate systems indicated
- in the diagram ([7]Fig. 2), this problem is solved by means of the
- equations :
- eq. 1: file eq01.gif
- y1 = y
- z1 = z
- eq. 2: file eq02.gif
- This system of equations is known as the " Lorentz transformation." *
- If in place of the law of transmission of light we had taken as our
- basis the tacit assumptions of the older mechanics as to the absolute
- character of times and lengths, then instead of the above we should
- have obtained the following equations:
- x1 = x - vt
- y1 = y
- z1 = z
- t1 = t
- This system of equations is often termed the " Galilei
- transformation." The Galilei transformation can be obtained from the
- Lorentz transformation by substituting an infinitely large value for
- the velocity of light c in the latter transformation.
- Aided by the following illustration, we can readily see that, in
- accordance with the Lorentz transformation, the law of the
- transmission of light in vacuo is satisfied both for the
- reference-body K and for the reference-body K1. A light-signal is sent
- along the positive x-axis, and this light-stimulus advances in
- accordance with the equation
- x = ct,
- i.e. with the velocity c. According to the equations of the Lorentz
- transformation, this simple relation between x and t involves a
- relation between x1 and t1. In point of fact, if we substitute for x
- the value ct in the first and fourth equations of the Lorentz
- transformation, we obtain:
- eq. 3: file eq03.gif
- eq. 4: file eq04.gif
- from which, by division, the expression
- x1 = ct1
- immediately follows. If referred to the system K1, the propagation of
- light takes place according to this equation. We thus see that the
- velocity of transmission relative to the reference-body K1 is also
- equal to c. The same result is obtained for rays of light advancing in
- any other direction whatsoever. Of cause this is not surprising, since
- the equations of the Lorentz transformation were derived conformably
- to this point of view.
- Notes
- *) A simple derivation of the Lorentz transformation is given in
- Appendix I.
- THE BEHAVIOUR OF MEASURING-RODS AND CLOCKS IN MOTION
- Place a metre-rod in the x1-axis of K1 in such a manner that one end
- (the beginning) coincides with the point x1=0 whilst the other end
- (the end of the rod) coincides with the point x1=I. What is the length
- of the metre-rod relatively to the system K? In order to learn this,
- we need only ask where the beginning of the rod and the end of the rod
- lie with respect to K at a particular time t of the system K. By means
- of the first equation of the Lorentz transformation the values of
- these two points at the time t = 0 can be shown to be
- eq. 05a: file eq05a.gif
- eq. 05b: file eq05b.gif
- the distance between the points being eq. 06 .
- But the metre-rod is moving with the velocity v relative to K. It
- therefore follows that the length of a rigid metre-rod moving in the
- direction of its length with a velocity v is eq. 06 of a metre.
- The rigid rod is thus shorter when in motion than when at rest, and
- the more quickly it is moving, the shorter is the rod. For the
- velocity v=c we should have eq. 06a ,
- and for stiII greater velocities the square-root becomes imaginary.
- From this we conclude that in the theory of relativity the velocity c
- plays the part of a limiting velocity, which can neither be reached
- nor exceeded by any real body.
- Of course this feature of the velocity c as a limiting velocity also
- clearly follows from the equations of the Lorentz transformation, for
- these became meaningless if we choose values of v greater than c.
- If, on the contrary, we had considered a metre-rod at rest in the
- x-axis with respect to K, then we should have found that the length of
- the rod as judged from K1 would have been eq. 06 ;
- this is quite in accordance with the principle of relativity which
- forms the basis of our considerations.
- A Priori it is quite clear that we must be able to learn something
- about the physical behaviour of measuring-rods and clocks from the
- equations of transformation, for the magnitudes z, y, x, t, are
- nothing more nor less than the results of measurements obtainable by
- means of measuring-rods and clocks. If we had based our considerations
- on the Galileian transformation we should not have obtained a
- contraction of the rod as a consequence of its motion.
- Let us now consider a seconds-clock which is permanently situated at
- the origin (x1=0) of K1. t1=0 and t1=I are two successive ticks of
- this clock. The first and fourth equations of the Lorentz
- transformation give for these two ticks :
- t = 0
- and
- eq. 07: file eq07.gif
- As judged from K, the clock is moving with the velocity v; as judged
- from this reference-body, the time which elapses between two strokes
- of the clock is not one second, but
- eq. 08: file eq08.gif
- seconds, i.e. a somewhat larger time. As a consequence of its motion
- the clock goes more slowly than when at rest. Here also the velocity c
- plays the part of an unattainable limiting velocity.
- THEOREM OF THE ADDITION OF VELOCITIES.
- THE EXPERIMENT OF FIZEAU
- Now in practice we can move clocks and measuring-rods only with
- velocities that are small compared with the velocity of light; hence
- we shall hardly be able to compare the results of the previous section
- directly with the reality. But, on the other hand, these results must
- strike you as being very singular, and for that reason I shall now
- draw another conclusion from the theory, one which can easily be
- derived from the foregoing considerations, and which has been most
- elegantly confirmed by experiment.
- In Section 6 we derived the theorem of the addition of velocities
- in one direction in the form which also results from the hypotheses of
- classical mechanics- This theorem can also be deduced readily horn the
- Galilei transformation (Section 11). In place of the man walking
- inside the carriage, we introduce a point moving relatively to the
- co-ordinate system K1 in accordance with the equation
- x1 = wt1
- By means of the first and fourth equations of the Galilei
- transformation we can express x1 and t1 in terms of x and t, and we
- then obtain
- x = (v + w)t
- This equation expresses nothing else than the law of motion of the
- point with reference to the system K (of the man with reference to the
- embankment). We denote this velocity by the symbol W, and we then
- obtain, as in Section 6,
- W=v+w A)
- But we can carry out this consideration just as well on the basis of
- the theory of relativity. In the equation
- x1 = wt1 B)
- we must then express x1and t1 in terms of x and t, making use of the
- first and fourth equations of the Lorentz transformation. Instead of
- the equation (A) we then obtain the equation
- eq. 09: file eq09.gif
- which corresponds to the theorem of addition for velocities in one
- direction according to the theory of relativity. The question now
- arises as to which of these two theorems is the better in accord with
- experience. On this point we axe enlightened by a most important
- experiment which the brilliant physicist Fizeau performed more than
- half a century ago, and which has been repeated since then by some of
- the best experimental physicists, so that there can be no doubt about
- its result. The experiment is concerned with the following question.
- Light travels in a motionless liquid with a particular velocity w. How
- quickly does it travel in the direction of the arrow in the tube T
- (see the accompanying diagram, Fig. 3) when the liquid above
- mentioned is flowing through the tube with a velocity v ?
- In accordance with the principle of relativity we shall certainly have
- to take for granted that the propagation of light always takes place
- with the same velocity w with respect to the liquid, whether the
- latter is in motion with reference to other bodies or not. The
- velocity of light relative to the liquid and the velocity of the
- latter relative to the tube are thus known, and we require the
- velocity of light relative to the tube.
- It is clear that we have the problem of Section 6 again before us. The
- tube plays the part of the railway embankment or of the co-ordinate
- system K, the liquid plays the part of the carriage or of the
- co-ordinate system K1, and finally, the light plays the part of the
- Figure 03: file fig03.gif
- man walking along the carriage, or of the moving point in the present
- section. If we denote the velocity of the light relative to the tube
- by W, then this is given by the equation (A) or (B), according as the
- Galilei transformation or the Lorentz transformation corresponds to
- the facts. Experiment * decides in favour of equation (B) derived
- from the theory of relativity, and the agreement is, indeed, very
- exact. According to recent and most excellent measurements by Zeeman,
- the influence of the velocity of flow v on the propagation of light is
- represented by formula (B) to within one per cent.
- Nevertheless we must now draw attention to the fact that a theory of
- this phenomenon was given by H. A. Lorentz long before the statement
- of the theory of relativity. This theory was of a purely
- electrodynamical nature, and was obtained by the use of particular
- hypotheses as to the electromagnetic structure of matter. This
- circumstance, however, does not in the least diminish the
- conclusiveness of the experiment as a crucial test in favour of the
- theory of relativity, for the electrodynamics of Maxwell-Lorentz, on
- which the original theory was based, in no way opposes the theory of
- relativity. Rather has the latter been developed trom electrodynamics
- as an astoundingly simple combination and generalisation of the
- hypotheses, formerly independent of each other, on which
- electrodynamics was built.
- Notes
- *) Fizeau found eq. 10 , where eq. 11
- is the index of refraction of the liquid. On the other hand, owing to
- the smallness of eq. 12 as compared with I,
- we can replace (B) in the first place by eq. 13 , or to the same order
- of approximation by
- eq. 14 , which agrees with Fizeau's result.
- THE HEURISTIC VALUE OF THE THEORY OF RELATIVITY
- Our train of thought in the foregoing pages can be epitomised in the
- following manner. Experience has led to the conviction that, on the
- one hand, the principle of relativity holds true and that on the other
- hand the velocity of transmission of light in vacuo has to be
- considered equal to a constant c. By uniting these two postulates we
- obtained the law of transformation for the rectangular co-ordinates x,
- y, z and the time t of the events which constitute the processes of
- nature. In this connection we did not obtain the Galilei
- transformation, but, differing from classical mechanics, the Lorentz
- transformation.
- The law of transmission of light, the acceptance of which is justified
- by our actual knowledge, played an important part in this process of
- thought. Once in possession of the Lorentz transformation, however, we
- can combine this with the principle of relativity, and sum up the
- theory thus:
- Every general law of nature must be so constituted that it is
- transformed into a law of exactly the same form when, instead of the
- space-time variables x, y, z, t of the original coordinate system K,
- we introduce new space-time variables x1, y1, z1, t1 of a co-ordinate
- system K1. In this connection the relation between the ordinary and
- the accented magnitudes is given by the Lorentz transformation. Or in
- brief : General laws of nature are co-variant with respect to Lorentz
- transformations.
- This is a definite mathematical condition that the theory of
- relativity demands of a natural law, and in virtue of this, the theory
- becomes a valuable heuristic aid in the search for general laws of
- nature. If a general law of nature were to be found which did not
- satisfy this condition, then at least one of the two fundamental
- assumptions of the theory would have been disproved. Let us now
- examine what general results the latter theory has hitherto evinced.
- GENERAL RESULTS OF THE THEORY
- It is clear from our previous considerations that the (special) theory
- of relativity has grown out of electrodynamics and optics. In these
- fields it has not appreciably altered the predictions of theory, but
- it has considerably simplified the theoretical structure, i.e. the
- derivation of laws, and -- what is incomparably more important -- it
- has considerably reduced the number of independent hypothese forming
- the basis of theory. The special theory of relativity has rendered the
- Maxwell-Lorentz theory so plausible, that the latter would have been
- generally accepted by physicists even if experiment had decided less
- unequivocally in its favour.
- Classical mechanics required to be modified before it could come into
- line with the demands of the special theory of relativity. For the
- main part, however, this modification affects only the laws for rapid
- motions, in which the velocities of matter v are not very small as
- compared with the velocity of light. We have experience of such rapid
- motions only in the case of electrons and ions; for other motions the
- variations from the laws of classical mechanics are too small to make
- themselves evident in practice. We shall not consider the motion of
- stars until we come to speak of the general theory of relativity. In
- accordance with the theory of relativity the kinetic energy of a
- material point of mass m is no longer given by the well-known
- expression
- eq. 15: file eq15.gif
- but by the expression
- eq. 16: file eq16.gif
- This expression approaches infinity as the velocity v approaches the
- velocity of light c. The velocity must therefore always remain less
- than c, however great may be the energies used to produce the
- acceleration. If we develop the expression for the kinetic energy in
- the form of a series, we obtain
- eq. 17: file eq17.gif
- When eq. 18 is small compared with unity, the third of these terms is
- always small in comparison with the second,
- which last is alone considered in classical mechanics. The first term
- mc^2 does not contain the velocity, and requires no consideration if
- we are only dealing with the question as to how the energy of a
- point-mass; depends on the velocity. We shall speak of its essential
- significance later.
- The most important result of a general character to which the special
- theory of relativity has led is concerned with the conception of mass.
- Before the advent of relativity, physics recognised two conservation
- laws of fundamental importance, namely, the law of the canservation of
- energy and the law of the conservation of mass these two fundamental
- laws appeared to be quite independent of each other. By means of the
- theory of relativity they have been united into one law. We shall now
- briefly consider how this unification came about, and what meaning is
- to be attached to it.
- The principle of relativity requires that the law of the concervation
- of energy should hold not only with reference to a co-ordinate system
- K, but also with respect to every co-ordinate system K1 which is in a
- state of uniform motion of translation relative to K, or, briefly,
- relative to every " Galileian " system of co-ordinates. In contrast to
- classical mechanics; the Lorentz transformation is the deciding factor
- in the transition from one such system to another.
- By means of comparatively simple considerations we are led to draw the
- following conclusion from these premises, in conjunction with the
- fundamental equations of the electrodynamics of Maxwell: A body moving
- with the velocity v, which absorbs * an amount of energy E[0] in
- the form of radiation without suffering an alteration in velocity in
- the process, has, as a consequence, its energy increased by an amount
- eq. 19: file eq19.gif
- In consideration of the expression given above for the kinetic energy
- of the body, the required energy of the body comes out to be
- eq. 20: file eq20.gif
- Thus the body has the same energy as a body of mass
- eq.21: file eq21.gif
- moving with the velocity v. Hence we can say: If a body takes up an
- amount of energy E[0], then its inertial mass increases by an amount
- eq. 22: file eq22.gif
- the inertial mass of a body is not a constant but varies according to
- the change in the energy of the body. The inertial mass of a system of
- bodies can even be regarded as a measure of its energy. The law of the
- conservation of the mass of a system becomes identical with the law of
- the conservation of energy, and is only valid provided that the system
- neither takes up nor sends out energy. Writing the expression for the
- energy in the form
- eq. 23: file eq23.gif
- we see that the term mc^2, which has hitherto attracted our attention,
- is nothing else than the energy possessed by the body ** before it
- absorbed the energy E[0].
