Read · Special relativity: from clock operations to electrodynamics · Results

§4 · rods, clocks, and the limit of the model

Clocks, rods and light first, then electricity and magnetism: follow the relativity paper through its introduction and all ten sections, including the later ones most retellings skip.

The paper's results

Page 903

A moving sphere measured as an ellipsoid

A rigid body that is a sphere of radius $R$ when measured at rest has, measured from the stationary system while it moves with speed $v$ , the shape of an ellipsoid whose axis along the motion is $R\sqrt{1-(v/V)^2}$ and whose two other axes stay $R$ .

As printed

Ein starrer Körper, welcher in ruhendem Zustande ausgemessen die Gestalt einer Kugel hat, hat also in bewegtem Zustande — vom ruhenden System aus betrachtet — die Gestalt eines Rotationsellipsoides mit den Achsen
$R \sqrt{1 - \left( \frac{v}{V} \right)^2} , \quad R , \quad R .$ Sphere radiusSpeed of the moving systemSpeed of lightExplain this equation
Während also die $Y$ - und $Z$ -Dimension der Kugel (also auch jedes starren Körpers von beliebiger Gestalt) durch die Bewegung nicht modifiziert erscheinen, erscheint die $X$ -Dimension im Verhältnis $1 : \sqrt{1 - (v/V)^2}$ verkürzt, also um so stärker, je größer $v$ ist. Für $v = V$ schrumpfen alle bewegten Objekte — vom „ruhenden“ System aus betrachtet — in flächenhafte Gebilde zusammen. Für Überlichtgeschwindigkeiten werden unsere Überlegungen sinnlos; wir werden übrigens in den folgenden Betrachtungen finden, daß die Lichtgeschwindigkeit in unserer Theorie physikalisch die Rolle der unendlich großen Geschwindigkeiten spielt.

Text on page 903 of the German source, § 4, paragraph 1 · Display on page 903 of the German source, § 4, display 3 · Text on page 903 of the German source, § 4, paragraph 3

In modern notation

L = \frac{L_0}{\gamma}

Shown in modern letters; Einstein's are in §4 of the German source face.

Premise: The shape is a set of positions read at one time of the stationary system. It is not what a camera would record, which also depends on when light from each point left it.

Decoder

$R\sqrt{1-(v/V)^2}$: The length of the axis along the motion, measured in the stationary system: the sphere is shortened in the ratio $1:\sqrt{1-(v/V)^2}$ in that direction only.
$R, R$: The two axes across the motion, which the motion leaves unchanged.
"vom ruhenden System aus betrachtet": Viewed from the stationary system: the shape is found from the positions of the body's points at one time $t=0$ of that system.
$v=V$: At the speed of light every moving object would be flat in the stationary system; for speeds greater than light the paper says its considerations become meaningless.

Try it

SR-03: The sphere moving at 0.6 of the speed of light: what are its three axes in the stationary system? Choose the preset “Moving Sphere Ellipsoid” there.

SR-03: Measure k's rod from K and K's rod from k: is the shortening the same both ways? Choose the preset “Reciprocal Measurement” there.

Where this stops: Each of two clocks in relative motion runs slow as judged from the other, because the two judgments use different distant clocks. Transforming one pair of events does not by itself give a length or a clock reading; the pair has to be the right one. Read the argument

Common wrong turns

Length contraction is what a camera would see: a moving sphere would look flattened in a photograph.

Argument: A consequence of the transformation equations
Model: Exact within the paper's kinematics, for a body the paper calls rigid
Evidence: A derivation, not evidence
History: FitzGerald in 1889 and Lorentz in 1892 had proposed that bodies moving through the ether contract along their motion, as a physical effect; here the same factor follows from the definition of simultaneity and the two principles, and the paper cites no one for it

Pages 903, 904, 905

A moving clock falls behind

A clock at rest in the moving system, viewed from the stationary system, loses $1-\sqrt{1-(v/V)^2}$ seconds each second, so a clock moved from A to B, or around a closed curve back to A, lags behind the clock that stayed, by $\frac{1}{2}tv^2/V^2$ seconds to second order.

