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THE SYSTEM OF CO-ORDINATES `
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On the basis of the physical interpretation of distance which has been `
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indicated, we are also in a position to establish the distance between `
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two points on a rigid body by means of measurements. For this purpose `
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we require a " distance " (rod S) which is to be used once and for `
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all, and which we employ as a standard measure. If, now, A and B are `
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two points on a rigid body, we can construct the line joining them `
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according to the rules of geometry ; then, starting from A, we can `
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mark off the distance S time after time until we reach B. The number `
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of these operations required is the numerical measure of the distance `
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AB. This is the basis of all measurement of length. * `
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Every description of the scene of an event or of the position of an `
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object in space is based on the specification of the point on a rigid `
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body (body of reference) with which that event or object coincides. `
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This applies not only to scientific description, but also to everyday `
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life. If I analyse the place specification " Times Square, New York," `
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**A I arrive at the following result. The earth is the rigid body `
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to which the specification of place refers; " Times Square, New York," `
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is a well-defined point, to which a name has been assigned, and with `
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which the event coincides in space.**B `
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This primitive method of place specification deals only with places on `
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the surface of rigid bodies, and is dependent on the existence of `
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points on this surface which are distinguishable from each other. But `
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we can free ourselves from both of these limitations without altering `
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the nature of our specification of position. If, for instance, a cloud `
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is hovering over Times Square, then we can determine its position `
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relative to the surface of the earth by erecting a pole `
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perpendicularly on the Square, so that it reaches the cloud. The `
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length of the pole measured with the standard measuring-rod, combined `
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with the specification of the position of the foot of the pole, `
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supplies us with a complete place specification. On the basis of this `
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illustration, we are able to see the manner in which a refinement of `
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the conception of position has been developed. `
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(a) We imagine the rigid body, to which the place specification is `
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referred, supplemented in such a manner that the object whose position `
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we require is reached by. the completed rigid body. `
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(b) In locating the position of the object, we make use of a number `
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(here the length of the pole measured with the measuring-rod) instead `
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of designated points of reference. `
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(c) We speak of the height of the cloud even when the pole which `
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reaches the cloud has not been erected. By means of optical `
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observations of the cloud from different positions on the ground, and `
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taking into account the properties of the propagation of light, we `
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determine the length of the pole we should have required in order to `
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reach the cloud. `
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From this consideration we see that it will be advantageous if, in the `
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description of position, it should be possible by means of numerical `
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measures to make ourselves independent of the existence of marked `
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positions (possessing names) on the rigid body of reference. In the `
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physics of measurement this is attained by the application of the `
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Cartesian system of co-ordinates. `
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This consists of three plane surfaces perpendicular to each other and `
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rigidly attached to a rigid body. Referred to a system of `
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co-ordinates, the scene of any event will be determined (for the main `
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part) by the specification of the lengths of the three perpendiculars `
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or co-ordinates (x, y, z) which can be dropped from the scene of the `
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event to those three plane surfaces. The lengths of these three `
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perpendiculars can be determined by a series of manipulations with `
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rigid measuring-rods performed according to the rules and methods laid `
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down by Euclidean geometry. `
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In practice, the rigid surfaces which constitute the system of `
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co-ordinates are generally not available ; furthermore, the magnitudes `
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of the co-ordinates are not actually determined by constructions with `
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rigid rods, but by indirect means. If the results of physics and `
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astronomy are to maintain their clearness, the physical meaning of `
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specifications of position must always be sought in accordance with `
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the above considerations. *** `
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We thus obtain the following result: Every description of events in `
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space involves the use of a rigid body to which such events have to be `
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referred. The resulting relationship takes for granted that the laws `
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of Euclidean geometry hold for "distances;" the "distance" being `
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represented physically by means of the convention of two marks on a `
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rigid body. `
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Notes `
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* Here we have assumed that there is nothing left over i.e. that `
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the measurement gives a whole number. This difficulty is got over by `
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the use of divided measuring-rods, the introduction of which does not `
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demand any fundamentally new method. `
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**A Einstein used "Potsdamer Platz, Berlin" in the original text. `
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In the authorised translation this was supplemented with "Tranfalgar `
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Square, London". We have changed this to "Times Square, New York", as `