- A direct comparison of this relation with experiment is not possible
- at the present time (1920; see *** Note, p. 48), owing to the fact that
- the changes in energy E[0] to which we can Subject a system are not
- large enough to make themselves perceptible as a change in the
- inertial mass of the system.
- eq. 22: file eq22.gif
- is too small in comparison with the mass m, which was present before
- the alteration of the energy. It is owing to this circumstance that
- classical mechanics was able to establish successfully the
- conservation of mass as a law of independent validity.
- Let me add a final remark of a fundamental nature. The success of the
- Faraday-Maxwell interpretation of electromagnetic action at a distance
- resulted in physicists becoming convinced that there are no such
- things as instantaneous actions at a distance (not involving an
- intermediary medium) of the type of Newton's law of gravitation.
- According to the theory of relativity, action at a distance with the
- velocity of light always takes the place of instantaneous action at a
- distance or of action at a distance with an infinite velocity of
- transmission. This is connected with the fact that the velocity c
- plays a fundamental role in this theory. In Part II we shall see in
- what way this result becomes modified in the general theory of
- relativity.
- Notes
- *) E[0] is the energy taken up, as judged from a co-ordinate system
- moving with the body.
- **) As judged from a co-ordinate system moving with the body.
- ***[Note] The equation E = mc^2 has been thoroughly proved time and
- again since this time.
- EXPERIENCE AND THE SPECIAL THEORY OF RELATIVITY
- To what extent is the special theory of relativity supported by
- experience? This question is not easily answered for the reason
- already mentioned in connection with the fundamental experiment of
- Fizeau. The special theory of relativity has crystallised out from the
- Maxwell-Lorentz theory of electromagnetic phenomena. Thus all facts of
- experience which support the electromagnetic theory also support the
- theory of relativity. As being of particular importance, I mention
- here the fact that the theory of relativity enables us to predict the
- effects produced on the light reaching us from the fixed stars. These
- results are obtained in an exceedingly simple manner, and the effects
- indicated, which are due to the relative motion of the earth with
- reference to those fixed stars are found to be in accord with
- experience. We refer to the yearly movement of the apparent position
- of the fixed stars resulting from the motion of the earth round the
- sun (aberration), and to the influence of the radial components of the
- relative motions of the fixed stars with respect to the earth on the
- colour of the light reaching us from them. The latter effect manifests
- itself in a slight displacement of the spectral lines of the light
- transmitted to us from a fixed star, as compared with the position of
- the same spectral lines when they are produced by a terrestrial source
- of light (Doppler principle). The experimental arguments in favour of
- the Maxwell-Lorentz theory, which are at the same time arguments in
- favour of the theory of relativity, are too numerous to be set forth
- here. In reality they limit the theoretical possibilities to such an
- extent, that no other theory than that of Maxwell and Lorentz has been
- able to hold its own when tested by experience.
- But there are two classes of experimental facts hitherto obtained
- which can be represented in the Maxwell-Lorentz theory only by the
- introduction of an auxiliary hypothesis, which in itself -- i.e.
- without making use of the theory of relativity -- appears extraneous.
- It is known that cathode rays and the so-called b-rays emitted by
- radioactive substances consist of negatively electrified particles
- (electrons) of very small inertia and large velocity. By examining the
- deflection of these rays under the influence of electric and magnetic
- fields, we can study the law of motion of these particles very
- exactly.
- In the theoretical treatment of these electrons, we are faced with the
- difficulty that electrodynamic theory of itself is unable to give an
- account of their nature. For since electrical masses of one sign repel
- each other, the negative electrical masses constituting the electron
- would necessarily be scattered under the influence of their mutual
- repulsions, unless there are forces of another kind operating between
- them, the nature of which has hitherto remained obscure to us.* If
- we now assume that the relative distances between the electrical
- masses constituting the electron remain unchanged during the motion of
- the electron (rigid connection in the sense of classical mechanics),
- we arrive at a law of motion of the electron which does not agree with
- experience. Guided by purely formal points of view, H. A. Lorentz was
- the first to introduce the hypothesis that the form of the electron
- experiences a contraction in the direction of motion in consequence of
- that motion. the contracted length being proportional to the
- expression
- eq. 05: file eq05.gif
- This, hypothesis, which is not justifiable by any electrodynamical
- facts, supplies us then with that particular law of motion which has
- been confirmed with great precision in recent years.
- The theory of relativity leads to the same law of motion, without
- requiring any special hypothesis whatsoever as to the structure and
- the behaviour of the electron. We arrived at a similar conclusion in
- Section 13 in connection with the experiment of Fizeau, the result
- of which is foretold by the theory of relativity without the necessity
- of drawing on hypotheses as to the physical nature of the liquid.
- The second class of facts to which we have alluded has reference to
- the question whether or not the motion of the earth in space can be
- made perceptible in terrestrial experiments. We have already remarked
- in Section 5 that all attempts of this nature led to a negative
- result. Before the theory of relativity was put forward, it was
- difficult to become reconciled to this negative result, for reasons
- now to be discussed. The inherited prejudices about time and space did
- not allow any doubt to arise as to the prime importance of the
- Galileian transformation for changing over from one body of reference
- to another. Now assuming that the Maxwell-Lorentz equations hold for a
- reference-body K, we then find that they do not hold for a
- reference-body K1 moving uniformly with respect to K, if we assume
- that the relations of the Galileian transformstion exist between the
- co-ordinates of K and K1. It thus appears that, of all Galileian
- co-ordinate systems, one (K) corresponding to a particular state of
- motion is physically unique. This result was interpreted physically by
- regarding K as at rest with respect to a hypothetical æther of space.
- On the other hand, all coordinate systems K1 moving relatively to K
- were to be regarded as in motion with respect to the æther. To this
- motion of K1 against the æther ("æther-drift " relative to K1) were
- attributed the more complicated laws which were supposed to hold
- relative to K1. Strictly speaking, such an æther-drift ought also to
- be assumed relative to the earth, and for a long time the efforts of
- physicists were devoted to attempts to detect the existence of an
- æther-drift at the earth's surface.
- In one of the most notable of these attempts Michelson devised a
- method which appears as though it must be decisive. Imagine two
- mirrors so arranged on a rigid body that the reflecting surfaces face
- each other. A ray of light requires a perfectly definite time T to
- pass from one mirror to the other and back again, if the whole system
- be at rest with respect to the æther. It is found by calculation,
- however, that a slightly different time T1 is required for this
- process, if the body, together with the mirrors, be moving relatively
- to the æther. And yet another point: it is shown by calculation that
- for a given velocity v with reference to the æther, this time T1 is
- different when the body is moving perpendicularly to the planes of the
- mirrors from that resulting when the motion is parallel to these
- planes. Although the estimated difference between these two times is
- exceedingly small, Michelson and Morley performed an experiment
- involving interference in which this difference should have been
- clearly detectable. But the experiment gave a negative result -- a
- fact very perplexing to physicists. Lorentz and FitzGerald rescued the
- theory from this difficulty by assuming that the motion of the body
- relative to the æther produces a contraction of the body in the
- direction of motion, the amount of contraction being just sufficient
- to compensate for the differeace in time mentioned above. Comparison
- with the discussion in Section 11 shows that also from the
- standpoint of the theory of relativity this solution of the difficulty
- was the right one. But on the basis of the theory of relativity the
- method of interpretation is incomparably more satisfactory. According
- to this theory there is no such thing as a " specially favoured "
- (unique) co-ordinate system to occasion the introduction of the
- æther-idea, and hence there can be no æther-drift, nor any experiment
- with which to demonstrate it. Here the contraction of moving bodies
- follows from the two fundamental principles of the theory, without the
- introduction of particular hypotheses ; and as the prime factor
- involved in this contraction we find, not the motion in itself, to
- which we cannot attach any meaning, but the motion with respect to the
- body of reference chosen in the particular case in point. Thus for a
- co-ordinate system moving with the earth the mirror system of
- Michelson and Morley is not shortened, but it is shortened for a
- co-ordinate system which is at rest relatively to the sun.
- Notes
- *) The general theory of relativity renders it likely that the
- electrical masses of an electron are held together by gravitational
- forces.
- MINKOWSKI'S FOUR-DIMENSIONAL SPACE
- The non-mathematician is seized by a mysterious shuddering when he
- hears of "four-dimensional" things, by a feeling not unlike that
- awakened by thoughts of the occult. And yet there is no more
- common-place statement than that the world in which we live is a
- four-dimensional space-time continuum.
- Space is a three-dimensional continuum. By this we mean that it is
- possible to describe the position of a point (at rest) by means of
- three numbers (co-ordinales) x, y, z, and that there is an indefinite
- number of points in the neighbourhood of this one, the position of
- which can be described by co-ordinates such as x[1], y[1], z[1], which
- may be as near as we choose to the respective values of the
- co-ordinates x, y, z, of the first point. In virtue of the latter
- property we speak of a " continuum," and owing to the fact that there
- are three co-ordinates we speak of it as being " three-dimensional."
- Similarly, the world of physical phenomena which was briefly called "
- world " by Minkowski is naturally four dimensional in the space-time
- sense. For it is composed of individual events, each of which is
- described by four numbers, namely, three space co-ordinates x, y, z,
- and a time co-ordinate, the time value t. The" world" is in this sense
- also a continuum; for to every event there are as many "neighbouring"
- events (realised or at least thinkable) as we care to choose, the
- co-ordinates x[1], y[1], z[1], t[1] of which differ by an indefinitely
- small amount from those of the event x, y, z, t originally considered.
- That we have not been accustomed to regard the world in this sense as
- a four-dimensional continuum is due to the fact that in physics,
- before the advent of the theory of relativity, time played a different
- and more independent role, as compared with the space coordinates. It
- is for this reason that we have been in the habit of treating time as
- an independent continuum. As a matter of fact, according to classical
- mechanics, time is absolute, i.e. it is independent of the position
- and the condition of motion of the system of co-ordinates. We see this
- expressed in the last equation of the Galileian transformation (t1 =
- t)
- The four-dimensional mode of consideration of the "world" is natural
- on the theory of relativity, since according to this theory time is
- robbed of its independence. This is shown by the fourth equation of
- the Lorentz transformation:
- eq. 24: file eq24.gif
- Moreover, according to this equation the time difference Dt1 of two
- events with respect to K1 does not in general vanish, even when the
- time difference Dt1 of the same events with reference to K vanishes.
- Pure " space-distance " of two events with respect to K results in "
- time-distance " of the same events with respect to K. But the
- discovery of Minkowski, which was of importance for the formal
- development of the theory of relativity, does not lie here. It is to
- be found rather in the fact of his recognition that the
- four-dimensional space-time continuum of the theory of relativity, in
- its most essential formal properties, shows a pronounced relationship
- to the three-dimensional continuum of Euclidean geometrical
- space.* In order to give due prominence to this relationship,
- however, we must replace the usual time co-ordinate t by an imaginary
- magnitude eq. 25 proportional to it. Under these conditions, the
- natural laws satisfying the demands of the (special) theory of
- relativity assume mathematical forms, in which the time co-ordinate
- plays exactly the same role as the three space co-ordinates. Formally,
- these four co-ordinates correspond exactly to the three space
- co-ordinates in Euclidean geometry. It must be clear even to the
- non-mathematician that, as a consequence of this purely formal
- addition to our knowledge, the theory perforce gained clearness in no
- mean measure.
- These inadequate remarks can give the reader only a vague notion of
- the important idea contributed by Minkowski. Without it the general
- theory of relativity, of which the fundamental ideas are developed in
- the following pages, would perhaps have got no farther than its long
- clothes. Minkowski's work is doubtless difficult of access to anyone
- inexperienced in mathematics, but since it is not necessary to have a
- very exact grasp of this work in order to understand the fundamental
- ideas of either the special or the general theory of relativity, I
- shall leave it here at present, and revert to it only towards the end
- of Part 2.
- Notes
- *) Cf. the somewhat more detailed discussion in Appendix II.
- PART II
- THE GENERAL THEORY OF RELATIVITY
- SPECIAL AND GENERAL PRINCIPLE OF RELATIVITY
- The basal principle, which was the pivot of all our previous
- considerations, was the special principle of relativity, i.e. the
- principle of the physical relativity of all uniform motion. Let as
- once more analyse its meaning carefully.
- It was at all times clear that, from the point of view of the idea it
- conveys to us, every motion must be considered only as a relative
- motion. Returning to the illustration we have frequently used of the
- embankment and the railway carriage, we can express the fact of the
- motion here taking place in the following two forms, both of which are
- equally justifiable :
- (a) The carriage is in motion relative to the embankment,
- (b) The embankment is in motion relative to the carriage.
- In (a) the embankment, in (b) the carriage, serves as the body of
- reference in our statement of the motion taking place. If it is simply
- a question of detecting or of describing the motion involved, it is in
- principle immaterial to what reference-body we refer the motion. As
- already mentioned, this is self-evident, but it must not be confused
- with the much more comprehensive statement called "the principle of
- relativity," which we have taken as the basis of our investigations.
- The principle we have made use of not only maintains that we may
- equally well choose the carriage or the embankment as our
- reference-body for the description of any event (for this, too, is
- self-evident). Our principle rather asserts what follows : If we
- formulate the general laws of nature as they are obtained from
- experience, by making use of
- (a) the embankment as reference-body,
- (b) the railway carriage as reference-body,
- then these general laws of nature (e.g. the laws of mechanics or the
- law of the propagation of light in vacuo) have exactly the same form
- in both cases. This can also be expressed as follows : For the
- physical description of natural processes, neither of the reference
- bodies K, K1 is unique (lit. " specially marked out ") as compared
- with the other. Unlike the first, this latter statement need not of
- necessity hold a priori; it is not contained in the conceptions of "
- motion" and " reference-body " and derivable from them; only
- experience can decide as to its correctness or incorrectness.
- Up to the present, however, we have by no means maintained the
- equivalence of all bodies of reference K in connection with the
- formulation of natural laws. Our course was more on the following
- Iines. In the first place, we started out from the assumption that
- there exists a reference-body K, whose condition of motion is such
- that the Galileian law holds with respect to it : A particle left to
- itself and sufficiently far removed from all other particles moves
- uniformly in a straight line. With reference to K (Galileian
- reference-body) the laws of nature were to be as simple as possible.
