Albert Einstein’s book Relativity: The Special and General Theory was originally published in German and translated into English in 1920. In the second chapter he introduces “The System of Co-ordinates”. The following post gives Einstein’s text followed by a revision that exchanges length with duration and space with time. First, Einstein’s text, with alternative wordings in square brackets:
End of Chapter I – 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.
Chapter II – On the basis of the physical interpretation of distance [line-interval] 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 [again and again] 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,” 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 [place], to which a name has been assigned, and with which the event coincides in space.
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 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.
What follows is a revision, with changes in italics, for three-dimensional motion, with a monorail transportation system over a rigid frame. Note: a rigid body clock is a linear clock, which measures duration in proportion to distance moved.
End of Chapter I – Let us supplement the propositions of Euclidean geometry by the single proposition that two instants on an inertial monorail moving over a practically rigid body always correspond to the same distime (time-interval), independently of any changes in position to which we may subject the monorail, the propositions of Euclidean geometry then resolve themselves into propositions on the possible relative time position of practically rigid body clocks.
Chapter II – On the basis of the physical interpretation of a time interval which has been indicated, we are also in a position to establish the time interval between two instants on an inertial monorail over a practically rigid body by means of measurements. For this purpose we require a “distime” (rod clock T) which is to be used once and for all, and which we employ as a standard measure. If, now, A and B are two instants on a rigid body clock, we can construct the line joining them according to the rules of geometry; then, starting from A, we can mark off the distime T over and over until we reach B. The number of these operations required is the numerical measure of the distime AB. This is the basis of all measurement of length-duration.
Every description of the context of an event or of the position of an object in time is based on the specification of the instant on an inertial monorail over a practically 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 time specification “New Year’s Eve, Times Square, New York,” I arrive at the following result. The earth is the rigid body clock to which the specification of time refers; “New Year’s Eve, Times Square, New York,” is a well-defined time, to which a name has been assigned, and with which the event coincides in time.
This primitive method of time specification deals only with times on the surface of rigid body clocks, and is dependent on the existence of instants 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 hot air balloon is rising over Times Square, then we can determine its position relative to an instant on the monorail over a rigid body by erecting a pole clock perpendicularly on the Square, so that it reaches the hot air balloon. The length-duration of the pole clock measured with the standard measuring-rod clock, combined with the specification of the position of the foot of the pole, supplies us with a complete time specification. On the basis of this illustration, we are able to see the manner in which a refinement of the conception of time position has been developed.
(a) We imagine the rigid body clock, to which the time specification is referred, supplemented in such a manner that the object whose time position we require is reached by the completed rigid body clock.
(b) In locating the time position of the object, we make use of a number (here the duration-length of the pole measured with the measuring-rod clock) instead of designated instants of reference.
(c) We speak of the duration-height of the cloud even when the pole clock which reaches the cloud has not been erected. By means of optical observations of the cloud from different time positions on the ground, and taking into account the properties of the propagation of light, we determine the duration-length of the pole clock 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 time position, it should be possible by means of numerical measures to make ourselves independent of the existence of marked time positions (possessing names) on the rigid body clock 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 clock. Referred to a system of co-ordinates, the context of any event will be determined (for the main part) by the specification of the duration-lengths of the three perpendiculars or co-ordinates which can be dropped from the context of the event to those three plane surfaces. The duration-lengths of these three perpendiculars can be determined by a series of manipulations with rigid measuring-rod clocks 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 rod clocks, but by indirect means. If the results of physics and astronomy are to maintain their clearness, the physical meaning of specifications of time positions must always be sought in accordance with the above considerations.
We thus obtain the following result: Every description of events in time involves the use of a rigid body clock to which such events have to be referred. The resulting relationship takes for granted that the laws of Euclidean geometry hold for “distimes;” the “distime” being represented physically by means of the convention of two marks on a rigid body clock.