As printed

Wir denken uns ferner eine der Uhren, welche relativ zum ruhenden System ruhend die Zeit $t$ , relativ zum bewegten System ruhend die Zeit $\tau$ anzugeben befähigt sind, im Koordinatenursprung von $k$ gelegen und so gerichtet, daß sie die Zeit $\tau$ angibt. Wie schnell geht diese Uhr, vom ruhenden System aus betrachtet?
$\tau = t \sqrt{1 - \left( \frac{v}{V} \right)^2} = t - \left( 1 - \sqrt{1 - \left( \frac{v}{V} \right)^2} \right) t ,$ Time in the moving systemTime in the stationary systemSpeed of the moving systemSpeed of lightExplain this equation
Sind in den Punkten $A$ und $B$ von $K$ ruhende, im ruhenden System betrachtet, synchron gehende Uhren vorhanden, und bewegt man die Uhr in $A$ mit der Geschwindigkeit $v$ auf der Verbindungslinie nach $B$ , so gehen nach Ankunft dieser Uhr in $B$ die beiden Uhren nicht mehr synchron, sondern die von $A$ nach $B$ bewegte Uhr geht gegenüber der von Anfang an in $B$ befindlichen um $\frac{1}{2} t v^2 / V^2$ Sek. (bis auf Größen vierter und höherer Ordnung) nach, wenn $t$ die Zeit ist, welche die Uhr von $A$ nach $B$ braucht.
Nimmt man an, daß das für eine polygonale Linie bewiesene Resultat auch für eine stetig gekrümmte Kurve gelte, so erhält man den Satz: Befinden sich in $A$ zwei synchron gehende Uhren und bewegt man die eine derselben auf einer geschlossenen Kurve mit konstanter Geschwindigkeit, bis sie wieder nach $A$ zurückkommt, was $t$ Sek. dauern möge, so geht die letztere Uhr bei ihrer Ankunft in $A$ gegenüber der unbewegt gebliebenen um $\frac{1}{2} t (v/V)^2$ Sek. nach. Man schließt daraus, daß eine am Erdäquator befindliche Unruhuhr um einen sehr kleinen Betrag langsamer laufen muß als eine genau gleich beschaffene, sonst gleichen Bedingungen unterworfene, an einem Erdpole befindliche Uhr.

Text on pages 903–904 of the German source, § 4, paragraph 5 · Display on page 904 of the German source, § 4, display 6 · Text on page 904 of the German source, § 4, paragraph 6 · Text on pages 904–905 of the German source, § 4, paragraph 8

In modern notation

\Delta t = \gamma\,\Delta\tau

1 - \frac{1}{\gamma} \approx \frac{1}{2}\,\frac{v^{2}}{c^{2}}

\Delta\tau = \int_{t_{0}}^{t_{1}} \sqrt{1 - \frac{u\left(t\right)^{2}}{c^{2}}}\,\mathrm{d}t

Shown in modern letters; Einstein's are in §4 of the German source face.

Approximation: The lag $\frac{1}{2}tv^2/V^2$ is printed to within quantities of the fourth and higher order; the exact loss follows from the square root.
Premise: The theorem for a closed curve assumes that the result proved for a polygonal line also holds for a continuously curved one, and the paper says it assumes this.
An inference beyond the argument: The equator remark applies the closed-curve theorem to the earth's rotation, and it holds in this paper's kinematics alone. The card keeps the remark as printed.
A comparison, not a premise: On the real earth the difference in gravitational potential cancels the effect, and clocks at sea level keep the same rate everywhere, which the later general theory of relativity explains.

Decoder

$\tau$: The reading of the clock at the origin of the moving system k: the time of k there. It is k's time coordinate, not a proper time.
$\tau = t\sqrt{1-(v/V)^2}$: At the place of that clock, $x=vt$ , the transformation gives this: while the stationary system's clocks advance by $t$ , the moving clock advances by less.
$\frac{1}{2}tv^2/V^2$: The lag after a trip lasting $t$ seconds of the stationary system, to within quantities of the fourth and higher order in $v/V$ .
Unruhuhr: A balance-wheel clock, the paper's example for the clock at the equator.

Try it

SR-05: A clock goes out and back at 0.6 of the speed of light: how far behind the clock that stayed is it at the reunion? Choose the preset “Out and back at 0.6c” there.

SR-05: At $v/V = 10^{-4}$ , compare the exact loss with the paper's second-order lag. Choose the preset “Low speed, beta = 10^-4” there.

Where this stops: Comparisons on the real Earth also need gravity. Acceleration changes the path a clock takes; it adds no extra slowing of its own to the formula. Read the argument

Common wrong turns

Argument: A consequence of the transformation equations, and an assumption for curved paths
Model: Exact for a straight path, second order as printed, and assumed for a curved one
Evidence: A derivation, not evidence
History: Introduced in this paper

§4 · rods, clocks, and the limit of the model

Use the same measurement protocol

A clock moving at constant speed runs slow: between two of its ticks, less time passes on it than on the resting clocks it passes. A moving rod, with both ends marked at the same time, is shorter than at rest. Both are results of measurement, not of how things look in a photograph.

Read the argument

A reunion compares whole paths

Two clocks that part and meet again can be compared directly, and the one that travelled shows less elapsed time. That comparison adds up each clock's whole path. Einstein's remark about a clock at the equator needs more than this model, because a real Earth has gravity.

Read the argument