- But in addition to K, all bodies of reference K1 should be given
- preference in this sense, and they should be exactly equivalent to K
- for the formulation of natural laws, provided that they are in a state
- of uniform rectilinear and non-rotary motion with respect to K ; all
- these bodies of reference are to be regarded as Galileian
- reference-bodies. The validity of the principle of relativity was
- assumed only for these reference-bodies, but not for others (e.g.
- those possessing motion of a different kind). In this sense we speak
- of the special principle of relativity, or special theory of
- relativity.
- In contrast to this we wish to understand by the "general principle of
- relativity" the following statement : All bodies of reference K, K1,
- etc., are equivalent for the description of natural phenomena
- (formulation of the general laws of nature), whatever may be their
- state of motion. But before proceeding farther, it ought to be pointed
- out that this formulation must be replaced later by a more abstract
- one, for reasons which will become evident at a later stage.
- Since the introduction of the special principle of relativity has been
- justified, every intellect which strives after generalisation must
- feel the temptation to venture the step towards the general principle
- of relativity. But a simple and apparently quite reliable
- consideration seems to suggest that, for the present at any rate,
- there is little hope of success in such an attempt; Let us imagine
- ourselves transferred to our old friend the railway carriage, which is
- travelling at a uniform rate. As long as it is moving unifromly, the
- occupant of the carriage is not sensible of its motion, and it is for
- this reason that he can without reluctance interpret the facts of the
- case as indicating that the carriage is at rest, but the embankment in
- motion. Moreover, according to the special principle of relativity,
- this interpretation is quite justified also from a physical point of
- view.
- If the motion of the carriage is now changed into a non-uniform
- motion, as for instance by a powerful application of the brakes, then
- the occupant of the carriage experiences a correspondingly powerful
- jerk forwards. The retarded motion is manifested in the mechanical
- behaviour of bodies relative to the person in the railway carriage.
- The mechanical behaviour is different from that of the case previously
- considered, and for this reason it would appear to be impossible that
- the same mechanical laws hold relatively to the non-uniformly moving
- carriage, as hold with reference to the carriage when at rest or in
- uniform motion. At all events it is clear that the Galileian law does
- not hold with respect to the non-uniformly moving carriage. Because of
- this, we feel compelled at the present juncture to grant a kind of
- absolute physical reality to non-uniform motion, in opposition to the
- general principle of relatvity. But in what follows we shall soon see
- that this conclusion cannot be maintained.
- THE GRAVITATIONAL FIELD
- "If we pick up a stone and then let it go, why does it fall to the
- ground ?" The usual answer to this question is: "Because it is
- attracted by the earth." Modern physics formulates the answer rather
- differently for the following reason. As a result of the more careful
- study of electromagnetic phenomena, we have come to regard action at a
- distance as a process impossible without the intervention of some
- intermediary medium. If, for instance, a magnet attracts a piece of
- iron, we cannot be content to regard this as meaning that the magnet
- acts directly on the iron through the intermediate empty space, but we
- are constrained to imagine -- after the manner of Faraday -- that the
- magnet always calls into being something physically real in the space
- around it, that something being what we call a "magnetic field." In
- its turn this magnetic field operates on the piece of iron, so that
- the latter strives to move towards the magnet. We shall not discuss
- here the justification for this incidental conception, which is indeed
- a somewhat arbitrary one. We shall only mention that with its aid
- electromagnetic phenomena can be theoretically represented much more
- satisfactorily than without it, and this applies particularly to the
- transmission of electromagnetic waves. The effects of gravitation also
- are regarded in an analogous manner.
- The action of the earth on the stone takes place indirectly. The earth
- produces in its surrounding a gravitational field, which acts on the
- stone and produces its motion of fall. As we know from experience, the
- intensity of the action on a body dimishes according to a quite
- definite law, as we proceed farther and farther away from the earth.
- From our point of view this means : The law governing the properties
- of the gravitational field in space must be a perfectly definite one,
- in order correctly to represent the diminution of gravitational action
- with the distance from operative bodies. It is something like this:
- The body (e.g. the earth) produces a field in its immediate
- neighbourhood directly; the intensity and direction of the field at
- points farther removed from the body are thence determined by the law
- which governs the properties in space of the gravitational fields
- themselves.
- In contrast to electric and magnetic fields, the gravitational field
- exhibits a most remarkable property, which is of fundamental
- importance for what follows. Bodies which are moving under the sole
- influence of a gravitational field receive an acceleration, which does
- not in the least depend either on the material or on the physical
- state of the body. For instance, a piece of lead and a piece of wood
- fall in exactly the same manner in a gravitational field (in vacuo),
- when they start off from rest or with the same initial velocity. This
- law, which holds most accurately, can be expressed in a different form
- in the light of the following consideration.
- According to Newton's law of motion, we have
- (Force) = (inertial mass) x (acceleration),
- where the "inertial mass" is a characteristic constant of the
- accelerated body. If now gravitation is the cause of the acceleration,
- we then have
- (Force) = (gravitational mass) x (intensity of the gravitational
- field),
- where the "gravitational mass" is likewise a characteristic constant
- for the body. From these two relations follows:
- eq. 26: file eq26.gif
- If now, as we find from experience, the acceleration is to be
- independent of the nature and the condition of the body and always the
- same for a given gravitational field, then the ratio of the
- gravitational to the inertial mass must likewise be the same for all
- bodies. By a suitable choice of units we can thus make this ratio
- equal to unity. We then have the following law: The gravitational mass
- of a body is equal to its inertial law.
- It is true that this important law had hitherto been recorded in
- mechanics, but it had not been interpreted. A satisfactory
- interpretation can be obtained only if we recognise the following fact
- : The same quality of a body manifests itself according to
- circumstances as " inertia " or as " weight " (lit. " heaviness '). In
- the following section we shall show to what extent this is actually
- the case, and how this question is connected with the general
- postulate of relativity.
- THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS
- AS AN ARGUMENT FOR THE GENERAL POSTULE OF RELATIVITY
- We imagine a large portion of empty space, so far removed from stars
- and other appreciable masses, that we have before us approximately the
- conditions required by the fundamental law of Galilei. It is then
- possible to choose a Galileian reference-body for this part of space
- (world), relative to which points at rest remain at rest and points in
- motion continue permanently in uniform rectilinear motion. As
- reference-body let us imagine a spacious chest resembling a room with
- an observer inside who is equipped with apparatus. Gravitation
- naturally does not exist for this observer. He must fasten himself
- with strings to the floor, otherwise the slightest impact against the
- floor will cause him to rise slowly towards the ceiling of the room.
- To the middle of the lid of the chest is fixed externally a hook with
- rope attached, and now a " being " (what kind of a being is immaterial
- to us) begins pulling at this with a constant force. The chest
- together with the observer then begin to move "upwards" with a
- uniformly accelerated motion. In course of time their velocity will
- reach unheard-of values -- provided that we are viewing all this from
- another reference-body which is not being pulled with a rope.
- But how does the man in the chest regard the Process ? The
- acceleration of the chest will be transmitted to him by the reaction
- of the floor of the chest. He must therefore take up this pressure by
- means of his legs if he does not wish to be laid out full length on
- the floor. He is then standing in the chest in exactly the same way as
- anyone stands in a room of a home on our earth. If he releases a body
- which he previously had in his land, the accelertion of the chest will
- no longer be transmitted to this body, and for this reason the body
- will approach the floor of the chest with an accelerated relative
- motion. The observer will further convince himself that the
- acceleration of the body towards the floor of the chest is always of
- the same magnitude, whatever kind of body he may happen to use for the
- experiment.
- Relying on his knowledge of the gravitational field (as it was
- discussed in the preceding section), the man in the chest will thus
- come to the conclusion that he and the chest are in a gravitational
- field which is constant with regard to time. Of course he will be
- puzzled for a moment as to why the chest does not fall in this
- gravitational field. just then, however, he discovers the hook in the
- middle of the lid of the chest and the rope which is attached to it,
- and he consequently comes to the conclusion that the chest is
- suspended at rest in the gravitational field.
- Ought we to smile at the man and say that he errs in his conclusion ?
- I do not believe we ought to if we wish to remain consistent ; we must
- rather admit that his mode of grasping the situation violates neither
- reason nor known mechanical laws. Even though it is being accelerated
- with respect to the "Galileian space" first considered, we can
- nevertheless regard the chest as being at rest. We have thus good
- grounds for extending the principle of relativity to include bodies of
- reference which are accelerated with respect to each other, and as a
- result we have gained a powerful argument for a generalised postulate
- of relativity.
- We must note carefully that the possibility of this mode of
- interpretation rests on the fundamental property of the gravitational
- field of giving all bodies the same acceleration, or, what comes to
- the same thing, on the law of the equality of inertial and
- gravitational mass. If this natural law did not exist, the man in the
- accelerated chest would not be able to interpret the behaviour of the
- bodies around him on the supposition of a gravitational field, and he
- would not be justified on the grounds of experience in supposing his
- reference-body to be " at rest."
- Suppose that the man in the chest fixes a rope to the inner side of
- the lid, and that he attaches a body to the free end of the rope. The
- result of this will be to strech the rope so that it will hang "
- vertically " downwards. If we ask for an opinion of the cause of
- tension in the rope, the man in the chest will say: "The suspended
- body experiences a downward force in the gravitational field, and this
- is neutralised by the tension of the rope ; what determines the
- magnitude of the tension of the rope is the gravitational mass of the
- suspended body." On the other hand, an observer who is poised freely
- in space will interpret the condition of things thus : " The rope must
- perforce take part in the accelerated motion of the chest, and it
- transmits this motion to the body attached to it. The tension of the
- rope is just large enough to effect the acceleration of the body. That
- which determines the magnitude of the tension of the rope is the
- inertial mass of the body." Guided by this example, we see that our
- extension of the principle of relativity implies the necessity of the
- law of the equality of inertial and gravitational mass. Thus we have
- obtained a physical interpretation of this law.
- From our consideration of the accelerated chest we see that a general
- theory of relativity must yield important results on the laws of
- gravitation. In point of fact, the systematic pursuit of the general
- idea of relativity has supplied the laws satisfied by the
- gravitational field. Before proceeding farther, however, I must warn
- the reader against a misconception suggested by these considerations.
- A gravitational field exists for the man in the chest, despite the
- fact that there was no such field for the co-ordinate system first
- chosen. Now we might easily suppose that the existence of a
- gravitational field is always only an apparent one. We might also
- think that, regardless of the kind of gravitational field which may be
- present, we could always choose another reference-body such that no
- gravitational field exists with reference to it. This is by no means
- true for all gravitational fields, but only for those of quite special
- form. It is, for instance, impossible to choose a body of reference
- such that, as judged from it, the gravitational field of the earth (in
- its entirety) vanishes.
- We can now appreciate why that argument is not convincing, which we
- brought forward against the general principle of relativity at theend
- of Section 18. It is certainly true that the observer in the
- railway carriage experiences a jerk forwards as a result of the
- application of the brake, and that he recognises, in this the
- non-uniformity of motion (retardation) of the carriage. But he is
- compelled by nobody to refer this jerk to a " real " acceleration
- (retardation) of the carriage. He might also interpret his experience
- thus: " My body of reference (the carriage) remains permanently at
- rest. With reference to it, however, there exists (during the period
- of application of the brakes) a gravitational field which is directed
- forwards and which is variable with respect to time. Under the
- influence of this field, the embankment together with the earth moves
- non-uniformly in such a manner that their original velocity in the
- backwards direction is continuously reduced."
- IN WHAT RESPECTS ARE THE FOUNDATIONS OF CLASSICAL MECHANICS AND OF THE
- SPECIAL THEORY OF RELATIVITY UNSATISFACTORY?
- We have already stated several times that classical mechanics starts
- out from the following law: Material particles sufficiently far
- removed from other material particles continue to move uniformly in a
- straight line or continue in a state of rest. We have also repeatedly
- emphasised that this fundamental law can only be valid for bodies of
- reference K which possess certain unique states of motion, and which
- are in uniform translational motion relative to each other. Relative
- to other reference-bodies K the law is not valid. Both in classical
- mechanics and in the special theory of relativity we therefore
- differentiate between reference-bodies K relative to which the
- recognised " laws of nature " can be said to hold, and
- reference-bodies K relative to which these laws do not hold.
- But no person whose mode of thought is logical can rest satisfied with
- this condition of things. He asks : " How does it come that certain
- reference-bodies (or their states of motion) are given priority over
- other reference-bodies (or their states of motion) ? What is the
- reason for this Preference? In order to show clearly what I mean by
- this question, I shall make use of a comparison.
- I am standing in front of a gas range. Standing alongside of each
- other on the range are two pans so much alike that one may be mistaken
- for the other. Both are half full of water. I notice that steam is
- being emitted continuously from the one pan, but not from the other. I
- am surprised at this, even if I have never seen either a gas range or
- a pan before. But if I now notice a luminous something of bluish
- colour under the first pan but not under the other, I cease to be
- astonished, even if I have never before seen a gas flame. For I can
- only say that this bluish something will cause the emission of the
- steam, or at least possibly it may do so. If, however, I notice the
- bluish something in neither case, and if I observe that the one
- continuously emits steam whilst the other does not, then I shall
- remain astonished and dissatisfied until I have discovered some
- circumstance to which I can attribute the different behaviour of the
- two pans.
- Analogously, I seek in vain for a real something in classical
- mechanics (or in the special theory of relativity) to which I can
- attribute the different behaviour of bodies considered with respect to
- the reference systems K and K1.* Newton saw this objection and
- attempted to invalidate it, but without success. But E. Mach recognsed
- it most clearly of all, and because of this objection he claimed that
- mechanics must be placed on a new basis. It can only be got rid of by
- means of a physics which is conformable to the general principle of
- relativity, since the equations of such a theory hold for every body
- of reference, whatever may be its state of motion.
- Notes
- *) The objection is of importance more especially when the state of
- motion of the reference-body is of such a nature that it does not
- require any external agency for its maintenance, e.g. in the case when
- the reference-body is rotating uniformly.
