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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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II. Electrodynamical Part.Show the German source

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§ 7. Theory of Doppler’s Principle and of Aberration.Show the German source

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In the system $K$, very far from the coordinate origin, let there be a source of electrodynamic waves which, in a part of space containing the coordinate origin, may be represented to a sufficient approximation by the equations:Show the German source $$\begin{aligned} X &= X_0 \sin \Phi , &\quad L &= L_0 \sin \Phi , \\ Y &= Y_0 \sin \Phi , &\quad M &= M_0 \sin \Phi , &\quad \Phi &= \omega \left( t - \frac{a x + b y + c z}{V} \right) . \\ Z &= Z_0 \sin \Phi , &\quad N &= N_0 \sin \Phi , \end{aligned}$$$X$Electric field, stationary system$X_0$Electric field amplitude (stationary system)$\Phi$Wave phase$L$Magnetic field, stationary system$L_0$Magnetic field amplitude (stationary system)$Y$Electric field, stationary system$Y_0$Electric field amplitude (stationary system)$M$Magnetic field, stationary system$M_0$Magnetic field amplitude (stationary system)$\omega$Wave angular frequency (stationary system)$t$Time in the stationary system$a$Direction cosine (stationary system)$x$Position in the stationary system$b$Direction cosine (stationary system)$y$Position in the stationary system$c$Direction cosine (stationary system)$z$Position in the stationary system$V$Speed of light$Z$Electric field, stationary system$Z_0$Electric field amplitude (stationary system)$N$Magnetic field, stationary system$N_0$Magnetic field amplitude (stationary system)Explain this equationShow the German source Here $(X_0, Y_0, Z_0)$ and $(L_0, M_0, N_0)$ are the vectors that determine the amplitude of the wave train, and $a$, $b$, $c$ the direction cosines of the wave normal.Show the German source

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We ask about the nature of these waves when they are examined by an observer at rest in the moving system $k$.Show the German source — By applying the transformation equations for the electric and magnetic forces found in § 6, and the transformation equations for the coordinates and the time found in § 3, we obtain directly:Show the German source $$\begin{aligned} X' &= X_0 \sin \Phi' , &\qquad L' &= L_0 \sin \Phi' , \\ Y' &= \beta \left( Y_0 - \frac{v}{V} N_0 \right) \sin \Phi' , &\qquad M' &= \beta \left( M_0 + \frac{v}{V} Z_0 \right) \sin \Phi' , \\ Z' &= \beta \left( Z_0 + \frac{v}{V} M_0 \right) \sin \Phi' , &\qquad N' &= \beta \left( N_0 - \frac{v}{V} Y_0 \right) \sin \Phi' , \\ \Phi' &= \omega' \left( \tau - \frac{a' \xi + b' \eta + c' \zeta}{V} \right) , \end{aligned}$$$X^{\prime }$Electric field, moving system$X_0$Electric field amplitude (stationary system)${\Phi }^{\prime }$Wave phase$L^{\prime }$Magnetic field, moving system$L_0$Magnetic field amplitude (stationary system)$Y^{\prime }$Electric field, moving system$\beta$Lorentz factor$Y_0$Electric field amplitude (stationary system)$v$Speed of the moving system$V$Speed of light$N_0$Magnetic field amplitude (stationary system)$M^{\prime }$Magnetic field, moving system$M_0$Magnetic field amplitude (stationary system)$Z_0$Electric field amplitude (stationary system)$Z^{\prime }$Electric field, moving system$N^{\prime }$Magnetic field, moving system${\omega }^{\prime }$Wave angular frequency (moving system)$\tau$Time in the moving system$a^{\prime }$Direction cosine (moving system)$\xi$Position in the moving system$b^{\prime }$Direction cosine (moving system)$\eta$Position in the moving system$c^{\prime }$Direction cosine (moving system)$\zeta$Position in the moving systemExplain this equationShow the German source whereShow the German source $$\omega' = \omega \beta \left( 1 - a \frac{v}{V} \right) ,$$${\omega }^{\prime }$Wave angular frequency (moving system)$\omega$Wave angular frequency (stationary system)$\beta$Lorentz factor$a$Direction cosine (stationary system)$v$Speed of the moving system$V$Speed of lightExplain this equationShow the German source $$a' = \frac{a - \frac{v}{V}}{1 - a \frac{v}{V}} ,$$$a^{\prime }$Direction cosine (moving system)$a$Direction cosine (stationary system)$v$Speed of the moving system$V$Speed of lightExplain this equationShow the German source $$b' = \frac{b}{\beta \left( 1 - a \frac{v}{V} \right)} ,$$$b^{\prime }$Direction cosine (moving system)$b$Direction cosine (stationary system)$\beta$Lorentz factor$a$Direction cosine (stationary system)$v$Speed of the moving system$V$Speed of lightExplain this equationShow the German source $$c' = \frac{c}{\beta \left( 1 - a \frac{v}{V} \right)}$$$c^{\prime }$Direction cosine (moving system)$c$Direction cosine (stationary system)$\beta$Lorentz factor$a$Direction cosine (stationary system)$v$Speed of the moving system$V$Speed of lightExplain this equationShow the German source have been set.Show the German source

