restricted theory of relativity

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restricted theory of relativity

    Albert Einstein (18791955). Relativity: The Special and General Theory. 1920.

I. Physical Meaning of Geometrical Propositions(几何命题)


    IN your schooldays most of you who read this book made

    acquaintance with the noble building of Euclid’s geometry, and you rememberperhaps with more respect than lovethe magnificent

    ;华丽的? structure, on the lofty;崇高的? staircase;梯子? of

    which you were chased about for uncounted hours by conscientious teachers. By reason of your past experience, you would certainly regard every one with disdain;蔑视? who should pronounce even

    the most out-of-the-way ;不合常规的?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 ;断言?that these propositions are true?” Let

    us proceed ;继续?to give this question a little consideration.


     Geometry sets out from 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);公理? which, in virtue of;由于? these ideas, we are inclined to

;倾向于? accept as “true.” Then, on the basis of a logical process,

    the justification;理由? of which we feel ourselves compelled ;强

    迫?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;导出 由来? in the recognised

    manner from the axioms. The question of the “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 line,” to each of which is ascribed;归因于? 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;指定?

    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 3

    constrained;被强迫的? 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;独有的? cause of the

    genesis ;起源?of those ideas. Geometry ought to refrain from ;避

    免?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;刚体? is something

    which is lodged;寄存 嵌入? 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 ;一

    致?for observation with one eye, under suitable choice of our place of observation.

     4 If, in pursuance of ;依 按照?our habit of thought, we now

    supplement;补充? 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. 1 Geometry which has been supplemented in this way is then to be treated as a branch of physics. We can now legitimately(正当

    ) ask as to;至于? 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;有效性? for a construction with ruler and


     5 Of course the conviction ;深信?of the “truth” of geometrical

    propositions in this sense is founded exclusively;专有的 独一的?

    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;在……范围内? its


    Note 1. 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 our present purpose.

II. The System of Co-ordinates

ON the basis of the physical interpretation of distance which has 1

    been indicated, we are also in a position to establish the distance between two points on a rigid body by means of ;依靠 采用?

    measurements. For this purpose we require a “distance” (rod S)

    which is to be used once and for all;最后一次 彻底的?, 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 ;划

    分出?the distance S time after time ;一次又一次?until we reach

    B. The number of these operations required is the numerical;数字

    的? measure of the distance AB. This is the basis of all

    measurement of length. 1

     Every description of the scene of an event or of the position of an 2

    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 “Trafalgar Square, London,” 2

    I arrive at the following result. The earth is the rigid body to which the specification of place refers; “Trafalgar Square, London” is a

    well-defined point, to which a name has been assigned, and with which the event coincides in space. 3

     This primitive method of place specification deals only with places 3

    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 (盘旋)over Trafalgar Square, then we can

    determine its position relative to the surface of the earth by erecting;竖起? a pole perpendicularly;直立的? 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;精炼 改进? of the

    conception of position has been developed.

     (a) We imagine the rigid body, to which the place specification is 4

    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 5

    number (here the length of the pole measured with the

measuring-rod) instead of designated;指定的? points of reference


     (c) We speak of the height of the cloud even when the pole which 6

    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 7

    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;笛卡尔的? system of co-ordinates.

     8 This consists of three plane surfaces perpendicular to;垂直?

    each other and rigidly;牢固的? 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;处理? 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 9

    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. 4

     We thus obtain the following result: Every description of events in 10

    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.

Note 1. 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. [back]

    Note 2. I have chosen this as being more familiar to the English reader than the “Potsdamer Platz, Berlin,” which is referred to in the original. (R. W. L.) [back]

    Note 3. It is not necessary here to investigate further the significance of the expression “coincidence in space.” This conception is sufficiently(

    分的) obvious to ensure that differences of opinion are scarcely likely to arise as to its applicability in practice. [back]

    Note 4. 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.

III. Space and Time in Classical Mechanics

“THE PURPOSE of mechanics is to describe how bodies change their 1

    position in space with time.” I should load my conscience with grave sins against the sacred (神圣的)spirit of lucidity(明朗) 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 2

    “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(忽略) the influence of the air resistance, I see

    the stone descend(下落) in a straight line. A pedestrian(行人) who

    observes the misdeed(罪行) from the footpath notices that the stone

    falls to earth in a parabolic curve;抛物线?. 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;避开? the vague;模

    糊的? 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

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