I. Kinematical Part.Show the German source

II. Electrodynamical Part.Show the German source

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

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

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

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

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

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

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

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

Bern, June 1905.Show the German source

(Received 30 June 1905.)Show the German source