- A FEW INFERENCES FROM THE GENERAL PRINCIPLE OF RELATIVITY
- The considerations of Section 20 show that the general principle of
- relativity puts us in a position to derive properties of the
- gravitational field in a purely theoretical manner. Let us suppose,
- for instance, that we know the space-time " course " for any natural
- process whatsoever, as regards the manner in which it takes place in
- the Galileian domain relative to a Galileian body of reference K. By
- means of purely theoretical operations (i.e. simply by calculation) we
- are then able to find how this known natural process appears, as seen
- from a reference-body K1 which is accelerated relatively to K. But
- since a gravitational field exists with respect to this new body of
- reference K1, our consideration also teaches us how the gravitational
- field influences the process studied.
- For example, we learn that a body which is in a state of uniform
- rectilinear motion with respect to K (in accordance with the law of
- Galilei) is executing an accelerated and in general curvilinear motion
- with respect to the accelerated reference-body K1 (chest). This
- acceleration or curvature corresponds to the influence on the moving
- body of the gravitational field prevailing relatively to K. It is
- known that a gravitational field influences the movement of bodies in
- this way, so that our consideration supplies us with nothing
- essentially new.
- However, we obtain a new result of fundamental importance when we
- carry out the analogous consideration for a ray of light. With respect
- to the Galileian reference-body K, such a ray of light is transmitted
- rectilinearly with the velocity c. It can easily be shown that the
- path of the same ray of light is no longer a straight line when we
- consider it with reference to the accelerated chest (reference-body
- K1). From this we conclude, that, in general, rays of light are
- propagated curvilinearly in gravitational fields. In two respects this
- result is of great importance.
- In the first place, it can be compared with the reality. Although a
- detailed examination of the question shows that the curvature of light
- rays required by the general theory of relativity is only exceedingly
- small for the gravitational fields at our disposal in practice, its
- estimated magnitude for light rays passing the sun at grazing
- incidence is nevertheless 1.7 seconds of arc. This ought to manifest
- itself in the following way. As seen from the earth, certain fixed
- stars appear to be in the neighbourhood of the sun, and are thus
- capable of observation during a total eclipse of the sun. At such
- times, these stars ought to appear to be displaced outwards from the
- sun by an amount indicated above, as compared with their apparent
- position in the sky when the sun is situated at another part of the
- heavens. The examination of the correctness or otherwise of this
- deduction is a problem of the greatest importance, the early solution
- of which is to be expected of astronomers.[2]*
- In the second place our result shows that, according to the general
- theory of relativity, the law of the constancy of the velocity of
- light in vacuo, which constitutes one of the two fundamental
- assumptions in the special theory of relativity and to which we have
- already frequently referred, cannot claim any unlimited validity. A
- curvature of rays of light can only take place when the velocity of
- propagation of light varies with position. Now we might think that as
- a consequence of this, the special theory of relativity and with it
- the whole theory of relativity would be laid in the dust. But in
- reality this is not the case. We can only conclude that the special
- theory of relativity cannot claim an unlinlited domain of validity ;
- its results hold only so long as we are able to disregard the
- influences of gravitational fields on the phenomena (e.g. of light).
- Since it has often been contended by opponents of the theory of
- relativity that the special theory of relativity is overthrown by the
- general theory of relativity, it is perhaps advisable to make the
- facts of the case clearer by means of an appropriate comparison.
- Before the development of electrodynamics the laws of electrostatics
- were looked upon as the laws of electricity. At the present time we
- know that electric fields can be derived correctly from electrostatic
- considerations only for the case, which is never strictly realised, in
- which the electrical masses are quite at rest relatively to each
- other, and to the co-ordinate system. Should we be justified in saying
- that for this reason electrostatics is overthrown by the
- field-equations of Maxwell in electrodynamics ? Not in the least.
- Electrostatics is contained in electrodynamics as a limiting case ;
- the laws of the latter lead directly to those of the former for the
- case in which the fields are invariable with regard to time. No fairer
- destiny could be allotted to any physical theory, than that it should
- of itself point out the way to the introduction of a more
- comprehensive theory, in which it lives on as a limiting case.
- In the example of the transmission of light just dealt with, we have
- seen that the general theory of relativity enables us to derive
- theoretically the influence of a gravitational field on the course of
- natural processes, the Iaws of which are already known when a
- gravitational field is absent. But the most attractive problem, to the
- solution of which the general theory of relativity supplies the key,
- concerns the investigation of the laws satisfied by the gravitational
- field itself. Let us consider this for a moment.
- We are acquainted with space-time domains which behave (approximately)
- in a " Galileian " fashion under suitable choice of reference-body,
- i.e. domains in which gravitational fields are absent. If we now refer
- such a domain to a reference-body K1 possessing any kind of motion,
- then relative to K1 there exists a gravitational field which is
- variable with respect to space and time.[3]** The character of this
- field will of course depend on the motion chosen for K1. According to
- the general theory of relativity, the general law of the gravitational
- field must be satisfied for all gravitational fields obtainable in
- this way. Even though by no means all gravitationial fields can be
- produced in this way, yet we may entertain the hope that the general
- law of gravitation will be derivable from such gravitational fields of
- a special kind. This hope has been realised in the most beautiful
- manner. But between the clear vision of this goal and its actual
- realisation it was necessary to surmount a serious difficulty, and as
- this lies deep at the root of things, I dare not withhold it from the
- reader. We require to extend our ideas of the space-time continuum
- still farther.
- Notes
- *) By means of the star photographs of two expeditions equipped by
- a Joint Committee of the Royal and Royal Astronomical Societies, the
- existence of the deflection of light demanded by theory was first
- confirmed during the solar eclipse of 29th May, 1919. (Cf. Appendix
- III.)
- **) This follows from a generalisation of the discussion in
- Section 20
- BEHAVIOUR OF CLOCKS AND MEASURING-RODS ON A ROTATING BODY OF REFERENCE
- Hitherto I have purposely refrained from speaking about the physical
- interpretation of space- and time-data in the case of the general
- theory of relativity. As a consequence, I am guilty of a certain
- slovenliness of treatment, which, as we know from the special theory
- of relativity, is far from being unimportant and pardonable. It is now
- high time that we remedy this defect; but I would mention at the
- outset, that this matter lays no small claims on the patience and on
- the power of abstraction of the reader.
- We start off again from quite special cases, which we have frequently
- used before. Let us consider a space time domain in which no
- gravitational field exists relative to a reference-body K whose state
- of motion has been suitably chosen. K is then a Galileian
- reference-body as regards the domain considered, and the results of
- the special theory of relativity hold relative to K. Let us supposse
- the same domain referred to a second body of reference K1, which is
- rotating uniformly with respect to K. In order to fix our ideas, we
- shall imagine K1 to be in the form of a plane circular disc, which
- rotates uniformly in its own plane about its centre. An observer who
- is sitting eccentrically on the disc K1 is sensible of a force which
- acts outwards in a radial direction, and which would be interpreted as
- an effect of inertia (centrifugal force) by an observer who was at
- rest with respect to the original reference-body K. But the observer
- on the disc may regard his disc as a reference-body which is " at rest
- " ; on the basis of the general principle of relativity he is
- justified in doing this. The force acting on himself, and in fact on
- all other bodies which are at rest relative to the disc, he regards as
- the effect of a gravitational field. Nevertheless, the
- space-distribution of this gravitational field is of a kind that would
- not be possible on Newton's theory of gravitation.* But since the
- observer believes in the general theory of relativity, this does not
- disturb him; he is quite in the right when he believes that a general
- law of gravitation can be formulated- a law which not only explains
- the motion of the stars correctly, but also the field of force
- experienced by himself.
- The observer performs experiments on his circular disc with clocks and
- measuring-rods. In doing so, it is his intention to arrive at exact
- definitions for the signification of time- and space-data with
- reference to the circular disc K1, these definitions being based on
- his observations. What will be his experience in this enterprise ?
- To start with, he places one of two identically constructed clocks at
- the centre of the circular disc, and the other on the edge of the
- disc, so that they are at rest relative to it. We now ask ourselves
- whether both clocks go at the same rate from the standpoint of the
- non-rotating Galileian reference-body K. As judged from this body, the
- clock at the centre of the disc has no velocity, whereas the clock at
- the edge of the disc is in motion relative to K in consequence of the
- rotation. According to a result obtained in Section 12, it follows
- that the latter clock goes at a rate permanently slower than that of
- the clock at the centre of the circular disc, i.e. as observed from K.
- It is obvious that the same effect would be noted by an observer whom
- we will imagine sitting alongside his clock at the centre of the
- circular disc. Thus on our circular disc, or, to make the case more
- general, in every gravitational field, a clock will go more quickly or
- less quickly, according to the position in which the clock is situated
- (at rest). For this reason it is not possible to obtain a reasonable
- definition of time with the aid of clocks which are arranged at rest
- with respect to the body of reference. A similar difficulty presents
- itself when we attempt to apply our earlier definition of simultaneity
- in such a case, but I do not wish to go any farther into this
- question.
- Moreover, at this stage the definition of the space co-ordinates also
- presents insurmountable difficulties. If the observer applies his
- standard measuring-rod (a rod which is short as compared with the
- radius of the disc) tangentially to the edge of the disc, then, as
- judged from the Galileian system, the length of this rod will be less
- than I, since, according to Section 12, moving bodies suffer a
- shortening in the direction of the motion. On the other hand, the
- measaring-rod will not experience a shortening in length, as judged
- from K, if it is applied to the disc in the direction of the radius.
- If, then, the observer first measures the circumference of the disc
- with his measuring-rod and then the diameter of the disc, on dividing
- the one by the other, he will not obtain as quotient the familiar
- number p = 3.14 . . ., but a larger number,[4]** whereas of course,
- for a disc which is at rest with respect to K, this operation would
- yield p exactly. This proves that the propositions of Euclidean
- geometry cannot hold exactly on the rotating disc, nor in general in a
- gravitational field, at least if we attribute the length I to the rod
- in all positions and in every orientation. Hence the idea of a
- straight line also loses its meaning. We are therefore not in a
- position to define exactly the co-ordinates x, y, z relative to the
- disc by means of the method used in discussing the special theory, and
- as long as the co- ordinates and times of events have not been
- defined, we cannot assign an exact meaning to the natural laws in
- which these occur.
- Thus all our previous conclusions based on general relativity would
- appear to be called in question. In reality we must make a subtle
- detour in order to be able to apply the postulate of general
- relativity exactly. I shall prepare the reader for this in the
- following paragraphs.
- Notes
- *) The field disappears at the centre of the disc and increases
- proportionally to the distance from the centre as we proceed outwards.
- **) Throughout this consideration we have to use the Galileian
- (non-rotating) system K as reference-body, since we may only assume
- the validity of the results of the special theory of relativity
- relative to K (relative to K1 a gravitational field prevails).
- EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM
- The surface of a marble table is spread out in front of me. I can get
- from any one point on this table to any other point by passing
- continuously from one point to a " neighbouring " one, and repeating
- this process a (large) number of times, or, in other words, by going
- from point to point without executing "jumps." I am sure the reader
- will appreciate with sufficient clearness what I mean here by "
- neighbouring " and by " jumps " (if he is not too pedantic). We
- express this property of the surface by describing the latter as a
- continuum.
- Let us now imagine that a large number of little rods of equal length
- have been made, their lengths being small compared with the dimensions
- of the marble slab. When I say they are of equal length, I mean that
- one can be laid on any other without the ends overlapping. We next lay
- four of these little rods on the marble slab so that they constitute a
- quadrilateral figure (a square), the diagonals of which are equally
- long. To ensure the equality of the diagonals, we make use of a little
- testing-rod. To this square we add similar ones, each of which has one
- rod in common with the first. We proceed in like manner with each of
- these squares until finally the whole marble slab is laid out with
- squares. The arrangement is such, that each side of a square belongs
- to two squares and each corner to four squares.
- It is a veritable wonder that we can carry out this business without
- getting into the greatest difficulties. We only need to think of the
- following. If at any moment three squares meet at a corner, then two
- sides of the fourth square are already laid, and, as a consequence,
- the arrangement of the remaining two sides of the square is already
- completely determined. But I am now no longer able to adjust the
- quadrilateral so that its diagonals may be equal. If they are equal of
- their own accord, then this is an especial favour of the marble slab
- and of the little rods, about which I can only be thankfully
- surprised. We must experience many such surprises if the construction
- is to be successful.
- If everything has really gone smoothly, then I say that the points of
- the marble slab constitute a Euclidean continuum with respect to the
- little rod, which has been used as a " distance " (line-interval). By
- choosing one corner of a square as " origin" I can characterise every
- other corner of a square with reference to this origin by means of two
- numbers. I only need state how many rods I must pass over when,
- starting from the origin, I proceed towards the " right " and then "
- upwards," in order to arrive at the corner of the square under
- consideration. These two numbers are then the " Cartesian co-ordinates
- " of this corner with reference to the " Cartesian co-ordinate system"
- which is determined by the arrangement of little rods.
- By making use of the following modification of this abstract
- experiment, we recognise that there must also be cases in which the
- experiment would be unsuccessful. We shall suppose that the rods "
- expand " by in amount proportional to the increase of temperature. We
- heat the central part of the marble slab, but not the periphery, in
- which case two of our little rods can still be brought into
- coincidence at every position on the table. But our construction of
- squares must necessarily come into disorder during the heating,
- because the little rods on the central region of the table expand,
- whereas those on the outer part do not.
- With reference to our little rods -- defined as unit lengths -- the
- marble slab is no longer a Euclidean continuum, and we are also no
- longer in the position of defining Cartesian co-ordinates directly
- with their aid, since the above construction can no longer be carried
- out. But since there are other things which are not influenced in a
- similar manner to the little rods (or perhaps not at all) by the
- temperature of the table, it is possible quite naturally to maintain
- the point of view that the marble slab is a " Euclidean continuum."
- This can be done in a satisfactory manner by making a more subtle
- stipulation about the measurement or the comparison of lengths.
