The Relativistic Doppler Effect - Maple Help

Relativistic Doppler Effect

 Main Concept The relativistic Doppler effect is a phenomenon in which the wavelength (and frequency) of electromagnetic waves changes due to the relative motion of their source and the observer. It has two components: the classical Doppler effect (analogous to the perceived change of pitch when the source of sound is in motion) and the Einstein redshift effect which has no counterpart in the Doppler effect for sound. The redshift effect contributes the characteristic dilation factor $\mathrm{γ}=\frac{1}{\sqrt[]{1-\frac{{v}^{2}}{{c}^{2}}}}$ where $v$ is the relative velocity of the observer with respect to the source and $c\doteq 3\cdot {10}^{8}m/s$ is the speed of light in vacuum. This factor is present even when the source and the observer are at the point of nearest approach, seeing the frequency of the emitted light shifted while the sound waves would remain unaffected in a similar situation.   For a source emitting electromagnetic light of wavelength ${\mathrm{λ}}_{0}$ moving with velocity $v$ relative to the observer, the observed wavelength of the waves is shifted according to ${\mathrm{λ}}_{\mathrm{shift}}={\mathrm{λ}}_{0}\cdot \mathrm{γ}\cdot \left(1+\frac{v}{c}\cdot {\mathrm{cosθ}}_{\mathrm{shift}}\right)$ where ${\mathrm{θ}}_{\mathrm{shift}}$ is the relative angle of the observer to the source at the time the light is emitted, as perceived by the observer. The angle ${\mathrm{θ}}_{\mathrm{shift}}=\frac{\mathrm{π}}{2}$ gives the point of nearest approach, resulting in the transverse Doppler effect ${\mathrm{λ}}_{\mathrm{shift}}/{\mathrm{λ}}_{0}=\mathrm{γ}$. As $\mathrm{γ}>1$, the observed light will be shifted towards higher wavelengths (and lower frequencies).   For visible spectrum (wavelength ~ 380 - 740 nanometers), the relativistic Doppler effect and the aberration of light result in a shift in colors and the perceived direction from which the light arrives. This animation shows both effects from the point of view of an observer moving at different (constant) velocities towards the upper edge of the frame, observing uniform monochromatic light source.

Select the initial wavelength using the slider or pick a color from the spectrum, select a resolution and click "Start" to view the animation.

 Velocity (in units of $c$):     ${\mathrm{λ}}_{0}=$ $v$ = Resolution: 400 x 300300 x 225200 x 150100 x 75 Choose the initial wavelength by selecting the corresponding color or by adjusting the slider: Initial wavelength (in nm)

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