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

§3 · deriving and inverting the coordinate map

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 901

The two principles are compatible

A spherical light wave sent out from the common origin at the time zero spreads with the speed $V$ in the stationary system and, transformed by the equations of § 3, also with the speed $V$ in the moving system, which shows that the principle of relativity and the constancy of the speed of light are compatible.

As printed

Zur Zeit $t = \tau = 0$ werde von dem zu dieser Zeit gemeinsamen Koordinatenursprung beider Systeme aus eine Kugelwelle ausgesandt, welche sich im System $K$ mit der Geschwindigkeit $V$ ausbreitet.
$x^2 + y^2 + z^2 = V^2 t^2 .$ Position in the stationary systemPosition in the stationary systemPosition in the stationary systemSpeed of lightTime in the stationary systemExplain this equation
$\xi^2 + \eta^2 + \zeta^2 = V^2 \tau^2 .$ Position in the moving systemPosition in the moving systemPosition in the moving systemSpeed of lightTime in the moving systemExplain this equation
Die betrachtete Welle ist also auch im bewegten System betrachtet eine Kugelwelle von der Ausbreitungsgeschwindigkeit $V$ . Hiermit ist gezeigt, daß unsere beiden Grundprinzipien miteinander vereinbar sind.

Text on page 901 of the German source, § 3, paragraph 16 · Display on page 901 of the German source, § 3, display 17 · Display on page 901 of the German source, § 3, display 18 · Text on page 901 of the German source, § 3, paragraph 17

Premise: The transformation equations were built from the constancy of the speed of light in the stationary system and the definition of time; the spherical wave checks that the same holds in the moving system, which is what the principle of relativity requires.

Decoder

$x^2+y^2+z^2=V^2t^2$: The points the wave has just reached at the time $t$ of the stationary system K: a sphere of radius $Vt$ .
$\xi^2+\eta^2+\zeta^2=V^2\tau^2$: The same wave front in the coordinates of the moving system k, at its time $\tau$ : again a sphere, growing at the speed $V$ .
$\xi, \eta, \zeta, \tau$: The coordinates and the time of the moving system k, as § 3 defines them; $\tau$ is k's time coordinate, not a proper time.

Try it

SR-04: With the constructed map at 0.6 of the speed of light, does a ray keep the speed of light in both systems? Choose the preset “The constructed map at 0.6c” there.

SR-04: Put the Galilean map on the same rays: what speed does it give the light in the moving system? Choose the preset “The Galilean shelf: slow case and light rays at 0.6c” there.

Where this stops: Keeping one light ray at speed c does not fix every coefficient of the map. The spacetime interval and matrices check the result later; the 1905 argument does not assume them. Read the argument

Argument: A check that the two postulates do not contradict each other
Model: Exact within the paper's kinematics
Evidence: A consistency argument, not evidence
History: Introduced in this paper

Page 902

The transformation equations

An event with coordinates $x, y, z$ and time $t$ in the stationary system has, in a system moving along x with speed $v$ , the time $\tau=\beta(t-vx/V^2)$ and the coordinates $\xi=\beta(x-vt)$ , $\eta=y$ , $\zeta=z$ , once the unknown factor $\varphi(v)$ of the derivation is shown to be 1.

As printed

Aus dieser und der vorhin gefundenen Relation folgt, daß $\varphi(v) = 1$ sein muß, so daß die gefundenen Transformationsgleichungen übergehen in: wobei
$\begin{aligned} \tau &= \beta\left( t - \frac{v}{V^2} x \right), \\ \xi &= \beta(x - v t), \\ \eta &= y, \\ \zeta &= z, \end{aligned}$ Time in the moving systemLorentz factorTime in the stationary systemSpeed of the moving systemSpeed of lightPosition in the stationary systemPosition in the moving systemPosition in the moving systemPosition in the stationary systemPosition in the moving systemPosition in the stationary systemExplain this equation
$\beta = \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} ,$ Lorentz factorSpeed of the moving systemSpeed of lightExplain this equation

Text on page 902 of the German source, § 3, paragraph 22 · Display on page 902 of the German source, § 3, display 25 · Display on page 902 of the German source, § 3, display 26

In modern notation

t' = \gamma\,\left(t - \frac{v\,x}{c^{2}}\right)

\tau = \beta\,\left(t - \frac{v\,x}{V^{2}}\right)

x' = \gamma\,\left(x - v\,t\right)

\xi = \beta\,\left(x - v\,t\right)

\gamma = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}

\beta = \frac{1}{\sqrt{1 - \frac{v^{2}}{V^{2}}}}

Premise: The equations follow from the two principles, from § 1's definition of time and from the symmetry arguments the paper states; the factor $\varphi(v)$ is fixed by symmetry, not by an observation.

Decoder

$\beta$: Einstein's β is $1/\sqrt{1-(v/V)^2}$ , the factor written γ today. It is not today's β, which means $v/c$ .
$\tau$: The time of the moving system k at the event, as its own clocks, set by § 1's rule, read it. It is k's time coordinate, not a proper time.
$\xi, \eta, \zeta$: The event's coordinates in k, whose axes are parallel to those of the stationary system K.
$V$: The speed of light, printed V; today it is written c.
$\varphi(v)=1$: The derivation leaves a factor $\varphi(v)$ open. A third system moving back with $-v$ gives $\varphi(v)\varphi(-v)=1$ , and a rod moving across its length, which cannot depend on the sense of the motion, gives $\varphi(v)=\varphi(-v)$ ; so $\varphi(v)=1$ .

Try it

SR-04: At 0.6 of the speed of light, what does the constructed map give for β, and which constraint fixes each coefficient? Choose the preset “The constructed map at 0.6c” there.

SR-03: Boost to 0.6 of the speed of light: what do the two systems say about the same two events? Choose the preset “Boost to 0.6c (Signature)” there.

Where this stops: Checking a ray along the motion cannot fix the sideways scale. Checks by determinant or interval confirm the result; they do not replace the physical premises it rests on. Read the argument

Common wrong turns

Argument: The derivation of § 3, completed by a symmetry argument
Model: Exact within the paper's kinematics
Evidence: A derivation, not evidence
History: Lorentz had published equivalent coordinate equations in 1904, as a change of variables with a local time, and Poincaré named them the Lorentz transformation in June 1905; this paper derives them from its two principles and cites neither

§3 · deriving and inverting the coordinate map

Light constraints leave a scale to determine

Ask for a linear map that keeps light at speed c in both directions along the motion, and space and time must mix. One overall factor is left over; going there and back, and the symmetry of space, fix it at gamma.

Read the argument

The sideways step needs its own condition

Light along the motion says nothing about lengths across it. Send light across instead and require it to move at c in the moving frame too; with going there and back, that leaves the sideways coordinates unchanged: y' = y, z' = z.

Read the argument