- But if rods of every kind (i.e. of every material) were to behave in
- the same way as regards the influence of temperature when they are on
- the variably heated marble slab, and if we had no other means of
- detecting the effect of temperature than the geometrical behaviour of
- our rods in experiments analogous to the one described above, then our
- best plan would be to assign the distance one to two points on the
- slab, provided that the ends of one of our rods could be made to
- coincide with these two points ; for how else should we define the
- distance without our proceeding being in the highest measure grossly
- arbitrary ? The method of Cartesian coordinates must then be
- discarded, and replaced by another which does not assume the validity
- of Euclidean geometry for rigid bodies.* The reader will notice
- that the situation depicted here corresponds to the one brought about
- by the general postitlate of relativity (Section 23).
- Notes
- *) Mathematicians have been confronted with our problem in the
- following form. If we are given a surface (e.g. an ellipsoid) in
- Euclidean three-dimensional space, then there exists for this surface
- a two-dimensional geometry, just as much as for a plane surface. Gauss
- undertook the task of treating this two-dimensional geometry from
- first principles, without making use of the fact that the surface
- belongs to a Euclidean continuum of three dimensions. If we imagine
- constructions to be made with rigid rods in the surface (similar to
- that above with the marble slab), we should find that different laws
- hold for these from those resulting on the basis of Euclidean plane
- geometry. The surface is not a Euclidean continuum with respect to the
- rods, and we cannot define Cartesian co-ordinates in the surface.
- Gauss indicated the principles according to which we can treat the
- geometrical relationships in the surface, and thus pointed out the way
- to the method of Riemman of treating multi-dimensional, non-Euclidean
- continuum. Thus it is that mathematicians long ago solved the formal
- problems to which we are led by the general postulate of relativity.
- GAUSSIAN CO-ORDINATES
- According to Gauss, this combined analytical and geometrical mode of
- handling the problem can be arrived at in the following way. We
- imagine a system of arbitrary curves (see Fig. 4) drawn on the surface
- of the table. These we designate as u-curves, and we indicate each of
- them by means of a number. The Curves u= 1, u= 2 and u= 3 are drawn in
- the diagram. Between the curves u= 1 and u= 2 we must imagine an
- infinitely large number to be drawn, all of which correspond to real
- numbers lying between 1 and 2. fig. 04 We have then a system of
- u-curves, and this "infinitely dense" system covers the whole surface
- of the table. These u-curves must not intersect each other, and
- through each point of the surface one and only one curve must pass.
- Thus a perfectly definite value of u belongs to every point on the
- surface of the marble slab. In like manner we imagine a system of
- v-curves drawn on the surface. These satisfy the same conditions as
- the u-curves, they are provided with numbers in a corresponding
- manner, and they may likewise be of arbitrary shape. It follows that a
- value of u and a value of v belong to every point on the surface of
- the table. We call these two numbers the co-ordinates of the surface
- of the table (Gaussian co-ordinates). For example, the point P in the
- diagram has the Gaussian co-ordinates u= 3, v= 1. Two neighbouring
- points P and P1 on the surface then correspond to the co-ordinates
- P: u,v
- P1: u + du, v + dv,
- where du and dv signify very small numbers. In a similar manner we may
- indicate the distance (line-interval) between P and P1, as measured
- with a little rod, by means of the very small number ds. Then
- according to Gauss we have
- ds2 = g[11]du2 + 2g[12]dudv = g[22]dv2
- where g[11], g[12], g[22], are magnitudes which depend in a perfectly
- definite way on u and v. The magnitudes g[11], g[12] and g[22],
- determine the behaviour of the rods relative to the u-curves and
- v-curves, and thus also relative to the surface of the table. For the
- case in which the points of the surface considered form a Euclidean
- continuum with reference to the measuring-rods, but only in this case,
- it is possible to draw the u-curves and v-curves and to attach numbers
- to them, in such a manner, that we simply have :
- ds2 = du2 + dv2
- Under these conditions, the u-curves and v-curves are straight lines
- in the sense of Euclidean geometry, and they are perpendicular to each
- other. Here the Gaussian coordinates are samply Cartesian ones. It is
- clear that Gauss co-ordinates are nothing more than an association of
- two sets of numbers with the points of the surface considered, of such
- a nature that numerical values differing very slightly from each other
- are associated with neighbouring points " in space."
- So far, these considerations hold for a continuum of two dimensions.
- But the Gaussian method can be applied also to a continuum of three,
- four or more dimensions. If, for instance, a continuum of four
- dimensions be supposed available, we may represent it in the following
- way. With every point of the continuum, we associate arbitrarily four
- numbers, x[1], x[2], x[3], x[4], which are known as " co-ordinates."
- Adjacent points correspond to adjacent values of the coordinates. If a
- distance ds is associated with the adjacent points P and P1, this
- distance being measurable and well defined from a physical point of
- view, then the following formula holds:
- ds2 = g[11]dx[1]^2 + 2g[12]dx[1]dx[2] . . . . g[44]dx[4]^2,
- where the magnitudes g[11], etc., have values which vary with the
- position in the continuum. Only when the continuum is a Euclidean one
- is it possible to associate the co-ordinates x[1] . . x[4]. with the
- points of the continuum so that we have simply
- ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.
- In this case relations hold in the four-dimensional continuum which
- are analogous to those holding in our three-dimensional measurements.
- However, the Gauss treatment for ds2 which we have given above is not
- always possible. It is only possible when sufficiently small regions
- of the continuum under consideration may be regarded as Euclidean
- continua. For example, this obviously holds in the case of the marble
- slab of the table and local variation of temperature. The temperature
- is practically constant for a small part of the slab, and thus the
- geometrical behaviour of the rods is almost as it ought to be
- according to the rules of Euclidean geometry. Hence the imperfections
- of the construction of squares in the previous section do not show
- themselves clearly until this construction is extended over a
- considerable portion of the surface of the table.
- We can sum this up as follows: Gauss invented a method for the
- mathematical treatment of continua in general, in which "
- size-relations " (" distances " between neighbouring points) are
- defined. To every point of a continuum are assigned as many numbers
- (Gaussian coordinates) as the continuum has dimensions. This is done
- in such a way, that only one meaning can be attached to the
- assignment, and that numbers (Gaussian coordinates) which differ by an
- indefinitely small amount are assigned to adjacent points. The
- Gaussian coordinate system is a logical generalisation of the
- Cartesian co-ordinate system. It is also applicable to non-Euclidean
- continua, but only when, with respect to the defined "size" or
- "distance," small parts of the continuum under consideration behave
- more nearly like a Euclidean system, the smaller the part of the
- continuum under our notice.
- THE SPACE-TIME CONTINUUM OF THE SPEICAL THEORY OF RELATIVITY CONSIDERED AS A
- EUCLIDEAN CONTINUUM
- We are now in a position to formulate more exactly the idea of
- Minkowski, which was only vaguely indicated in Section 17. In
- accordance with the special theory of relativity, certain co-ordinate
- systems are given preference for the description of the
- four-dimensional, space-time continuum. We called these " Galileian
- co-ordinate systems." For these systems, the four co-ordinates x, y,
- z, t, which determine an event or -- in other words, a point of the
- four-dimensional continuum -- are defined physically in a simple
- manner, as set forth in detail in the first part of this book. For the
- transition from one Galileian system to another, which is moving
- uniformly with reference to the first, the equations of the Lorentz
- transformation are valid. These last form the basis for the derivation
- of deductions from the special theory of relativity, and in themselves
- they are nothing more than the expression of the universal validity of
- the law of transmission of light for all Galileian systems of
- reference.
- Minkowski found that the Lorentz transformations satisfy the following
- simple conditions. Let us consider two neighbouring events, the
- relative position of which in the four-dimensional continuum is given
- with respect to a Galileian reference-body K by the space co-ordinate
- differences dx, dy, dz and the time-difference dt. With reference to a
- second Galileian system we shall suppose that the corresponding
- differences for these two events are dx1, dy1, dz1, dt1. Then these
- magnitudes always fulfil the condition*
- dx2 + dy2 + dz2 - c^2dt2 = dx1 2 + dy1 2 + dz1 2 - c^2dt1 2.
- The validity of the Lorentz transformation follows from this
- condition. We can express this as follows: The magnitude
- ds2 = dx2 + dy2 + dz2 - c^2dt2,
- which belongs to two adjacent points of the four-dimensional
- space-time continuum, has the same value for all selected (Galileian)
- reference-bodies. If we replace x, y, z, sq. rt. -I . ct , by x[1],
- x[2], x[3], x[4], we also obtaill the result that
- ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.
- is independent of the choice of the body of reference. We call the
- magnitude ds the " distance " apart of the two events or
- four-dimensional points.
- Thus, if we choose as time-variable the imaginary variable sq. rt. -I
- . ct instead of the real quantity t, we can regard the space-time
- contintium -- accordance with the special theory of relativity -- as a
- ", Euclidean " four-dimensional continuum, a result which follows from
- the considerations of the preceding section.
- Notes
- *) Cf. Appendixes I and 2. The relations which are derived
- there for the co-ordlnates themselves are valid also for co-ordinate
- differences, and thus also for co-ordinate differentials (indefinitely
- small differences).
- THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF REALTIIVTY IS NOT A
- ECULIDEAN CONTINUUM
- In the first part of this book we were able to make use of space-time
- co-ordinates which allowed of a simple and direct physical
- interpretation, and which, according to Section 26, can be regarded
- as four-dimensional Cartesian co-ordinates. This was possible on the
- basis of the law of the constancy of the velocity of tight. But
- according to Section 21 the general theory of relativity cannot
- retain this law. On the contrary, we arrived at the result that
- according to this latter theory the velocity of light must always
- depend on the co-ordinates when a gravitational field is present. In
- connection with a specific illustration in Section 23, we found
- that the presence of a gravitational field invalidates the definition
- of the coordinates and the ifine, which led us to our objective in the
- special theory of relativity.
- In view of the resuIts of these considerations we are led to the
- conviction that, according to the general principle of relativity, the
- space-time continuum cannot be regarded as a Euclidean one, but that
- here we have the general case, corresponding to the marble slab with
- local variations of temperature, and with which we made acquaintance
- as an example of a two-dimensional continuum. Just as it was there
- impossible to construct a Cartesian co-ordinate system from equal
- rods, so here it is impossible to build up a system (reference-body)
- from rigid bodies and clocks, which shall be of such a nature that
- measuring-rods and clocks, arranged rigidly with respect to one
- another, shaIll indicate position and time directly. Such was the
- essence of the difficulty with which we were confronted in Section
- 23.
- But the considerations of Sections 25 and 26 show us the way to
- surmount this difficulty. We refer the fourdimensional space-time
- continuum in an arbitrary manner to Gauss co-ordinates. We assign to
- every point of the continuum (event) four numbers, x[1], x[2], x[3],
- x[4] (co-ordinates), which have not the least direct physical
- significance, but only serve the purpose of numbering the points of
- the continuum in a definite but arbitrary manner. This arrangement
- does not even need to be of such a kind that we must regard x[1],
- x[2], x[3], as "space" co-ordinates and x[4], as a " time "
- co-ordinate.
- The reader may think that such a description of the world would be
- quite inadequate. What does it mean to assign to an event the
- particular co-ordinates x[1], x[2], x[3], x[4], if in themselves these
- co-ordinates have no significance ? More careful consideration shows,
- however, that this anxiety is unfounded. Let us consider, for
- instance, a material point with any kind of motion. If this point had
- only a momentary existence without duration, then it would to
- described in space-time by a single system of values x[1], x[2], x[3],
- x[4]. Thus its permanent existence must be characterised by an
- infinitely large number of such systems of values, the co-ordinate
- values of which are so close together as to give continuity;
- corresponding to the material point, we thus have a (uni-dimensional)
- line in the four-dimensional continuum. In the same way, any such
- lines in our continuum correspond to many points in motion. The only
- statements having regard to these points which can claim a physical
- existence are in reality the statements about their encounters. In our
- mathematical treatment, such an encounter is expressed in the fact
- that the two lines which represent the motions of the points in
- question have a particular system of co-ordinate values, x[1], x[2],
- x[3], x[4], in common. After mature consideration the reader will
- doubtless admit that in reality such encounters constitute the only
- actual evidence of a time-space nature with which we meet in physical
- statements.
- When we were describing the motion of a material point relative to a
- body of reference, we stated nothing more than the encounters of this
- point with particular points of the reference-body. We can also
- determine the corresponding values of the time by the observation of
- encounters of the body with clocks, in conjunction with the
- observation of the encounter of the hands of clocks with particular
- points on the dials. It is just the same in the case of
- space-measurements by means of measuring-rods, as a litttle
- consideration will show.
- The following statements hold generally : Every physical description
- resolves itself into a number of statements, each of which refers to
- the space-time coincidence of two events A and B. In terms of Gaussian
- co-ordinates, every such statement is expressed by the agreement of
- their four co-ordinates x[1], x[2], x[3], x[4]. Thus in reality, the
- description of the time-space continuum by means of Gauss co-ordinates
- completely replaces the description with the aid of a body of
- reference, without suffering from the defects of the latter mode of
- description; it is not tied down to the Euclidean character of the
- continuum which has to be represented.
- EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY
- We are now in a position to replace the pro. visional formulation of
- the general principle of relativity given in Section 18 by an exact
- formulation. The form there used, "All bodies of reference K, K1,
- etc., are equivalent for the description of natural phenomena
- (formulation of the general laws of nature), whatever may be their
- state of motion," cannot be maintained, because the use of rigid
- reference-bodies, in the sense of the method followed in the special
- theory of relativity, is in general not possible in space-time
- description. The Gauss co-ordinate system has to take the place of the
- body of reference. The following statement corresponds to the
- fundamental idea of the general principle of relativity: "All Gaussian
- co-ordinate systems are essentially equivalent for the formulation of
- the general laws of nature."
- We can state this general principle of relativity in still another
- form, which renders it yet more clearly intelligible than it is when
- in the form of the natural extension of the special principle of
- relativity. According to the special theory of relativity, the
- equations which express the general laws of nature pass over into
- equations of the same form when, by making use of the Lorentz
- transformation, we replace the space-time variables x, y, z, t, of a
- (Galileian) reference-body K by the space-time variables x1, y1, z1,
- t1, of a new reference-body K1. According to the general theory of
- relativity, on the other hand, by application of arbitrary
- substitutions of the Gauss variables x[1], x[2], x[3], x[4], the
- equations must pass over into equations of the same form; for every
- transformation (not only the Lorentz transformation) corresponds to
- the transition of one Gauss co-ordinate system into another.
