Editing Relativistic Doppler effect (section)

===Motion in an arbitrary direction===
[[File:Doppler shift with source and receiver moving at arbitrary angles.svg|thumb|Figure 7. Doppler shift with source moving at an arbitrary angle with respect to the line between source and receiver.]]

The analysis used in section [[#Relativistic longitudinal Doppler effect|Relativistic longitudinal Doppler effect]] can be extended in a straightforward fashion to calculate the Doppler shift for the case where the inertial motions of the source and receiver are at any specified angle.<ref name=Gill/><ref name=Brown_a>
{{cite web
| last=Brown |first=Kevin S.
| title = Doppler Shift for Sound and Light
| url = http://www.mathpages.com/rr/s2-04/2-04.htm
| publisher = Mathpages 
| access-date = 6 August 2015
}}</ref>
Fig.&nbsp;7 presents the scenario from the frame of the receiver, with the source moving at speed <math>v</math> at an angle <math>\theta_r</math> measured in the frame of the receiver. The radial component of the source's motion along the line of sight is equal to <math>v \cos{\theta_r}.</math>

The equation below can be interpreted as the classical Doppler shift for a stationary and moving source modified by the Lorentz factor <math>\gamma :</math>

{{NumBlk||<math display="block"> f_r = \frac{f_s}{\gamma\left(1 + \beta \cos\theta_r\right)}.</math>|{{EquationRef|6|Eq. 6}}}}

In the case when <math>\theta_r = 90^{\circ}</math>, one obtains the [[#Transverse Doppler effect|transverse Doppler effect]]:
<math display="block">f_r = \frac {f_s}  {\gamma}. </math>

{{anchor|Einstein Doppler shift equation}}
In his 1905 paper on special relativity,<ref group=p name=Einstein1905/> Einstein obtained a somewhat different looking equation for the Doppler shift equation. After changing the variable names in Einstein's equation to be consistent with those used here, his equation reads
{{NumBlk||<math display="block"> f_r = \gamma \left(  1 - \beta \cos \theta_s \right)  f_s.</math>|{{EquationRef|7|Eq. 7}}}}

The differences stem from the fact that Einstein evaluated the angle <math>\theta_s</math> with respect to the source rest frame rather than the receiver rest frame. <math> \theta_s</math> is not equal to <math> \theta_r </math> because of the effect of [[relativistic aberration]]. The relativistic aberration equation is:

{{NumBlk||<math display="block">\cos \theta_r=\frac{\cos \theta_s-\beta }{1-\beta \cos \theta_s} </math>|{{EquationRef|8|Eq. 8}}}}

Substituting the relativistic aberration equation {{EquationNote|8|Equation&nbsp;8}} into  {{EquationNote|6|Equation&nbsp;6}} yields  {{EquationNote|7|Equation&nbsp;7}}, demonstrating the consistency of these alternate equations for the Doppler shift.<ref name=Brown_a/>

Setting <math>\theta_r = 0</math> in {{EquationNote|6|Equation&nbsp;6}} or <math>\theta_s = 0</math> in {{EquationNote|7|Equation&nbsp;7}} yields {{EquationNote|1|Equation&nbsp;1}}, the expression for relativistic longitudinal Doppler shift.

A four-vector approach to deriving these results may be found in Landau and Lifshitz (2005).<ref name=Landau>{{cite book| title=The Classical Theory of Fields | series=Course of Theoretical Physics: Volume 2 | last1=Landau | first1=L.D. | author-link1=Lev Landau | last2=Lifshitz | first2=E.M. | author-link2=Evgeny Lifshitz | others=Trans. Morton Hamermesh | publisher=Elsevier Butterworth-Heinemann | year=2005 | edition=Fourth revised English | pages=116–117 | isbn=9780750627689}}</ref>

In electromagnetic waves both the electric and the magnetic field amplitudes ''E'' and ''B'' transform in a similar manner as the frequency:<ref>{{cite book |title=Einstein's Physics: Atoms, Quanta, and Relativity - Derived, Explained, and Appraised |author1=Ta-Pei Cheng |edition=illustrated, reprinted |publisher=OUP Oxford |year=2013 |isbn=978-0-19-966991-2 |page=164 |url=https://books.google.com/books?id=thXT19cY9jsC}} [https://books.google.com/books?id=thXT19cY9jsC&pg=PA164 Extract of page 164]</ref>
<math display="block">\begin{align}
 E_r &= \gamma \left( 1 - \beta \cos \theta_s \right) E_s \\
 B_r &= \gamma \left( 1 - \beta \cos \theta_s \right) B_s.
\end{align} </math>