Relativity - The Special and General Theory - Albert Einstein (e books for reading txt) 📗

; 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 measuringrod, 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 measuringrod) 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 coordinates.

This consists of three plane surfaces perpendicular to each other and rigidly attached to a rigid body. Referred to a system of coordinates, the scene of any event will be determined (for the main part) by the specification of the lengths of the three perpendiculars or coordinates (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 measuringrods performed according to the rules and methods laid down by Euclidean geometry.

In practice, the rigid surfaces which constitute the system of coordinates are generally not available ; furthermore, the magnitudes of the coordinates 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 measuringrods, 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 coordinates,” 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 coordinates rigidly attached to the carriage, but relative to a system of coordinates 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 COORDINATES

 

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 coordinates 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 coordinates of which the state of motion is such that the law of inertia holds relative to it is called a ” Galileian system of coordinates.” The laws of the mechanics of Galflei-Newton can be regarded as valid only for a Galileian system of coordinates.

 

THE PRINCIPLE OF RELATIVITY (IN THE RESTRICTED SENSE)

 

In order to attain the greatest possible clearness, let us return to our example of the railway carriage supposed to be travelling uniformly. We call its motion a uniform translation (“uniform” because it is of constant velocity and direction, ” translation ” because although the carriage changes its position relative to the embankment yet it does not rotate in so doing). Let us imagine a raven flying through the air in such a manner that its motion, as observed from the embankment, is uniform and in a straight line. If we were to observe the flying raven from the moving railway carriage. we should find that the motion of the raven would be one of different velocity and direction, but that it would still be uniform and in a straight line. Expressed in an abstract manner we may say : If a mass m is moving uniformly in a straight line with respect to a coordinate system K, then it will also be moving uniformly and in a straight line relative to a second coordinate system K1 provided that the latter is executing a uniform translatory motion with respect to K. In accordance with the discussion contained in the preceding section, it follows that:

If K is a Galileian coordinate system. then every other coordinate system K’ is a Galileian one, when, in relation to K, it is in a condition of uniform motion of translation. Relative to K1 the mechanical laws of Galilei-Newton hold good exactly as they do with respect to K.

We advance a step farther in our generalisation when we express the tenet thus: If, relative to K, K1 is a uniformly moving coordinate system devoid of rotation, then natural phenomena run their course with respect to K1 according to exactly the same general laws as with respect to K. This statement is called the principle of relativity (in the restricted sense).

As long as one was convinced that all natural phenomena were capable of representation with the help of classical mechanics, there was no need to doubt the validity of this principle of relativity. But in view of the more recent development of electrodynamics and optics it became