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 `
`