Editing Special relativity (section)

== Optical effects ==
=== Dragging effects ===
{{main|Fizeau experiment}}
[[File:Fizeau experiment schematic.svg|thumb|300px|Figure 5–1. Highly simplified diagram of Fizeau's 1851 experiment.]]
In 1850, [[Hippolyte Fizeau]] and [[Léon Foucault]] independently established that light travels more slowly in water than in air, thus validating a prediction of [[Augustin-Jean Fresnel|Fresnel's]] [[wave theory of light]] and invalidating the corresponding prediction of Newton's [[Corpuscular theory of light|corpuscular theory]].<ref name=Lauginie2004>{{cite journal |last1=Lauginie |first1=P. |title=Measuring Speed of Light: Why? Speed of what? |journal=Proceedings of the Fifth International Conference for History of Science in Science Education |date=2004 |url=http://sci-ed.org/documents/Lauginie-M.pdf|access-date=3 July 2015 |archive-url=https://web.archive.org/web/20150704043700/http://sci-ed.org/documents/Lauginie-M.pdf |archive-date=4 July 2015}}</ref> The speed of light was measured in still water. What would be the speed of light in flowing water?

In 1851, Fizeau conducted an experiment to answer this question, a simplified representation of which is illustrated in Fig.&nbsp;5-1. A beam of light is divided by a beam splitter, and the split beams are passed in opposite directions through a tube of flowing water. They are recombined to form interference fringes, indicating a difference in optical path length, that an observer can view. The experiment demonstrated that dragging of the light by the flowing water caused a displacement of the fringes, showing that the motion of the water had affected the speed of the light.

According to the theories prevailing at the time, light traveling through a moving medium would be a simple sum of its speed ''through'' the medium plus the speed ''of'' the medium. Contrary to expectation, Fizeau found that although light appeared to be dragged by the water, the magnitude of the dragging was much lower than expected. If <math>u' = c/n</math> is the speed of light in still water, and <math>v</math> is the speed of the water, and <math> u_{\pm} </math> is the water-borne speed of light in the lab frame with the flow of water adding to or subtracting from the speed of light, then
<math display="block">u_{\pm} =\frac{c}{n} \pm v\left(1-\frac{1}{n^2}\right) \ . </math>

Fizeau's results, although consistent with Fresnel's earlier hypothesis of [[Aether drag hypothesis|partial aether dragging]], were extremely disconcerting to physicists of the time. Among other things, the presence of an index of refraction term meant that, since <math>n</math> depends on wavelength, ''the aether must be capable of sustaining different motions at the same time''.{{refn|group=note|The refractive index dependence of the presumed partial aether-drag was eventually confirmed by [[Pieter Zeeman]] in 1914–1915, long after special relativity had been accepted by the mainstream. Using a scaled-up version of Michelson's apparatus connected directly to [[Amsterdam]]'s main water conduit, Zeeman was able to perform extended measurements using monochromatic light ranging from violet (4358 Å) through red (6870 Å).<ref name=zee1 group=p>{{Cite journal|author=Zeeman, Pieter |title=Fresnel's coefficient for light of different colours. (First part) |journal=Proc. Kon. Acad. Van Weten.|volume=17|year=1914|pages=445–451|url=https://archive.org/details/p1proceedingsofs17akad|bibcode=1914KNAB...17..445Z}}</ref><ref name=zee2 group=p>{{Cite journal|author=Zeeman, Pieter |title=Fresnel's coefficient for light of different colours. (Second part)|journal=Proc. Kon. Acad. Van Weten.|volume=18|year=1915 |pages=398–408 |url=https://archive.org/details/proceedingsofsec181koni|bibcode=1915KNAB...18..398Z}}</ref>}} A variety of theoretical explanations were proposed to explain [[Fizeau experiment#Fresnel drag coefficient|Fresnel's dragging coefficient]], that were completely at odds with each other. Even before the Michelson–Morley experiment, Fizeau's experimental results were among a number of observations that created a critical situation in explaining the optics of moving bodies.<ref name=Stachel2005>{{cite book |last=Stachel |first=J. |title=The universe of general relativity |year=2005 |publisher=Birkhäuser |location=Boston |isbn=978-0-8176-4380-5 |pages=1–13 |chapter-url=https://books.google.com/books?id=-KlBhDwUKF8C&pg=PA1 |editor=Kox, A.J. |editor2=Eisenstaedt, J |access-date=17 April 2012 |chapter=Fresnel's (dragging) coefficient as a challenge to 19th century optics of moving bodies}}</ref>

From the point of view of special relativity, Fizeau's result is nothing but an approximation to {{EquationNote|10|Equation&nbsp;10}}, the relativistic formula for composition of velocities.<ref name=Rindler1977/>
: <math>u_{\pm} = \frac{u' \pm v}{ 1 \pm u'v/c^2 } =</math> <math> \frac {c/n \pm v}{ 1 \pm v/cn } \approx</math> <math> c \left( \frac{1}{n} \pm \frac{v}{c} \right) \left( 1 \mp \frac{v}{cn} \right) \approx </math> <math> \frac{c}{n} \pm v \left( 1 - \frac{1}{n^2} \right) </math>

=== 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>

=== Relativistic Doppler effect ===
{{main|Relativistic Doppler effect}}

==== Relativistic longitudinal Doppler effect====
The classical Doppler effect depends on whether the source, receiver, or both are in motion with respect to the medium. The relativistic Doppler effect is independent of any medium. Nevertheless, relativistic Doppler shift for the longitudinal case, with source and receiver moving directly towards or away from each other, can be derived as if it were the classical phenomenon, but modified by the addition of a [[time dilation]] term, and that is the treatment described here.<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>

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). Assume that the source is stationary in the medium. Then
<math display="block">f_{r} = \left(1 - \frac v {c_s} \right) f_s</math>
where <math>c_s</math> is the speed of sound.

