37-5 Doppler Effect for Light

37-5 Doppler Effect for Light#

Prompts

  • How does the Doppler effect for light differ from that for sound? Why does it depend only on the relative velocity between source and detector?

  • When the source–detector separation is increasing (receding), is the detected wavelength red-shifted or blue-shifted? Write the formula for \(f\) and \(\lambda\) in terms of \(f_0\), \(\lambda_0\), and \(\beta = v/c\).

  • For low speeds (\(v \ll c\)), show that \(|\Delta\lambda|/\lambda_0 \approx v/c\). How is this used in astronomy to measure radial velocities?

  • Explain the transverse Doppler effect: when the source moves perpendicular to the line of sight, why is there still a frequency shift? What is the formula? (Hint: time dilation.)

  • A police radar uses a microwave beam. A car reflects it. Why does the radar unit detect a frequency shifted up when the car approaches? (Consider both the car as “detector” and as “re-emitter.”)

Lecture Notes#

Overview#

  • The Doppler effect for light is a shift in detected frequency (or wavelength) when source and detector move relative to each other. Unlike sound, it depends only on the relative velocity—no medium is required.

  • Proper frequency \(f_0\) and proper wavelength \(\lambda_0\) are measured in the rest frame of the source.

  • Receding (separation increasing): red shift—\(\lambda > \lambda_0\), \(f < f_0\). Approaching (separation decreasing): blue shift—\(\lambda < \lambda_0\), \(f > f_0\).

  • Transverse Doppler effect: when the source moves perpendicular to the line of sight, there is still a shift (to lower frequency) due to time dilation—a purely relativistic effect with no classical analog.

Longitudinal Doppler effect#

Let \(f_0\) be the proper frequency (measured in the source’s rest frame) and \(f\) the frequency detected by an observer moving with speed \(v\) relative to the source. The radial speed \(v\) is the component of velocity along the line joining source and detector.

Source and detector receding (separation increasing):

(334)#\[ f = f_0\sqrt{\frac{1 - \beta}{1 + \beta}}, \quad \lambda = \lambda_0\sqrt{\frac{1 + \beta}{1 - \beta}} \]

Source and detector approaching (separation decreasing): reverse the signs in front of \(\beta\):

(335)#\[ f = f_0\sqrt{\frac{1 + \beta}{1 - \beta}}, \quad \lambda = \lambda_0\sqrt{\frac{1 - \beta}{1 + \beta}} \]

where \(\beta = v/c\). Red shift = longer \(\lambda\); blue shift = shorter \(\lambda\). (The terms refer to the visible spectrum but apply to any wavelength.)

Low-speed approximation#

For \(v \ll c\) (\(\beta \ll 1\)), the wavelength shift magnitude is approximately

(336)#\[ |\Delta\lambda| = |\lambda - \lambda_0| \approx \lambda_0\,\frac{v}{c} \]

This is used in astronomy to measure the radial velocity of stars and galaxies: \(v \approx c\,|\Delta\lambda|/\lambda_0\). Only the radial component of velocity produces a Doppler shift; transverse motion does not contribute to the longitudinal formula.

Transverse Doppler effect#

When the source’s velocity is perpendicular to the line joining source and detector (e.g., source passes the detector at closest approach), the classical Doppler effect would predict no shift. In relativity, there is still a shift because of time dilation: the source’s “clock” (its emission rate) runs slow as measured by the detector.

(337)#\[ f = \frac{f_0}{\gamma} = f_0\sqrt{1 - \beta^2} \]

The detected frequency is lower than the proper frequency. This effect is purely relativistic; it does not exist for sound.

Applications#

Police radar: A radar unit emits microwaves at frequency \(f_0\). A car approaching reflects the beam. The car “sees” a blue-shifted frequency (approaching source); it re-emits that frequency. The radar unit receives a further blue-shifted signal (approaching detector). The unit compares the detected frequency with \(f_0\) to compute the car’s speed.

Astronomy: Spectral lines from stars and galaxies are shifted. Red shift indicates recession (e.g., expanding universe); blue shift indicates approach. The radial velocity is \(v \approx c\,\Delta\lambda/\lambda_0\) for \(v \ll c\).

Example: galactic red shift

A galaxy’s hydrogen line (rest wavelength \(\lambda_0 = 656\) nm) is observed at \(\lambda = 662\) nm. Red shift \(\Delta\lambda = 6\) nm. Radial recession speed: \(v \approx c\,\Delta\lambda/\lambda_0 = 3\times 10^8 \times (6/656) \approx 2.7\times 10^6\) m/s \(\approx 0.009c\). (For larger \(v\), use the exact relativistic formula.)

Summary#

  • Longitudinal Doppler: receding \(f = f_0\sqrt{(1-\beta)/(1+\beta)}\); approaching—reverse \(\beta\) signs.

  • Red shift (longer \(\lambda\)): separation increasing. Blue shift (shorter \(\lambda\)): separation decreasing.

  • Low-speed: \(|\Delta\lambda|/\lambda_0 \approx v/c\); used for radial velocities in astronomy.

  • Transverse Doppler: \(f = f_0/\gamma\); due to time dilation; no classical analog.