Editing Special relativity (section)

=== Relativistic aberration of light ===
{{main|Aberration of light|Light-time correction}}

[[File:Stellar aberration illustration.svg|thumb|Figure 5–2. Illustration of stellar aberration]]
Because of the finite speed of light, if the relative motions of a source and receiver include a transverse component, then the direction from which light arrives at the receiver will be displaced from the geometric position in space of the source relative to the receiver. The classical calculation of the displacement takes two forms and makes different predictions depending on whether the receiver, the source, or both are in motion with respect to the medium. (1) If the receiver is in motion, the displacement would be the consequence of the [[aberration of light]]. The incident angle of the beam relative to the receiver would be calculable from the vector sum of the receiver's motions and the velocity of the incident light.<ref name=Mould>{{cite book |title=Basic Relativity |page=8 |url=https://books.google.com/books?id=lfGE-wyJYIUC&pg=PA8 |isbn=978-0-387-95210-9 |date=2001 |publisher=Springer |author=Richard A. Mould |edition=2nd}}</ref> (2) If the source is in motion, the displacement would be the consequence of [[light-time correction]]. The displacement of the apparent position of the source from its geometric position would be the result of the source's motion during the time that its light takes to reach the receiver.<ref name="Seidelmann">{{cite book |editor1-last=Seidelmann |editor1-first=P. Kenneth |title=Explanatory Supplement to the Astronomical Almanac |date=1992 |publisher=University Science Books |location=ill Valley, Calif. |isbn=978-0-935702-68-2 |page=393 |url=https://archive.org/details/131123ExplanatorySupplementAstronomicalAlmanac/page/n209}}</ref>

The classical explanation failed experimental test. Since the aberration angle depends on the relationship between the velocity of the receiver and the speed of the incident light, passage of the incident light through a refractive medium should change the aberration angle. In 1810, [[François Arago|Arago]] used this expected phenomenon in a failed attempt to measure the speed of light,<ref name="Ferraro">{{cite journal |last1=Ferraro |first1=Rafael |last2=Sforza |first2=Daniel M. |title=European Physical Society logo Arago (1810): the first experimental result against the ether |journal=European Journal of Physics |volume=26 |pages=195–204 |doi=10.1088/0143-0807/26/1/020 |arxiv=physics/0412055 |year=2005 |issue=1 |bibcode=2005EJPh...26..195F |s2cid=119528074 }}</ref> and in 1870, [[George Airy]] tested the hypothesis using a water-filled telescope, finding that, against expectation, the measured aberration was identical to the aberration measured with an air-filled telescope.<ref name="Dolan">{{cite web |last1=Dolan |first1=Graham |title=Airy's Water Telescope (1870) |url=http://www.royalobservatorygreenwich.org/articles.php?article=1069 |publisher=The Royal Observatory Greenwich |access-date=20 November 2018}}</ref> A "cumbrous" attempt to explain these results used the hypothesis of partial aether-drag,<ref name="Hollis">{{cite journal |last1=Hollis |first1=H. P. |title=Airy's water telescope |journal=The Observatory |date=1937 |volume=60 |pages=103–107 |url=http://adsbit.harvard.edu//full/1937Obs....60..103H/0000105.000.html |access-date=20 November 2018|bibcode=1937Obs....60..103H }}</ref> but was incompatible with the results of the Michelson–Morley experiment, which apparently demanded ''complete'' aether-drag.<ref>{{cite book |author1=Janssen, Michel |author2=Stachel, John |editor1-last=Stachel |editor1-first=John |title=Going Critical |date=2004 |publisher=Springer |isbn=978-1-4020-1308-9 |chapter=The Optics and Electrodynamics of Moving Bodies |chapter-url=https://www.mpiwg-berlin.mpg.de/Preprints/P265.PDF}}</ref>

Assuming inertial frames, the relativistic expression for the aberration of light is applicable to both the receiver moving and source moving cases. A variety of trigonometrically equivalent formulas have been published. Expressed in terms of the variables in Fig.&nbsp;5-2, these include<ref name="Rindler1977"/>{{rp|57–60}}
: <math>\cos \theta ' = \frac{ \cos \theta + v/c}{ 1 + (v/c)\cos \theta}</math> &nbsp;&nbsp;'''OR'''&nbsp;&nbsp; <math> \sin \theta ' = \frac{\sin \theta}{\gamma [ 1 + (v/c) \cos \theta ]}</math> &nbsp;&nbsp;'''OR'''&nbsp;&nbsp; <math> \tan \frac{\theta '}{2} = \left( \frac{c - v}{c + v} \right)^{1/2} \tan \frac {\theta}{2}</math>