Editing Relativistic Doppler effect (section)

=== {{anchor|Motion along the line of sight}} Relativistic longitudinal Doppler effect===

Relativistic Doppler shift for the longitudinal case, with source and receiver moving directly towards or away from each other, is often derived as if it were the classical phenomenon, but modified by the addition of a [[time dilation]] term.<ref>{{cite journal |last1=Sher |first1=D. |title=The Relativistic Doppler Effect |journal=Journal of the Royal Astronomical Society of Canada |date=1968 |volume=62 |pages=105–111 |bibcode=1968JRASC..62..105S |url=http://adsbit.harvard.edu//full/1968JRASC..62..105S/0000105.000.html |access-date=11 October 2018}}</ref><ref name="Gill">{{cite book |last1=Gill |first1=T. P. |title=The Doppler Effect |date=1965 |publisher=Logos Press Limited |location=London |pages=6–9 |ol=5947329M }}</ref> This is the approach employed in first-year physics or mechanics textbooks such as those by Feynman<ref name=Feynman>{{cite book| title=The Feynman Lectures on Physics: Volume 1 | publisher=[[Addison-Wesley]] | location=Reading, Massachusetts |date=February 1977 | last1=Feynman | first1=Richard P. | author-link1=Richard Feynman | last2=Leighton | first2=Robert B. | author-link2=Robert B. Leighton | last3=Sands | first3=Matthew | author-link3=Matthew Sands | lccn=2010938208 | isbn=9780201021165 | pages=34–7 f |chapter-url=http://www.feynmanlectures.caltech.edu/I_34.html |chapter=Relativistic Effects in Radiation}}</ref> or Morin.<ref name=Morin/>

Following this approach towards deriving the relativistic longitudinal Doppler effect, assume the receiver and the source are moving ''away'' from each other with a relative speed <math>v\,</math> as measured by an observer on the receiver or the source (The sign convention adopted here is that <math>v\,</math> is ''negative'' if the receiver and the source are moving ''towards'' each other).

Consider the problem in the [[Frame of reference|reference frame]] of the source.

Suppose one [[wavefront]] arrives at the receiver. The next wavefront is then at a distance <math>\lambda_s = c/f_s\,</math> away from the receiver (where <math>\lambda_s\,</math> is the [[wavelength]], <math>f_s\,</math> is the [[frequency]] of the waves that the source emits, and <math>c\,</math> is the [[speed of light]]).

The wavefront moves with speed <math>c\,</math>, but at the same time the receiver moves away with speed <math>v</math> during a time <math>t_{r,s}</math>, which is the period of light waves impinging on the receiver, ''as observed in the frame of the source.'' So, <math display="block">\lambda_s + vt_{r,s} = ct_{r,s} \Longleftrightarrow \lambda_s = ct_{r,s}(1-v/c) \Longleftrightarrow t_{r,s} = \frac{1}{f_s (1-\beta)},</math>where <math>\beta = v / c\,</math> is the speed of the receiver in terms of the speed of light. The corresponding <math>f_{r,s}</math>, the frequency of at which wavefronts impinge on the receiver in the source's frame, is: <math display="block">f_{r,s} = 1/t_{r,s} = f_s(1-\beta).</math>

Thus far, the equations have been identical to those of the classical Doppler effect with a stationary source and a moving receiver.

However, due to relativistic effects, clocks on the receiver are [[time dilation|time dilated]] relative to clocks at the source: <math>t_r = t_{r,s}/ \gamma</math>, where <math display="inline">\gamma = 1/\sqrt{1 - \beta^2}</math> is the [[Lorentz factor]]. In order to know which time is dilated, we recall that <math>t_{r,s}</math> is the time in the frame in which the source is at rest. The receiver will measure the received frequency to be

{{NumBlk|| <math display="block">f_r = f_{r,s} \gamma = \frac{1 - \beta}{\sqrt{1-\beta^2}} f_s = \sqrt{\frac{1 - \beta}{1 + \beta}}\,f_s.</math>|{{EquationRef|1|Eq. 1}}}}

The ratio
<math display="block">\frac{f_s}{f_r} = \sqrt{\frac{1 + \beta}{1 - \beta}}</math>
is called the '''Doppler factor''' of the source relative to the receiver. (This terminology is particularly prevalent in the subject of [[astrophysics]]: see [[relativistic beaming]].)

The corresponding [[wavelength]]s are related by
{{NumBlk||<math display="block">\frac{\lambda_r}{\lambda_s} = \frac{f_s}{f_r} = \sqrt{\frac{1 + \beta}{1 - \beta}},</math>|{{EquationRef|2|Eq. 2}}}}

Identical expressions for relativistic Doppler shift are obtained when performing the analysis in the reference frame of the ''receiver'' with a moving source. This matches up with the expectations of the [[principle of relativity]], which dictates that the result can not depend on which object is considered to be the one at rest. In contrast, the classic nonrelativistic Doppler effect ''is'' dependent on whether it is the source or the receiver that is stationary with respect to the medium.<ref name=Feynman/><ref name=Morin/>