The Principle of Relativity

by Albert Einstein; H. Minkowski

Before we enter into the discussion, a very important point must be noticed. Suppose we have individualised time and space in any manner; then a world-line parallel to the t-axis will correspond to a stationary point; a world-line inclined to the t-axis will correspond to a point moving uniformly; and a world-curve will correspond to a point moving in any manner. Let us now picture to our mind the world-line passing through any world point x, y, z, t; now if we find the world-line parallel to the radius vector OA′ of the hyperboloidal sheet, then we can introduce OA′ as a new time-axis, and then according to the new conceptions of time and space the substance will appear to be at rest in the world point concerned. We shall now introduce this fundamental axiom:—

The substance existing at any world point can always be conceived to be at rest, if we establish our time and space suitably. The axiom denotes that in a world-point, the expression

c²dt² - dx² - dy² - dz²

shall always be positive or what is equivalent to the same thing, every velocity V should be smaller than c. c shall therefore be the upper limit for all substantial velocities and herein lies a deep significance for the quantity c. At the first impression, the axiom seems to be rather unsatisfactory. It is to be remembered that only a modified mechanics will occur, in which the square root of this differential combination takes the place of time, so that cases in which the velocity is greater than c will play no part, something like imaginary coordinates in geometry.

The impulse and real cause of inducement for the assumption of the group-transformation G_c is the fact that the differential equation for the propagation of light in vacant space possesses the group-transformation G_c. On the other hand, the idea of rigid bodies has any sense only in a system mechanics with the group G_infinity. Now if we have an optics with G_c, and on the other hand if there are rigid bodies, it is easy to see that a t-direction can be defined by the two hyperboloidal shells common to the groups G_∞, and G_c, which has got the further consequence, that by means of suitable rigid instruments in the laboratory, we can perceive a change in natural phenomena, in case of different orientations, with regard to the direction of progressive motion of the earth. But all efforts directed towards this object, and even the celebrated interference-experiment of Michelson have given negative results. In order to supply an explanation for this result, H. A. Lorentz formed a hypothesis which practically amounts to an invariance of optics for the group G_c. According to Lorentz every substance shall suffer a contraction

1:(√(1 - v²/c²)) in length, in the direction of its motion

l/l′ = 1/√(1 - v²/c²) l′ = l(1 - v²/c²).

This hypothesis sounds rather phantastical. For the contraction is not to be thought of as a consequence of the resistance of ether, but purely as a gift from the skies, as a sort of condition always accompanying a state of motion.

I shall show in our figure that Lorentz’s hypothesis is fully equivalent to the new conceptions about time and space. Thereby it may appear more intelligible. Let us now, for the sake of simplicity, neglect (y, z) and fix our attention on a two dimensional world, in which let upright strips parallel to the t-axis represent a state of rest and another parallel strip inclined to the t-axis represent a state of uniform motion for a body, which has a constant spatial extension (see fig. 1). If OA′ is parallel to the second strip, we can take t′ as the t-axis and x′ as the x-axis, then the second body will appear to be at rest, and the first body in uniform motion. We shall now assume that the first body supposed to be at rest, has the length l, i.e., the cross section PP of the first strip upon the x-axis = l^. OC, where OC is the unit measuring rod upon the x-axis—and the second body also, when supposed to be at rest, has the same length l, this means that, the cross section Q′Q′ of the second strip has a cross-section l^· OC′, when measured parallel to the x′-axis. In these two bodies, we have now images of two Lorentz-electrons, one of which is at rest and the other moves uniformly. Now if we stick to our original coordinates, then the extension of the second electron is given by the cross section QQ of the strip belonging to it measured parallel to the x-axis. Now it is clear since Q′Q′ = l^· OC′, that QQ = l^· OD′.

If (dc/dt) = v, an easy calculation gives that