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On the Electrodynamics of Moving BodiesShow the German source

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by A. EinsteinShow the German source

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I. Kinematical Part.Show the German source

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§ 5. The Addition Theorem of Velocities.Show the German source

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In the system $k$, moving along the $X$-axis of the system $K$ with the velocity $v$, let a point move according to the equations:Show the German source $$\begin{aligned} \xi &= w_\xi \tau , \\ \eta &= w_\eta \tau , \\ \zeta &= 0 , \end{aligned}$$$\xi$Position in the moving system$w_{\xi }$Velocity along x, moving system$\tau$Time in the moving system$\eta$Position in the moving system$w_{\eta }$Velocity along y, moving system$\zeta$Position in the moving systemExplain this equationShow the German source where $w_\xi$ and $w_\eta$ denote constants.Show the German source

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The motion of the point relative to the system $K$ is sought.Show the German source If, with the help of the transformation equations developed in § 3, we introduce the quantities $x$, $y$, $z$, $t$ into the equations of motion of the point, we obtain:Show the German source $$\begin{aligned} x &= \frac{w_\xi + v}{1 + \frac{v w_\xi}{V^2}} t , \\ y &= \frac{\sqrt{1 - \left( \frac{v}{V} \right)^2}}{1 + \frac{v w_\xi}{V^2}} w_\eta t , \\ z &= 0 . \end{aligned}$$$x$Position in the stationary system$w_{\xi }$Velocity along x, moving system$v$Speed of the moving system$V$Speed of light$t$Time in the stationary system$y$Position in the stationary system$w_{\eta }$Velocity along y, moving system$z$Position in the stationary systemExplain this equationShow the German source The law of the parallelogram of velocities therefore holds, according to our theory, only to a first approximation.Show the German source We put:Show the German source $$\begin{aligned} U^2 &= \left( \frac{d x}{d t} \right)^2 + \left( \frac{d y}{d t} \right)^2 , \\ w^2 &= w_\xi^2 + w_\eta^2 \end{aligned}$$$U$Composed velocity (stationary system)$x$Position in the stationary system$t$Time in the stationary system$y$Position in the stationary system$w$Velocity in the moving frame$w_{\xi }$Velocity along x, moving system$w_{\eta }$Velocity along y, moving systemExplain this equationShow the German source andShow the German source $$\alpha = \operatorname{arctg} \frac{w_y}{w_x} ;$$$\alpha$Velocity direction angle: alpha$w_y$Velocity along y, moving system$w_x$Velocity along x, moving systemExplain this equationShow the German source $\alpha$ is then to be regarded as the angle between the velocities $v$ and $w$.Show the German source After a simple calculation we obtain:Show the German source $$U = \frac{\sqrt{(v^2 + w^2 + 2 v w \cos \alpha) - \left( \frac{v w \sin \alpha}{V} \right)^2}}{1 + \frac{v w \cos \alpha}{V^2}} .$$$U$Composed velocity (stationary system)$v$Speed of the moving system$w$Velocity in the moving frame$\alpha$Velocity direction angle: alpha$V$Speed of lightExplain this equationShow the German source It is noteworthy that $v$ and $w$ enter the expression for the resultant velocity symmetrically.Show the German source If $w$ also has the direction of the $X$-axis ($\Xi$-axis), we obtain:Show the German source $$U = \frac{v + w}{1 + \frac{v w}{V^2}} .$$$U$Composed velocity (stationary system)$v$Speed of the moving system$w$Velocity in the moving frame$V$Speed of lightExplain this equationShow the German source From this equation it follows that from the composition of two velocities which are smaller than $V$ there always results a velocity smaller than $V$.Show the German source For if we put $v = V - \varkappa$, $w = V - \lambda$, where $\varkappa$ and $\lambda$ are positive and smaller than $V$, then:Show the German source $$U = V \frac{2 V - \varkappa - \lambda}{2 V - \varkappa - \lambda + \frac{\varkappa \lambda}{V}} < V .$$$U$Composed velocity (stationary system)$V$Speed of light$\varkappa$Speed deficit from light speed: kappa, lambda$\lambda$Speed deficit from light speed: kappa, lambdaExplain this equationShow the German source

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It follows further that the velocity of light $V$ cannot be altered by composition with a “sub-light velocity”.Show the German source For this case we obtain:Show the German source $$U = \frac{V + w}{1 + \frac{w}{V}} = V .$$$U$Composed velocity (stationary system)$V$Speed of light$w$Velocity in the moving frameExplain this equationShow the German source We could also have obtained the formula for $U$, for the case in which $v$ and $w$ have the same direction, by composing two transformations according to § 3.Show the German source If, besides the systems $K$ and $k$ figuring in § 3, we introduce a third coordinate system $k'$, in parallel motion relative to $k$, whose origin moves on the $\Xi$-axis with the velocity $w$, we obtain between the quantities $x$, $y$, $z$, $t$ and the corresponding quantities of $k'$ equations which differ from those found in § 3 only in that the quantityShow the German source $$\frac{v + w}{1 + \frac{v w}{V^2}}$$$v$Speed of the moving system$w$Velocity in the moving frame$V$Speed of lightExplain this equationShow the German source takes the place of “$v$”; one sees from this that such parallel transformations — as must be the case — form a group.Show the German source

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We have now derived the theorems, necessary for our purposes, of the kinematics corresponding to our two principles, and we proceed to show their application in electrodynamics.Show the German source

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II. Electrodynamical Part.Show the German source

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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

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Bern, June 1905.Show the German source

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(Received 30 June 1905.)Show the German source