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From the equation for $\omega'$ it follows: If an observer moves with the velocity $v$ relative to an infinitely distant light source of frequency $\nu$ in such a way that the line joining “light source–observer” makes the angle $\varphi$ with the velocity of the observer, referred to a coordinate system at rest relative to the light source, then the frequency $\nu'$ of the light perceived by the observer is given by the equation:Show the German source $$\nu' = \nu \frac{1 - \cos \varphi \frac{v}{V}}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} .$$${\nu }^{\prime }$Frequency, moving system$\nu$Frequency, stationary system$\phi$Angle to the motion, stationary system$v$Speed of the moving system$V$Speed of lightExplain this equationShow the German source This is Doppler’s principle for arbitrary velocities.Show the German source For $\varphi = 0$ the equation takes the perspicuous form:Show the German source $$\nu' = \nu \sqrt{\frac{1 - \frac{v}{V}}{1 + \frac{v}{V}}} .$$${\nu }^{\prime }$Frequency, moving system$\nu$Frequency, stationary system$v$Speed of the moving system$V$Speed of lightExplain this equationShow the German source One sees that — contrary to the usual view — for $v = -\infty$, $\nu = \infty$.Show the German source

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If we call $\varphi'$ the angle between the wave normal (direction of the ray) in the moving system and the line joining “light source–observer”, the equation for $a'$ takes the form:Show the German source $$\cos \varphi' = \frac{\cos \varphi - \frac{v}{V}}{1 - \frac{v}{V} \cos \varphi} .$$${\phi }^{\prime }$Angle to the motion, moving system$\phi$Angle to the motion, stationary system$v$Speed of the moving system$V$Speed of lightExplain this equationShow the German source This equation expresses the law of aberration in its most general form.Show the German source If $\varphi = \pi/2$, the equation takes the simple form:Show the German source $$\cos \varphi' = - \frac{v}{V} .$$${\phi }^{\prime }$Angle to the motion, moving system$v$Speed of the moving system$V$Speed of lightExplain this equationShow the German source

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We still have to find the amplitude of the waves as it appears in the moving system.Show the German source If $A$ or $A'$ denotes the amplitude of the electric or magnetic force measured in the stationary or the moving system respectively, we obtain:Show the German source $$A'^2 = A^2 \frac{\left( 1 - \frac{v}{V} \cos \varphi \right)^2}{1 - \left( \frac{v}{V} \right)^2} ,$$$A^{\prime }$Light amplitude (moving system)$A$Light amplitude (stationary system)$v$Speed of the moving system$V$Speed of light$\phi$Angle to the motion, stationary systemExplain this equationShow the German source an equation which for $\varphi = 0$ passes over into the simpler one:Show the German source $$A'^2 = A^2 \frac{1 - \frac{v}{V}}{1 + \frac{v}{V}} .$$$A^{\prime }$Light amplitude (moving system)$A$Light amplitude (stationary system)$v$Speed of the moving system$V$Speed of lightExplain this equationShow the German source

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It follows from the equations developed that, for an observer approaching a light source with the velocity $V$, this light source would have to appear infinitely intense.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