# Reading Help Relativity: The Special and General Theory

`

` `

` `

` 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 `

`

` `

` `

` 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 `

`