- If we desire to adhere to our "old-time" three-dimensional view of
- things, then we can characterise the development which is being
- undergone by the fundamental idea of the general theory of relativity
- as follows : The special theory of relativity has reference to
- Galileian domains, i.e. to those in which no gravitational field
- exists. In this connection a Galileian reference-body serves as body
- of reference, i.e. a rigid body the state of motion of which is so
- chosen that the Galileian law of the uniform rectilinear motion of
- "isolated" material points holds relatively to it.
- Certain considerations suggest that we should refer the same Galileian
- domains to non-Galileian reference-bodies also. A gravitational field
- of a special kind is then present with respect to these bodies (cf.
- Sections 20 and 23).
- In gravitational fields there are no such things as rigid bodies with
- Euclidean properties; thus the fictitious rigid body of reference is
- of no avail in the general theory of relativity. The motion of clocks
- is also influenced by gravitational fields, and in such a way that a
- physical definition of time which is made directly with the aid of
- clocks has by no means the same degree of plausibility as in the
- special theory of relativity.
- For this reason non-rigid reference-bodies are used, which are as a
- whole not only moving in any way whatsoever, but which also suffer
- alterations in form ad lib. during their motion. Clocks, for which the
- law of motion is of any kind, however irregular, serve for the
- definition of time. We have to imagine each of these clocks fixed at a
- point on the non-rigid reference-body. These clocks satisfy only the
- one condition, that the "readings" which are observed simultaneously
- on adjacent clocks (in space) differ from each other by an
- indefinitely small amount. This non-rigid reference-body, which might
- appropriately be termed a "reference-mollusc", is in the main
- equivalent to a Gaussian four-dimensional co-ordinate system chosen
- arbitrarily. That which gives the "mollusc" a certain
- comprehensibility as compared with the Gauss co-ordinate system is the
- (really unjustified) formal retention of the separate existence of the
- space co-ordinates as opposed to the time co-ordinate. Every point on
- the mollusc is treated as a space-point, and every material point
- which is at rest relatively to it as at rest, so long as the mollusc
- is considered as reference-body. The general principle of relativity
- requires that all these molluscs can be used as reference-bodies with
- equal right and equal success in the formulation of the general laws
- of nature; the laws themselves must be quite independent of the choice
- of mollusc.
- The great power possessed by the general principle of relativity lies
- in the comprehensive limitation which is imposed on the laws of nature
- in consequence of what we have seen above.
- THE SOLUTION OF THE PROBLEM OF GRAVITATION ON THE BASIS OF THE GENERAL
- PRINCIPLE OF RELATIVITY
- If the reader has followed all our previous considerations, he will
- have no further difficulty in understanding the methods leading to the
- solution of the problem of gravitation.
- We start off on a consideration of a Galileian domain, i.e. a domain
- in which there is no gravitational field relative to the Galileian
- reference-body K. The behaviour of measuring-rods and clocks with
- reference to K is known from the special theory of relativity,
- likewise the behaviour of "isolated" material points; the latter move
- uniformly and in straight lines.
- Now let us refer this domain to a random Gauss coordinate system or to
- a "mollusc" as reference-body K1. Then with respect to K1 there is a
- gravitational field G (of a particular kind). We learn the behaviour
- of measuring-rods and clocks and also of freely-moving material points
- with reference to K1 simply by mathematical transformation. We
- interpret this behaviour as the behaviour of measuring-rods, docks and
- material points tinder the influence of the gravitational field G.
- Hereupon we introduce a hypothesis: that the influence of the
- gravitational field on measuringrods, clocks and freely-moving
- material points continues to take place according to the same laws,
- even in the case where the prevailing gravitational field is not
- derivable from the Galfleian special care, simply by means of a
- transformation of co-ordinates.
- The next step is to investigate the space-time behaviour of the
- gravitational field G, which was derived from the Galileian special
- case simply by transformation of the coordinates. This behaviour is
- formulated in a law, which is always valid, no matter how the
- reference-body (mollusc) used in the description may be chosen.
- This law is not yet the general law of the gravitational field, since
- the gravitational field under consideration is of a special kind. In
- order to find out the general law-of-field of gravitation we still
- require to obtain a generalisation of the law as found above. This can
- be obtained without caprice, however, by taking into consideration the
- following demands:
- (a) The required generalisation must likewise satisfy the general
- postulate of relativity.
- (b) If there is any matter in the domain under consideration, only its
- inertial mass, and thus according to Section 15 only its energy is
- of importance for its etfect in exciting a field.
- (c) Gravitational field and matter together must satisfy the law of
- the conservation of energy (and of impulse).
- Finally, the general principle of relativity permits us to determine
- the influence of the gravitational field on the course of all those
- processes which take place according to known laws when a
- gravitational field is absent i.e. which have already been fitted into
- the frame of the special theory of relativity. In this connection we
- proceed in principle according to the method which has already been
- explained for measuring-rods, clocks and freely moving material
- points.
- The theory of gravitation derived in this way from the general
- postulate of relativity excels not only in its beauty ; nor in
- removing the defect attaching to classical mechanics which was brought
- to light in Section 21; nor in interpreting the empirical law of
- the equality of inertial and gravitational mass ; but it has also
- already explained a result of observation in astronomy, against which
- classical mechanics is powerless.
- If we confine the application of the theory to the case where the
- gravitational fields can be regarded as being weak, and in which all
- masses move with respect to the coordinate system with velocities
- which are small compared with the velocity of light, we then obtain as
- a first approximation the Newtonian theory. Thus the latter theory is
- obtained here without any particular assumption, whereas Newton had to
- introduce the hypothesis that the force of attraction between mutually
- attracting material points is inversely proportional to the square of
- the distance between them. If we increase the accuracy of the
- calculation, deviations from the theory of Newton make their
- appearance, practically all of which must nevertheless escape the test
- of observation owing to their smallness.
- We must draw attention here to one of these deviations. According to
- Newton's theory, a planet moves round the sun in an ellipse, which
- would permanently maintain its position with respect to the fixed
- stars, if we could disregard the motion of the fixed stars themselves
- and the action of the other planets under consideration. Thus, if we
- correct the observed motion of the planets for these two influences,
- and if Newton's theory be strictly correct, we ought to obtain for the
- orbit of the planet an ellipse, which is fixed with reference to the
- fixed stars. This deduction, which can be tested with great accuracy,
- has been confirmed for all the planets save one, with the precision
- that is capable of being obtained by the delicacy of observation
- attainable at the present time. The sole exception is Mercury, the
- planet which lies nearest the sun. Since the time of Leverrier, it has
- been known that the ellipse corresponding to the orbit of Mercury,
- after it has been corrected for the influences mentioned above, is not
- stationary with respect to the fixed stars, but that it rotates
- exceedingly slowly in the plane of the orbit and in the sense of the
- orbital motion. The value obtained for this rotary movement of the
- orbital ellipse was 43 seconds of arc per century, an amount ensured
- to be correct to within a few seconds of arc. This effect can be
- explained by means of classical mechanics only on the assumption of
- hypotheses which have little probability, and which were devised
- solely for this purponse.
- On the basis of the general theory of relativity, it is found that the
- ellipse of every planet round the sun must necessarily rotate in the
- manner indicated above ; that for all the planets, with the exception
- of Mercury, this rotation is too small to be detected with the
- delicacy of observation possible at the present time ; but that in the
- case of Mercury it must amount to 43 seconds of arc per century, a
- result which is strictly in agreement with observation.
- Apart from this one, it has hitherto been possible to make only two
- deductions from the theory which admit of being tested by observation,
- to wit, the curvature of light rays by the gravitational field of the
- sun,*x and a displacement of the spectral lines of light reaching
- us from large stars, as compared with the corresponding lines for
- light produced in an analogous manner terrestrially (i.e. by the same
- kind of atom).** These two deductions from the theory have both
- been confirmed.
- Notes
- *) First observed by Eddington and others in 1919. (Cf. Appendix
- III, pp. 126-129).
- **) Established by Adams in 1924. (Cf. p. 132)
- PART III
- CONSIDERATIONS ON THE UNIVERSE AS A WHOLE
- COSMOLOGICAL DIFFICULTIES OF NEWTON'S THEORY
- Part from the difficulty discussed in Section 21, there is a second
- fundamental difficulty attending classical celestial mechanics, which,
- to the best of my knowledge, was first discussed in detail by the
- astronomer Seeliger. If we ponder over the question as to how the
- universe, considered as a whole, is to be regarded, the first answer
- that suggests itself to us is surely this: As regards space (and time)
- the universe is infinite. There are stars everywhere, so that the
- density of matter, although very variable in detail, is nevertheless
- on the average everywhere the same. In other words: However far we
- might travel through space, we should find everywhere an attenuated
- swarm of fixed stars of approrimately the same kind and density.
- This view is not in harmony with the theory of Newton. The latter
- theory rather requires that the universe should have a kind of centre
- in which the density of the stars is a maximum, and that as we proceed
- outwards from this centre the group-density of the stars should
- diminish, until finally, at great distances, it is succeeded by an
- infinite region of emptiness. The stellar universe ought to be a
- finite island in the infinite ocean of space.*
- This conception is in itself not very satisfactory. It is still less
- satisfactory because it leads to the result that the light emitted by
- the stars and also individual stars of the stellar system are
- perpetually passing out into infinite space, never to return, and
- without ever again coming into interaction with other objects of
- nature. Such a finite material universe would be destined to become
- gradually but systematically impoverished.
- In order to escape this dilemma, Seeliger suggested a modification of
- Newton's law, in which he assumes that for great distances the force
- of attraction between two masses diminishes more rapidly than would
- result from the inverse square law. In this way it is possible for the
- mean density of matter to be constant everywhere, even to infinity,
- without infinitely large gravitational fields being produced. We thus
- free ourselves from the distasteful conception that the material
- universe ought to possess something of the nature of a centre. Of
- course we purchase our emancipation from the fundamental difficulties
- mentioned, at the cost of a modification and complication of Newton's
- law which has neither empirical nor theoretical foundation. We can
- imagine innumerable laws which would serve the same purpose, without
- our being able to state a reason why one of them is to be preferred to
- the others ; for any one of these laws would be founded just as little
- on more general theoretical principles as is the law of Newton.
- Notes
- *) Proof -- According to the theory of Newton, the number of "lines
- of force" which come from infinity and terminate in a mass m is
- proportional to the mass m. If, on the average, the Mass density p[0]
- is constant throughout tithe universe, then a sphere of volume V will
- enclose the average man p[0]V. Thus the number of lines of force
- passing through the surface F of the sphere into its interior is
- proportional to p[0] V. For unit area of the surface of the sphere the
- number of lines of force which enters the sphere is thus proportional
- to p[0] V/F or to p[0]R. Hence the intensity of the field at the
- surface would ultimately become infinite with increasing radius R of
- the sphere, which is impossible.
- THE POSSIBILITY OF A "FINITE" AND YET "UNBOUNDED" UNIVERSE
- But speculations on the structure of the universe also move in quite
- another direction. The development of non-Euclidean geometry led to
- the recognition of the fact, that we can cast doubt on the
- infiniteness of our space without coming into conflict with the laws
- of thought or with experience (Riemann, Helmholtz). These questions
- have already been treated in detail and with unsurpassable lucidity by
- Helmholtz and Poincaré, whereas I can only touch on them briefly here.
- In the first place, we imagine an existence in two dimensional space.
- Flat beings with flat implements, and in particular flat rigid
- measuring-rods, are free to move in a plane. For them nothing exists
- outside of this plane: that which they observe to happen to themselves
- and to their flat " things " is the all-inclusive reality of their
- plane. In particular, the constructions of plane Euclidean geometry
- can be carried out by means of the rods e.g. the lattice construction,
- considered in Section 24. In contrast to ours, the universe of
- these beings is two-dimensional; but, like ours, it extends to
- infinity. In their universe there is room for an infinite number of
- identical squares made up of rods, i.e. its volume (surface) is
- infinite. If these beings say their universe is " plane," there is
- sense in the statement, because they mean that they can perform the
- constructions of plane Euclidean geometry with their rods. In this
- connection the individual rods always represent the same distance,
- independently of their position.
- Let us consider now a second two-dimensional existence, but this time
- on a spherical surface instead of on a plane. The flat beings with
- their measuring-rods and other objects fit exactly on this surface and
- they are unable to leave it. Their whole universe of observation
- extends exclusively over the surface of the sphere. Are these beings
- able to regard the geometry of their universe as being plane geometry
- and their rods withal as the realisation of " distance " ? They cannot
- do this. For if they attempt to realise a straight line, they will
- obtain a curve, which we " three-dimensional beings " designate as a
- great circle, i.e. a self-contained line of definite finite length,
- which can be measured up by means of a measuring-rod. Similarly, this
- universe has a finite area that can be compared with the area, of a
- square constructed with rods. The great charm resulting from this
- consideration lies in the recognition of the fact that the universe of
- these beings is finite and yet has no limits.
- But the spherical-surface beings do not need to go on a world-tour in
- order to perceive that they are not living in a Euclidean universe.
- They can convince themselves of this on every part of their " world,"
- provided they do not use too small a piece of it. Starting from a
- point, they draw " straight lines " (arcs of circles as judged in
- three dimensional space) of equal length in all directions. They will
- call the line joining the free ends of these lines a " circle." For a
- plane surface, the ratio of the circumference of a circle to its
- diameter, both lengths being measured with the same rod, is, according
- to Euclidean geometry of the plane, equal to a constant value p, which
- is independent of the diameter of the circle. On their spherical
- surface our flat beings would find for this ratio the value
- eq. 27: file eq27.gif
- i.e. a smaller value than p, the difference being the more
- considerable, the greater is the radius of the circle in comparison
- with the radius R of the " world-sphere." By means of this relation
- the spherical beings can determine the radius of their universe ("
- world "), even when only a relatively small part of their worldsphere
- is available for their measurements. But if this part is very small
- indeed, they will no longer be able to demonstrate that they are on a
- spherical " world " and not on a Euclidean plane, for a small part of
- a spherical surface differs only slightly from a piece of a plane of
- the same size.
