I. Kinematical Part.Show the German source

§ 2. On the Relativity of Lengths and Times.Show the German source

The following considerations are based on the principle of relativity and on the principle of the constancy of the velocity of light, which two principles we define as follows.Show the German source
1. The laws according to which the states of physical systems change are independent of which of two coordinate systems in uniform translational motion relative to each other these changes of state are referred to.Show the German source
2. Every ray of light moves in the “stationary” coordinate system with the definite velocity $V$, independently of whether this ray of light is emitted by a stationary or by a moving body.Show the German source Here [velocity = light path / time interval]Show the German source $$\text{Geschwindigkeit} = \frac{\text{Lichtweg}}{\text{Zeitdauer}} ,$$Explain this equationShow the German source where “Zeitdauer” [time interval] is to be understood in the sense of the definition in § 1.Show the German source

Let a stationary rigid rod be given; let it possess, measured with a measuring rod that is likewise stationary, the length $l$.Show the German source We now imagine the axis of the rod laid along the $X$-axis of the stationary coordinate system, and the rod then given a uniform motion of parallel translation (velocity $v$) along the $X$-axis in the direction of increasing $x$.Show the German source We now inquire into the length of the moving rod, which we imagine to be ascertained by the following two operations:Show the German source
a) The observer moves, together with the measuring rod just mentioned, with the rod to be measured, and measures the length of the rod directly by applying the measuring rod, just as if the rod to be measured, the observer and the measuring rod were at rest.Show the German source
b) By means of clocks at rest set up in the stationary system and synchronous according to § 1, the observer ascertains at which points of the stationary system the beginning and the end of the rod to be measured are located at a definite time $t$.Show the German source
The distance between these two points, measured with the measuring rod already used, in this case at rest, is likewise a length, which may be called the “length of the rod”.Show the German source

According to the principle of relativity, the length to be found by operation a), which we shall call “the length of the rod in the moving system”, must be equal to the length $l$ of the stationary rod.Show the German source

The length to be found by operation b), which we shall call “the length of the (moving) rod in the stationary system”, we shall determine on the basis of our two principles, and we shall find that it differs from $l$.Show the German source

The kinematics in general use tacitly assumes that the lengths determined by these two operations are exactly equal to each other, or in other words, that a moving rigid body at the epoch $t$ may in geometrical respects be completely replaced by the same body when it is at rest in a definite position.Show the German source

We imagine further that at the two ends of the rod ($A$ and $B$) clocks are placed which are synchronous with the clocks of the stationary system, that is, whose readings correspond at any moment to the “time of the stationary system” at the places where they happen to be; these clocks are therefore “synchronous in the stationary system”.Show the German source

We imagine further that with each clock there is an observer moving with it, and that these observers apply to the two clocks the criterion established in § 1 for the synchronous running of two clocks.Show the German source At the time^[1)] $t_A$ let a ray of light go out from $A$, let it be reflected at the time $t_B$ at $B$, and let it arrive back at the time $t'_A$ at $A$.Show the German source Taking into account the principle of the constancy of the velocity of light, we find:Show the German source $$t_B - t_A = \frac{r_{AB}}{V - v}$$$t_B$Time in the stationary system$t_A$Time in the stationary system$r_{AB}$Length of the moving rod$V$Speed of light$v$Speed of the moving systemExplain this equationShow the German source andShow the German source $$t'_A - t_B = \frac{r_{AB}}{V + v} ,$$$t_A^{\prime }$Time in the stationary system$t_B$Time in the stationary system$r_{AB}$Length of the moving rod$V$Speed of light$v$Speed of the moving systemExplain this equationShow the German source where $r_{AB}$ denotes the length — measured in the stationary system — of the moving rod.Show the German source Observers moving with the moving rod would thus find that the two clocks do not run synchronously, while observers located in the stationary system would declare the clocks to be running synchronously.Show the German source

We see, then, that we must not attach any absolute meaning to the concept of simultaneity, but that two events which are simultaneous when viewed from one coordinate system are no longer to be regarded as simultaneous events when viewed from a system moving relative to that system.Show the German source

II. Electrodynamical Part.Show the German source

In conclusion I remark that in working on the problem treated here my friend and colleague M. Besso stood loyally by my side, and that I owe him several valuable suggestions.Show the German source

Bern, June 1905.Show the German source

(Received 30 June 1905.)Show the German source