Results · §5 · velocity composition

Pages 905, 906

The addition theorem of velocities

A point that moves with the velocity $w$ in the system k, while k moves with $v$ relative to K, has in K a velocity $U$ that for two velocities along one line is $U=(v+w)/(1+vw/V^2)$ , so that two velocities smaller than $V$ always compose to one smaller than $V$ , and the velocity of light composed with a smaller one stays $V$ .

As printed

Das Gesetz vom Parallelogramm der Geschwindigkeiten gilt also nach unserer Theorie nur in erster Annäherung.
$U = \frac{\sqrt{(v^2 + w^2 + 2 v w \cos \alpha) - \left( \frac{v w \sin \alpha}{V} \right)^2}}{1 + \frac{v w \cos \alpha}{V^2}} .$ Composed velocity (stationary system)Speed of the moving systemVelocity in the moving frameVelocity direction angle: alphaSpeed of lightExplain this equation
Hat auch $w$ die Richtung der $X$ -Achse ( $\Xi$ -Achse), so erhalten wir:
$U = \frac{v + w}{1 + \frac{v w}{V^2}} .$ Composed velocity (stationary system)Speed of the moving systemVelocity in the moving frameSpeed of lightExplain this equation
Aus dieser Gleichung folgt, daß aus der Zusammensetzung zweier Geschwindigkeiten, welche kleiner sind als $V$ , stets eine Geschwindigkeit kleiner als $V$ resultiert.
Es folgt ferner, daß die Lichtgeschwindigkeit $V$ durch Zusammensetzung mit einer „Unterlichtgeschwindigkeit“ nicht geändert werden kann.
$U = \frac{V + w}{1 + \frac{w}{V}} = V .$ Composed velocity (stationary system)Speed of lightVelocity in the moving frameExplain this equation

Text on page 905 of the German source, § 5, paragraph 2 · Display on page 906 of the German source, § 5, display 5 · Text on page 906 of the German source, § 5, paragraph 2 · Display on page 906 of the German source, § 5, display 6 · Text on page 906 of the German source, § 5, paragraph 2 · Text on page 906 of the German source, § 5, paragraph 6 · Display on page 906 of the German source, § 5, display 8

In modern notation

u_x' = \frac{u_x - v}{1 - \frac{v\,u_x}{c^{2}}}

u_y' = \frac{u_y}{\gamma\,\left(1 - \frac{v\,u_x}{c^{2}}\right)}

Shown in modern letters; Einstein's are in §5 of the German source face.

Premise: The theorem applies the transformation equations of § 3 to a point moving uniformly in k.
Conditional: The bound is proved for two velocities each smaller than $V$ ; § 4 says that for velocities greater than that of light the paper's considerations become meaningless.
A comparison, not a premise: Laue showed in 1907 that the formula for velocities along one line gives Fresnel's drag coefficient $1-1/n^2$ , which Fizeau had measured in flowing water in 1851. That is a later use of the theorem, not part of the paper.

Decoder

$U$: The velocity of the point relative to the stationary system K.
$w$ , $\alpha$: The velocity of the point relative to the moving system k, and the angle between the velocities $v$ and $w$ .
$\varkappa$ , $\lambda$: The amounts by which $v$ and $w$ fall short of $V$ , in $v=V-\varkappa$ and $w=V-\lambda$ , both positive and smaller than $V$ . They are speeds; $\lambda$ is not a wavelength here.
"nur in erster Annäherung": Only to a first approximation: when $v$ and $w$ are small compared with $V$ , the terms divided by $V^2$ are small, and $U$ is close to what the parallelogram of velocities gives.
the modern layer: The theorem finds the velocity in K from the velocity in k, with a plus sign. The modern equations run the other way, from K to k, with a minus sign, and write $c$ for $V$ .

Try it

SR-06: Compose 0.6 of the speed of light with 0.6 along one line: what is $U$ , and what would the parallelogram of velocities give? Choose the preset “0.6c along 0.6c” there.

SR-06: Compose 0.99 of the speed of light with 0.99: how close to $V$ does $U$ come? Choose the preset “0.99c with 0.99c” there.

SR-06: A ray of light crosses k at right angles to the motion: what speed does it have in K? Choose the preset “A transverse light ray” there.

Where this stops: The plus-sign formula for combining speeds and the minus-sign change of frame run in opposite directions. Two boosts in different directions combine into a boost and a rotation. Read the argument

Argument: A consequence of the transformation equations
Model: Exact within the paper's kinematics
Evidence: A derivation, not evidence
History: Introduced in this paper

Reading preferences

§5 · velocity composition

The paper's results