- Thus if the spherical surface beings are living on a planet of which
- the solar system occupies only a negligibly small part of the
- spherical universe, they have no means of determining whether they are
- living in a finite or in an infinite universe, because the " piece of
- universe " to which they have access is in both cases practically
- plane, or Euclidean. It follows directly from this discussion, that
- for our sphere-beings the circumference of a circle first increases
- with the radius until the " circumference of the universe " is
- reached, and that it thenceforward gradually decreases to zero for
- still further increasing values of the radius. During this process the
- area of the circle continues to increase more and more, until finally
- it becomes equal to the total area of the whole " world-sphere."
- Perhaps the reader will wonder why we have placed our " beings " on a
- sphere rather than on another closed surface. But this choice has its
- justification in the fact that, of all closed surfaces, the sphere is
- unique in possessing the property that all points on it are
- equivalent. I admit that the ratio of the circumference c of a circle
- to its radius r depends on r, but for a given value of r it is the
- same for all points of the " worldsphere "; in other words, the "
- world-sphere " is a " surface of constant curvature."
- To this two-dimensional sphere-universe there is a three-dimensional
- analogy, namely, the three-dimensional spherical space which was
- discovered by Riemann. its points are likewise all equivalent. It
- possesses a finite volume, which is determined by its "radius"
- (2p2R3). Is it possible to imagine a spherical space? To imagine a
- space means nothing else than that we imagine an epitome of our "
- space " experience, i.e. of experience that we can have in the
- movement of " rigid " bodies. In this sense we can imagine a spherical
- space.
- Suppose we draw lines or stretch strings in all directions from a
- point, and mark off from each of these the distance r with a
- measuring-rod. All the free end-points of these lengths lie on a
- spherical surface. We can specially measure up the area (F) of this
- surface by means of a square made up of measuring-rods. If the
- universe is Euclidean, then F = 4pR2 ; if it is spherical, then F is
- always less than 4pR2. With increasing values of r, F increases from
- zero up to a maximum value which is determined by the " world-radius,"
- but for still further increasing values of r, the area gradually
- diminishes to zero. At first, the straight lines which radiate from
- the starting point diverge farther and farther from one another, but
- later they approach each other, and finally they run together again at
- a "counter-point" to the starting point. Under such conditions they
- have traversed the whole spherical space. It is easily seen that the
- three-dimensional spherical space is quite analogous to the
- two-dimensional spherical surface. It is finite (i.e. of finite
- volume), and has no bounds.
- It may be mentioned that there is yet another kind of curved space: "
- elliptical space." It can be regarded as a curved space in which the
- two " counter-points " are identical (indistinguishable from each
- other). An elliptical universe can thus be considered to some extent
- as a curved universe possessing central symmetry.
- It follows from what has been said, that closed spaces without limits
- are conceivable. From amongst these, the spherical space (and the
- elliptical) excels in its simplicity, since all points on it are
- equivalent. As a result of this discussion, a most interesting
- question arises for astronomers and physicists, and that is whether
- the universe in which we live is infinite, or whether it is finite in
- the manner of the spherical universe. Our experience is far from being
- sufficient to enable us to answer this question. But the general
- theory of relativity permits of our answering it with a moduate degree
- of certainty, and in this connection the difficulty mentioned in
- Section 30 finds its solution.
- THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY
- According to the general theory of relativity, the geometrical
- properties of space are not independent, but they are determined by
- matter. Thus we can draw conclusions about the geometrical structure
- of the universe only if we base our considerations on the state of the
- matter as being something that is known. We know from experience that,
- for a suitably chosen co-ordinate system, the velocities of the stars
- are small as compared with the velocity of transmission of light. We
- can thus as a rough approximation arrive at a conclusion as to the
- nature of the universe as a whole, if we treat the matter as being at
- rest.
- We already know from our previous discussion that the behaviour of
- measuring-rods and clocks is influenced by gravitational fields, i.e.
- by the distribution of matter. This in itself is sufficient to exclude
- the possibility of the exact validity of Euclidean geometry in our
- universe. But it is conceivable that our universe differs only
- slightly from a Euclidean one, and this notion seems all the more
- probable, since calculations show that the metrics of surrounding
- space is influenced only to an exceedingly small extent by masses even
- of the magnitude of our sun. We might imagine that, as regards
- geometry, our universe behaves analogously to a surface which is
- irregularly curved in its individual parts, but which nowhere departs
- appreciably from a plane: something like the rippled surface of a
- lake. Such a universe might fittingly be called a quasi-Euclidean
- universe. As regards its space it would be infinite. But calculation
- shows that in a quasi-Euclidean universe the average density of matter
- would necessarily be nil. Thus such a universe could not be inhabited
- by matter everywhere ; it would present to us that unsatisfactory
- picture which we portrayed in Section 30.
- If we are to have in the universe an average density of matter which
- differs from zero, however small may be that difference, then the
- universe cannot be quasi-Euclidean. On the contrary, the results of
- calculation indicate that if matter be distributed uniformly, the
- universe would necessarily be spherical (or elliptical). Since in
- reality the detailed distribution of matter is not uniform, the real
- universe will deviate in individual parts from the spherical, i.e. the
- universe will be quasi-spherical. But it will be necessarily finite.
- In fact, the theory supplies us with a simple connection * between
- the space-expanse of the universe and the average density of matter in
- it.
- Notes
- *) For the radius R of the universe we obtain the equation
- eq. 28: file eq28.gif
- The use of the C.G.S. system in this equation gives 2/k = 1^.08.10^27;
- p is the average density of the matter and k is a constant connected
- with the Newtonian constant of gravitation.
- APPENDIX I
- SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION
- (SUPPLEMENTARY TO SECTION 11)
- For the relative orientation of the co-ordinate systems indicated in
- Fig. 2, the x-axes of both systems pernumently coincide. In the
- present case we can divide the problem into parts by considering first
- only events which are localised on the x-axis. Any such event is
- represented with respect to the co-ordinate system K by the abscissa x
- and the time t, and with respect to the system K1 by the abscissa x'
- and the time t'. We require to find x' and t' when x and t are given.
- A light-signal, which is proceeding along the positive axis of x, is
- transmitted according to the equation
- x = ct
- or
- x - ct = 0 . . . (1).
- Since the same light-signal has to be transmitted relative to K1 with
- the velocity c, the propagation relative to the system K1 will be
- represented by the analogous formula
- x' - ct' = O . . . (2)
- Those space-time points (events) which satisfy (x) must also satisfy
- (2). Obviously this will be the case when the relation
- (x' - ct') = l (x - ct) . . . (3).
- is fulfilled in general, where l indicates a constant ; for, according
- to (3), the disappearance of (x - ct) involves the disappearance of
- (x' - ct').
- If we apply quite similar considerations to light rays which are being
- transmitted along the negative x-axis, we obtain the condition
- (x' + ct') = µ(x + ct) . . . (4).
- By adding (or subtracting) equations (3) and (4), and introducing for
- convenience the constants a and b in place of the constants l and µ,
- where
- eq. 29: file eq29.gif
- and
- eq. 30: file eq30.gif
- we obtain the equations
- eq. 31: file eq31.gif
- We should thus have the solution of our problem, if the constants a
- and b were known. These result from the following discussion.
- For the origin of K1 we have permanently x' = 0, and hence according
- to the first of the equations (5)
- eq. 32: file eq32.gif
- If we call v the velocity with which the origin of K1 is moving
- relative to K, we then have
- eq. 33: file eq33.gif
- The same value v can be obtained from equations (5), if we calculate
- the velocity of another point of K1 relative to K, or the velocity
- (directed towards the negative x-axis) of a point of K with respect to
- K'. In short, we can designate v as the relative velocity of the two
- systems.
- Furthermore, the principle of relativity teaches us that, as judged
- from K, the length of a unit measuring-rod which is at rest with
- reference to K1 must be exactly the same as the length, as judged from
- K', of a unit measuring-rod which is at rest relative to K. In order
- to see how the points of the x-axis appear as viewed from K, we only
- require to take a " snapshot " of K1 from K; this means that we have
- to insert a particular value of t (time of K), e.g. t = 0. For this
- value of t we then obtain from the first of the equations (5)
- x' = ax
- Two points of the x'-axis which are separated by the distance Dx' = I
- when measured in the K1 system are thus separated in our instantaneous
- photograph by the distance
- eq. 34: file eq34.gif
- But if the snapshot be taken from K'(t' = 0), and if we eliminate t
- from the equations (5), taking into account the expression (6), we
- obtain
- eq. 35: file eq35.gif
- From this we conclude that two points on the x-axis separated by the
- distance I (relative to K) will be represented on our snapshot by the
- distance
- eq. 36: file eq36.gif
- But from what has been said, the two snapshots must be identical;
- hence Dx in (7) must be equal to Dx' in (7a), so that we obtain
- eq. 37: file eq37.gif
- The equations (6) and (7b) determine the constants a and b. By
- inserting the values of these constants in (5), we obtain the first
- and the fourth of the equations given in Section 11.
- eq. 38: file eq38.gif
- Thus we have obtained the Lorentz transformation for events on the
- x-axis. It satisfies the condition
- x'2 - c^2t'2 = x2 - c^2t2 . . . (8a).
- The extension of this result, to include events which take place
- outside the x-axis, is obtained by retaining equations (8) and
- supplementing them by the relations
- eq. 39: file eq39.gif
- In this way we satisfy the postulate of the constancy of the velocity
- of light in vacuo for rays of light of arbitrary direction, both for
- the system K and for the system K'. This may be shown in the following
- manner.
- We suppose a light-signal sent out from the origin of K at the time t
- = 0. It will be propagated according to the equation
- eq. 40: file eq40.gif
- or, if we square this equation, according to the equation
- x2 + y2 + z2 = c^2t2 = 0 . . . (10).
- It is required by the law of propagation of light, in conjunction with
- the postulate of relativity, that the transmission of the signal in
- question should take place -- as judged from K1 -- in accordance with
- the corresponding formula
- r' = ct'
- or,
- x'2 + y'2 + z'2 - c^2t'2 = 0 . . . (10a).
- In order that equation (10a) may be a consequence of equation (10), we
- must have
- x'2 + y'2 + z'2 - c^2t'2 = s (x2 + y2 + z2 - c^2t2) (11).
- Since equation (8a) must hold for points on the x-axis, we thus have s
- = I. It is easily seen that the Lorentz transformation really
- satisfies equation (11) for s = I; for (11) is a consequence of (8a)
- and (9), and hence also of (8) and (9). We have thus derived the
- Lorentz transformation.
- The Lorentz transformation represented by (8) and (9) still requires
- to be generalised. Obviously it is immaterial whether the axes of K1
- be chosen so that they are spatially parallel to those of K. It is
- also not essential that the velocity of translation of K1 with respect
- to K should be in the direction of the x-axis. A simple consideration
- shows that we are able to construct the Lorentz transformation in this
- general sense from two kinds of transformations, viz. from Lorentz
- transformations in the special sense and from purely spatial
- transformations. which corresponds to the replacement of the
- rectangular co-ordinate system by a new system with its axes pointing
- in other directions.
- Mathematically, we can characterise the generalised Lorentz
- transformation thus :
- It expresses x', y', x', t', in terms of linear homogeneous functions
- of x, y, x, t, of such a kind that the relation
- x'2 + y'2 + z'2 - c^2t'2 = x2 + y2 + z2 - c^2t2 (11a).
- is satisficd identically. That is to say: If we substitute their
- expressions in x, y, x, t, in place of x', y', x', t', on the
- left-hand side, then the left-hand side of (11a) agrees with the
- right-hand side.
- APPENDIX II
- MINKOWSKI'S FOUR-DIMENSIONAL SPACE ("WORLD")
- (SUPPLEMENTARY TO SECTION 17)
- We can characterise the Lorentz transformation still more simply if we
- introduce the imaginary eq. 25 in place of t, as time-variable. If, in
- accordance with this, we insert
- x[1] = x
- x[2] = y
- x[3] = z
- x[4] = eq. 25
- and similarly for the accented system K1, then the condition which is
- identically satisfied by the transformation can be expressed thus :
- x[1]'2 + x[2]'2 + x[3]'2 + x[4]'2 = x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2
- (12).
- That is, by the afore-mentioned choice of " coordinates," (11a) [see
- the end of Appendix II] is transformed into this equation.
- We see from (12) that the imaginary time co-ordinate x[4], enters into
- the condition of transformation in exactly the same way as the space
- co-ordinates x[1], x[2], x[3]. It is due to this fact that, according
- to the theory of relativity, the " time "x[4], enters into natural
- laws in the same form as the space co ordinates x[1], x[2], x[3].
- A four-dimensional continuum described by the "co-ordinates" x[1],
- x[2], x[3], x[4], was called "world" by Minkowski, who also termed a
- point-event a " world-point." From a "happening" in three-dimensional
- space, physics becomes, as it were, an " existence " in the
- four-dimensional " world."
- This four-dimensional " world " bears a close similarity to the
- three-dimensional " space " of (Euclidean) analytical geometry. If we
- introduce into the latter a new Cartesian co-ordinate system (x'[1],
- x'[2], x'[3]) with the same origin, then x'[1], x'[2], x'[3], are
- linear homogeneous functions of x[1], x[2], x[3] which identically
- satisfy the equation
- x'[1]^2 + x'[2]^2 + x'[3]^2 = x[1]^2 + x[2]^2 + x[3]^2
- The analogy with (12) is a complete one. We can regard Minkowski's "
- world " in a formal manner as a four-dimensional Euclidean space (with
- an imaginary time coordinate) ; the Lorentz transformation corresponds
- to a " rotation " of the co-ordinate system in the fourdimensional "
- world."
- APPENDIX III
- THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY
- From a systematic theoretical point of view, we may imagine the
- process of evolution of an empirical science to be a continuous
- process of induction. Theories are evolved and are expressed in short
- compass as statements of a large number of individual observations in
- the form of empirical laws, from which the general laws can be
- ascertained by comparison. Regarded in this way, the development of a
- science bears some resemblance to the compilation of a classified
- catalogue. It is, as it were, a purely empirical enterprise.