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this is the most well known/identifiable location to English speakers `
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in the present day. [Note by the janitor.] `
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**B It is not necessary here to investigate further the significance `
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of the expression "coincidence in space." This conception is `
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sufficiently obvious to ensure that differences of opinion are `
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scarcely likely to arise as to its applicability in practice. `
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*** A refinement and modification of these views does not become `
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necessary until we come to deal with the general theory of relativity, `
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treated in the second part of this book. `
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SPACE AND TIME IN CLASSICAL MECHANICS `
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The purpose of mechanics is to describe how bodies change their `
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position in space with "time." I should load my conscience with grave `
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sins against the sacred spirit of lucidity were I to formulate the `
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aims of mechanics in this way, without serious reflection and detailed `
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explanations. Let us proceed to disclose these sins. `
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It is not clear what is to be understood here by "position" and `
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"space." I stand at the window of a railway carriage which is `
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travelling uniformly, and drop a stone on the embankment, without `
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throwing it. Then, disregarding the influence of the air resistance, I `
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see the stone descend in a straight line. A pedestrian who observes `
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the misdeed from the footpath notices that the stone falls to earth in `
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a parabolic curve. I now ask: Do the "positions" traversed by the `
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stone lie "in reality" on a straight line or on a parabola? Moreover, `
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what is meant here by motion "in space" ? From the considerations of `
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the previous section the answer is self-evident. In the first place we `
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entirely shun the vague word "space," of which, we must honestly `
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acknowledge, we cannot form the slightest conception, and we replace `
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it by "motion relative to a practically rigid body of reference." The `
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positions relative to the body of reference (railway carriage or `
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embankment) have already been defined in detail in the preceding `
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section. If instead of " body of reference " we insert " system of `
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co-ordinates," which is a useful idea for mathematical description, we `
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are in a position to say : The stone traverses a straight line `
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relative to a system of co-ordinates rigidly attached to the carriage, `
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but relative to a system of co-ordinates rigidly attached to the `
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ground (embankment) it describes a parabola. With the aid of this `
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example it is clearly seen that there is no such thing as an `
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independently existing trajectory (lit. "path-curve"*), but only `
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a trajectory relative to a particular body of reference. `
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In order to have a complete description of the motion, we must specify `
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how the body alters its position with time ; i.e. for every point on `
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the trajectory it must be stated at what time the body is situated `
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there. These data must be supplemented by such a definition of time `
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that, in virtue of this definition, these time-values can be regarded `
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essentially as magnitudes (results of measurements) capable of `
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observation. If we take our stand on the ground of classical `
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mechanics, we can satisfy this requirement for our illustration in the `
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following manner. We imagine two clocks of identical construction ; `
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the man at the railway-carriage window is holding one of them, and the `
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man on the footpath the other. Each of the observers determines the `
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position on his own reference-body occupied by the stone at each tick `
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of the clock he is holding in his hand. In this connection we have not `
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taken account of the inaccuracy involved by the finiteness of the `
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velocity of propagation of light. With this and with a second `
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difficulty prevailing here we shall have to deal in detail later. `
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Notes `
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*) That is, a curve along which the body moves. `
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THE GALILEIAN SYSTEM OF CO-ORDINATES `
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As is well known, the fundamental law of the mechanics of `
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Galilei-Newton, which is known as the law of inertia, can be stated `
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thus: A body removed sufficiently far from other bodies continues in a `
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state of rest or of uniform motion in a straight line. This law not `
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only says something about the motion of the bodies, but it also `
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indicates the reference-bodies or systems of coordinates, permissible `
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in mechanics, which can be used in mechanical description. The visible `
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fixed stars are bodies for which the law of inertia certainly holds to `
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a high degree of approximation. Now if we use a system of co-ordinates `
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which is rigidly attached to the earth, then, relative to this system, `
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every fixed star describes a circle of immense radius in the course of `
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an astronomical day, a result which is opposed to the statement of the `
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law of inertia. So that if we adhere to this law we must refer these `
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motions only to systems of coordinates relative to which the fixed `
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stars do not move in a circle. A system of co-ordinates of which the `
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state of motion is such that the law of inertia holds relative to it `
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is called a " Galileian system of co-ordinates." The laws of the `
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mechanics of Galflei-Newton can be regarded as valid only for a `
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Galileian system of co-ordinates. `
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THE PRINCIPLE OF RELATIVITY `
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(IN THE RESTRICTED SENSE) `
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In order to attain the greatest possible clearness, let us return to `
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