Results · §2 · lengths and measurement events

Pages 896, 897

The relativity of simultaneity

Clocks at the two ends of a moving rod, which the stationary system calls synchronous, are not synchronous by the moving observers' own test, so two events that are simultaneous in one coordinate system are in general not simultaneous in a system moving relative to it.

As printed

$t_B - t_A = \frac{r_{AB}}{V - v}$ Time in the stationary systemTime in the stationary systemLength of the moving rodSpeed of lightSpeed of the moving systemExplain this equation
$t'_A - t_B = \frac{r_{AB}}{V + v} ,$ Time in the stationary systemTime in the stationary systemLength of the moving rodSpeed of lightSpeed of the moving systemExplain this equation
Mit dem bewegten Stabe bewegte Beobachter würden also die beiden Uhren nicht synchron gehend finden, während im ruhenden System befindliche Beobachter die Uhren als synchron laufend erklären würden.
Wir sehen also, daß wir dem Begriffe der Gleichzeitigkeit keine absolute Bedeutung beimessen dürfen, sondern daß zwei Ereignisse, welche, von einem Koordinatensystem aus betrachtet, gleichzeitig sind, von einem relativ zu diesem System bewegten System aus betrachtet, nicht mehr als gleichzeitige Ereignisse aufzufassen sind.

Display on page 896 of the German source, § 2, display 2 · Display on page 897 of the German source, § 2, display 3 · Text on page 897 of the German source, § 2, paragraph 7 · Text on page 897 of the German source, § 2, paragraph 8

In modern notation

\Delta t' = -\frac{\gamma\,v\,\Delta x}{c^{2}}

Shown in modern letters; Einstein's are in §2 of the German source face.

Premise: The result rests on § 1's definition of synchronous clocks, by a ray of light sent out and reflected, and on the principle of the constancy of the velocity of light. Simultaneity at a distance is defined by that procedure; the paper does not observe it.

Decoder

$t_A$ , $t_B$ , $t'_A$: The times, read on the stationary system's clocks, at which a ray of light leaves the end A of the rod, is reflected at the end B, and arrives back at A.
$r_{AB}$: The length of the moving rod, measured in the stationary system.
$v$: The speed of the rod along the X-axis of the stationary system.
$V$: The speed of light, printed V; today it is written c.
why the two times differ: In the stationary system the ray chases the end B, which moves away at $v$ , and meets the end A coming toward it, so the way out takes $r_{AB}/(V-v)$ and the way back $r_{AB}/(V+v)$ . The test of § 1 that the moving observers apply calls the clocks synchronous only when the two times are equal.

Try it

SR-01: Two clocks 10 light-seconds apart move at 0.6 of the speed of light and are set by the moving observers' own test: how far apart do they read in the stationary system, and which one is ahead? Choose the preset “Moving pair: L = 10 ls at 0.6c (6 s desynchronization, trailing ahead)” there.

SR-03: Take two events that are simultaneous in K: at what times does k assign them? Choose the preset “Valid Frame-Simultaneous Pair” there.

Where this stops: The measured length is not what a photograph of the moving rod would show. Choosing a different pair of end events for each frame is the point of the measurement, not a correction to it. Read the argument

Common wrong turns

Einstein's 1905 argument about simultaneity uses lightning striking the two ends of a moving train.

Argument: A consequence of the definition of synchronous clocks and the two principles
Model: Exact within the paper's kinematics
Evidence: A derivation, not evidence
History: Introduced in this paper. In 1900 Poincaré had described observers moving through the ether who set their clocks by light signals and so obtain Lorentz's local time, to first order

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§2 · lengths and measurement events

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