- But this point of view by no means embraces the whole of the actual
- process ; for it slurs over the important part played by intuition and
- deductive thought in the development of an exact science. As soon as a
- science has emerged from its initial stages, theoretical advances are
- no longer achieved merely by a process of arrangement. Guided by
- empirical data, the investigator rather develops a system of thought
- which, in general, is built up logically from a small number of
- fundamental assumptions, the so-called axioms. We call such a system
- of thought a theory. The theory finds the justification for its
- existence in the fact that it correlates a large number of single
- observations, and it is just here that the " truth " of the theory
- lies.
- Corresponding to the same complex of empirical data, there may be
- several theories, which differ from one another to a considerable
- extent. But as regards the deductions from the theories which are
- capable of being tested, the agreement between the theories may be so
- complete that it becomes difficult to find any deductions in which the
- two theories differ from each other. As an example, a case of general
- interest is available in the province of biology, in the Darwinian
- theory of the development of species by selection in the struggle for
- existence, and in the theory of development which is based on the
- hypothesis of the hereditary transmission of acquired characters.
- We have another instance of far-reaching agreement between the
- deductions from two theories in Newtonian mechanics on the one hand,
- and the general theory of relativity on the other. This agreement goes
- so far, that up to the preseat we have been able to find only a few
- deductions from the general theory of relativity which are capable of
- investigation, and to which the physics of pre-relativity days does
- not also lead, and this despite the profound difference in the
- fundamental assumptions of the two theories. In what follows, we shall
- again consider these important deductions, and we shall also discuss
- the empirical evidence appertaining to them which has hitherto been
- obtained.
- (a) Motion of the Perihelion of Mercury
- According to Newtonian mechanics and Newton's law of gravitation, a
- planet which is revolving round the sun would describe an ellipse
- round the latter, or, more correctly, round the common centre of
- gravity of the sun and the planet. In such a system, the sun, or the
- common centre of gravity, lies in one of the foci of the orbital
- ellipse in such a manner that, in the course of a planet-year, the
- distance sun-planet grows from a minimum to a maximum, and then
- decreases again to a minimum. If instead of Newton's law we insert a
- somewhat different law of attraction into the calculation, we find
- that, according to this new law, the motion would still take place in
- such a manner that the distance sun-planet exhibits periodic
- variations; but in this case the angle described by the line joining
- sun and planet during such a period (from perihelion--closest
- proximity to the sun--to perihelion) would differ from 360^0. The line
- of the orbit would not then be a closed one but in the course of time
- it would fill up an annular part of the orbital plane, viz. between
- the circle of least and the circle of greatest distance of the planet
- from the sun.
- According also to the general theory of relativity, which differs of
- course from the theory of Newton, a small variation from the
- Newton-Kepler motion of a planet in its orbit should take place, and
- in such away, that the angle described by the radius sun-planet
- between one perhelion and the next should exceed that corresponding to
- one complete revolution by an amount given by
- eq. 41: file eq41.gif
- (N.B. -- One complete revolution corresponds to the angle 2p in the
- absolute angular measure customary in physics, and the above
- expression giver the amount by which the radius sun-planet exceeds
- this angle during the interval between one perihelion and the next.)
- In this expression a represents the major semi-axis of the ellipse, e
- its eccentricity, c the velocity of light, and T the period of
- revolution of the planet. Our result may also be stated as follows :
- According to the general theory of relativity, the major axis of the
- ellipse rotates round the sun in the same sense as the orbital motion
- of the planet. Theory requires that this rotation should amount to 43
- seconds of arc per century for the planet Mercury, but for the other
- Planets of our solar system its magnitude should be so small that it
- would necessarily escape detection. *
- In point of fact, astronomers have found that the theory of Newton
- does not suffice to calculate the observed motion of Mercury with an
- exactness corresponding to that of the delicacy of observation
- attainable at the present time. After taking account of all the
- disturbing influences exerted on Mercury by the remaining planets, it
- was found (Leverrier: 1859; and Newcomb: 1895) that an unexplained
- perihelial movement of the orbit of Mercury remained over, the amount
- of which does not differ sensibly from the above mentioned +43 seconds
- of arc per century. The uncertainty of the empirical result amounts to
- a few seconds only.
- (b) Deflection of Light by a Gravitational Field
- In Section 22 it has been already mentioned that according to the
- general theory of relativity, a ray of light will experience a
- curvature of its path when passing through a gravitational field, this
- curvature being similar to that experienced by the path of a body
- which is projected through a gravitational field. As a result of this
- theory, we should expect that a ray of light which is passing close to
- a heavenly body would be deviated towards the latter. For a ray of
- light which passes the sun at a distance of D sun-radii from its
- centre, the angle of deflection (a) should amount to
- eq. 42: file eq42.gif
- It may be added that, according to the theory, half of Figure 05 this
- deflection is produced by the Newtonian field of attraction of the
- sun, and the other half by the geometrical modification (" curvature
- ") of space caused by the sun.
- This result admits of an experimental test by means of the
- photographic registration of stars during a total eclipse of the sun.
- The only reason why we must wait for a total eclipse is because at
- every other time the atmosphere is so strongly illuminated by the
- light from the sun that the stars situated near the sun's disc are
- invisible. The predicted effect can be seen clearly from the
- accompanying diagram. If the sun (S) were not present, a star which is
- practically infinitely distant would be seen in the direction D[1], as
- observed front the earth. But as a consequence of the deflection of
- light from the star by the sun, the star will be seen in the direction
- D[2], i.e. at a somewhat greater distance from the centre of the sun
- than corresponds to its real position.
- In practice, the question is tested in the following way. The stars in
- the neighbourhood of the sun are photographed during a solar eclipse.
- In addition, a second photograph of the same stars is taken when the
- sun is situated at another position in the sky, i.e. a few months
- earlier or later. As compared whh the standard photograph, the
- positions of the stars on the eclipse-photograph ought to appear
- displaced radially outwards (away from the centre of the sun) by an
- amount corresponding to the angle a.
- We are indebted to the [British] Royal Society and to the Royal
- Astronomical Society for the investigation of this important
- deduction. Undaunted by the [first world] war and by difficulties of
- both a material and a psychological nature aroused by the war, these
- societies equipped two expeditions -- to Sobral (Brazil), and to the
- island of Principe (West Africa) -- and sent several of Britain's most
- celebrated astronomers (Eddington, Cottingham, Crommelin, Davidson),
- in order to obtain photographs of the solar eclipse of 29th May, 1919.
- The relative discrepancies to be expected between the stellar
- photographs obtained during the eclipse and the comparison photographs
- amounted to a few hundredths of a millimetre only. Thus great accuracy
- was necessary in making the adjustments required for the taking of the
- photographs, and in their subsequent measurement.
- The results of the measurements confirmed the theory in a thoroughly
- satisfactory manner. The rectangular components of the observed and of
- the calculated deviations of the stars (in seconds of arc) are set
- forth in the following table of results :
- Table 01: file table01.gif
- (c) Displacement of Spectral Lines Towards the Red
- In Section 23 it has been shown that in a system K1 which is in
- rotation with regard to a Galileian system K, clocks of identical
- construction, and which are considered at rest with respect to the
- rotating reference-body, go at rates which are dependent on the
- positions of the clocks. We shall now examine this dependence
- quantitatively. A clock, which is situated at a distance r from the
- centre of the disc, has a velocity relative to K which is given by
- V = wr
- where w represents the angular velocity of rotation of the disc K1
- with respect to K. If v[0], represents the number of ticks of the
- clock per unit time (" rate " of the clock) relative to K when the
- clock is at rest, then the " rate " of the clock (v) when it is moving
- relative to K with a velocity V, but at rest with respect to the disc,
- will, in accordance with Section 12, be given by
- eq. 43: file eq43.gif
- or with sufficient accuracy by
- eq. 44: file eq44.gif
- This expression may also be stated in the following form:
- eq. 45: file eq45.gif
- If we represent the difference of potential of the centrifugal force
- between the position of the clock and the centre of the disc by f,
- i.e. the work, considered negatively, which must be performed on the
- unit of mass against the centrifugal force in order to transport it
- from the position of the clock on the rotating disc to the centre of
- the disc, then we have
- eq. 46: file eq46.gif
- From this it follows that
- eq. 47: file eq47.gif
- In the first place, we see from this expression that two clocks of
- identical construction will go at different rates when situated at
- different distances from the centre of the disc. This result is aiso
- valid from the standpoint of an observer who is rotating with the
- disc.
- Now, as judged from the disc, the latter is in a gravititional field
- of potential f, hence the result we have obtained will hold quite
- generally for gravitational fields. Furthermore, we can regard an atom
- which is emitting spectral lines as a clock, so that the following
- statement will hold:
- An atom absorbs or emits light of a frequency which is dependent on
- the potential of the gravitational field in which it is situated.
- The frequency of an atom situated on the surface of a heavenly body
- will be somewhat less than the frequency of an atom of the same
- element which is situated in free space (or on the surface of a
- smaller celestial body).
- Now f = - K (M/r), where K is Newton's constant of gravitation, and M
- is the mass of the heavenly body. Thus a displacement towards the red
- ought to take place for spectral lines produced at the surface of
- stars as compared with the spectral lines of the same element produced
- at the surface of the earth, the amount of this displacement being
- eq. 48: file eq48.gif
- For the sun, the displacement towards the red predicted by theory
- amounts to about two millionths of the wave-length. A trustworthy
- calculation is not possible in the case of the stars, because in
- general neither the mass M nor the radius r are known.
- It is an open question whether or not this effect exists, and at the
- present time (1920) astronomers are working with great zeal towards
- the solution. Owing to the smallness of the effect in the case of the
- sun, it is difficult to form an opinion as to its existence. Whereas
- Grebe and Bachem (Bonn), as a result of their own measurements and
- those of Evershed and Schwarzschild on the cyanogen bands, have placed
- the existence of the effect almost beyond doubt, while other
- investigators, particularly St. John, have been led to the opposite
- opinion in consequence of their measurements.
- Mean displacements of lines towards the less refrangible end of the
- spectrum are certainly revealed by statistical investigations of the
- fixed stars ; but up to the present the examination of the available
- data does not allow of any definite decision being arrived at, as to
- whether or not these displacements are to be referred in reality to
- the effect of gravitation. The results of observation have been
- collected together, and discussed in detail from the standpoint of the
- question which has been engaging our attention here, in a paper by E.
- Freundlich entitled "Zur Prüfung der allgemeinen
- Relativit¨aut;ts-Theorie" (Die Naturwissenschaften, 1919, No. 35,
- p. 520: Julius Springer, Berlin).
- At all events, a definite decision will be reached during the next few
- years. If the displacement of spectral lines towards the red by the
- gravitational potential does not exist, then the general theory of
- relativity will be untenable. On the other hand, if the cause of the
- displacement of spectral lines be definitely traced to the
- gravitational potential, then the study of this displacement will
- furnish us with important information as to the mass of the heavenly
- bodies. [5][A]
- Notes
- *) Especially since the next planet Venus has an orbit that is
- almost an exact circle, which makes it more difficult to locate the
- perihelion with precision.
- The displacentent of spectral lines towards the red end of the
- spectrum was definitely established by Adams in 1924, by observations
- on the dense companion of Sirius, for which the effect is about thirty
- times greater than for the Sun. R.W.L. -- translator
- APPENDIX IV
- THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY
- (SUPPLEMENTARY TO SECTION 32)
- Since the publication of the first edition of this little book, our
- knowledge about the structure of space in the large (" cosmological
- problem ") has had an important development, which ought to be
- mentioned even in a popular presentation of the subject.
- My original considerations on the subject were based on two
- hypotheses:
- (1) There exists an average density of matter in the whole of space
- which is everywhere the same and different from zero.
- (2) The magnitude (" radius ") of space is independent of time.
- Both these hypotheses proved to be consistent, according to the
- general theory of relativity, but only after a hypothetical term was
- added to the field equations, a term which was not required by the
- theory as such nor did it seem natural from a theoretical point of
- view (" cosmological term of the field equations ").
- Hypothesis (2) appeared unavoidable to me at the time, since I thought
- that one would get into bottomless speculations if one departed from
- it.
- However, already in the 'twenties, the Russian mathematician Friedman
- showed that a different hypothesis was natural from a purely
- theoretical point of view. He realized that it was possible to
- preserve hypothesis (1) without introducing the less natural
- cosmological term into the field equations of gravitation, if one was
- ready to drop hypothesis (2). Namely, the original field equations
- admit a solution in which the " world radius " depends on time
- (expanding space). In that sense one can say, according to Friedman,
- that the theory demands an expansion of space.
- A few years later Hubble showed, by a special investigation of the
- extra-galactic nebulae (" milky ways "), that the spectral lines
- emitted showed a red shift which increased regularly with the distance
- of the nebulae. This can be interpreted in regard to our present
- knowledge only in the sense of Doppler's principle, as an expansive
- motion of the system of stars in the large -- as required, according
- to Friedman, by the field equations of gravitation. Hubble's discovery
- can, therefore, be considered to some extent as a confirmation of the
- theory.
- There does arise, however, a strange difficulty. The interpretation of
- the galactic line-shift discovered by Hubble as an expansion (which
- can hardly be doubted from a theoretical point of view), leads to an
- origin of this expansion which lies " only " about 10^9 years ago,
- while physical astronomy makes it appear likely that the development
- of individual stars and systems of stars takes considerably longer. It
- is in no way known how this incongruity is to be overcome.
- I further want to rernark that the theory of expanding space, together
- with the empirical data of astronomy, permit no decision to be reached
- about the finite or infinite character of (three-dimensional) space,
- while the original " static " hypothesis of space yielded the closure
- (finiteness) of space.
- K = co-ordinate system
- x, y = two-dimensional co-ordinates
- x, y, z = three-dimensional co-ordinates
- x, y, z, t = four-dimensional co-ordinates
- t = time
- I = distance
- v = velocity
- F = force
- G = gravitational field
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