For light, and with the receiver moving at relativistic speeds, clocks on the receiver are [[time dilation|time dilated]] relative to clocks at the source. The receiver will measure the received frequency to be
<math display="block">f_r = \gamma\left(1 - \beta\right) f_s = \sqrt{\frac{1 - \beta}{1 + \beta}}\,f_s.</math>
where
* <math>\beta = v/c </math>&nbsp; and
* <math>\gamma = \frac{1}{\sqrt{1 - \beta^2}}</math> is the [[Lorentz factor]].

An identical expression for relativistic Doppler shift is obtained when performing the analysis in the reference frame of the ''receiver'' with a moving source.<ref name=Feynman1977>{{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=https://feynmanlectures.caltech.edu/I_34.html |chapter=Relativistic Effects in Radiation}}</ref><ref name=Morin2007/>

==== Transverse Doppler effect ====
[[File:Transverse Doppler effect scenarios 5.svg|thumb|300px|Figure 5–3. Transverse Doppler effect for two scenarios: (a) receiver moving in a circle around the source; (b) source moving in a circle around the receiver.]]

The transverse [[Doppler effect]] is one of the main novel predictions of the special theory of relativity.

Classically, one might expect that if source and receiver are moving transversely with respect to each other with no longitudinal component to their relative motions, that there should be no Doppler shift in the light arriving at the receiver.

Special relativity predicts otherwise. Fig.&nbsp;5-3 illustrates two common variants of this scenario. Both variants can be analyzed using simple time dilation arguments.<ref name=Morin2007/> In Fig.&nbsp;5-3a, the receiver observes light from the source as being blueshifted by a factor of {{tmath|1= \gamma }}. In Fig.&nbsp;5-3b, the light is redshifted by the same factor.

=== Measurement versus visual appearance ===
{{main|Terrell rotation}}
[[File:Animated Terrell Rotation - Cube.gif|thumb|330px|Figure 5–4. Comparison of the measured length contraction of a cube versus its visual appearance.]]
Time dilation and length contraction are not optical illusions, but genuine effects. Measurements of these effects are not an artifact of [[Doppler shift]], nor are they the result of neglecting to take into account the time it takes light to travel from an event to an observer.

Scientists make a fundamental distinction between ''measurement'' or ''observation'' on the one hand, versus ''visual appearance'', or what one ''sees''. The measured shape of an object is a hypothetical snapshot of all of the object's points as they exist at a single moment in time. But the visual appearance of an object is affected by the varying lengths of time that light takes to travel from different points on the object to one's eye.

[[File:Terrell Rotation Sphere.gif|thumb|330px|Figure 5–5. Comparison of the measured length contraction of a globe versus its visual appearance, as viewed from a distance of three diameters of the globe from the eye to the red cross.]]
For many years, the distinction between the two had not been generally appreciated, and it had generally been thought that a length contracted object passing by an observer would in fact actually be ''seen'' as length contracted. In 1959, James Terrell and [[Roger Penrose]] independently pointed out that differential time lag effects in signals reaching the observer from the different parts of a moving object result in a fast moving object's visual appearance being quite different from its measured shape. For example, a receding object would ''appear'' contracted, an approaching object would ''appear'' elongated, and a passing object would have a skew appearance that has been likened to a rotation.<ref name="Terrell" group=p>{{cite journal |last1=Terrell |first1=James |title=Invisibility of the Lorentz Contraction |journal=[[Physical Review]] |date=15 November 1959 |volume=116 |issue=4 |pages=1041–1045 |doi=10.1103/PhysRev.116.1041 |bibcode=1959PhRv..116.1041T }}</ref><ref name="Penrose" group=p>{{cite journal |last1=Penrose |first1=Roger | authorlink1=Roger Penrose |title=The Apparent Shape of a Relativistically Moving Sphere |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] |date=24 October 2008 |volume=55 |issue=1 |pages=137–139 |doi=10.1017/S0305004100033776|bibcode=1959PCPS...55..137P |s2cid=123023118 }}</ref><ref>{{cite web |last1=Cook|first1=Helen |title=Relativistic Distortion |url=http://www.math.ubc.ca/~cass/courses/m309-01a/cook/ |publisher=Mathematics Department, University of British Columbia |access-date=12 April 2017 }}</ref><ref>{{cite web |last1=Signell |first1=Peter |title=Appearances at Relativistic Speeds |url=https://stuff.mit.edu/afs/athena/course/8/8.20/www/m44.pdf |website=Project PHYSNET |publisher=Michigan State University, East Lansing, MI |access-date=12 April 2017 |archive-url=https://web.archive.org/web/20170413153459/https://stuff.mit.edu/afs/athena/course/8/8.20/www/m44.pdf |archive-date=13 April 2017 |url-status=dead }}</ref> A sphere in motion retains the circular outline for all speeds, for any distance, and for all view angles, although
the surface of the sphere and the images on it will appear distorted.<ref>{{cite web |last1=Kraus |first1=Ute |title=The Ball is Round |url=http://www.spacetimetravel.org/fussball/fussball.html |website=Space Time Travel: Relativity visualized |publisher=Institut für Physik Universität Hildesheim |access-date=16 April 2017 |archive-url=https://web.archive.org/web/20170512165032/http://www.spacetimetravel.org/fussball/fussball.html |archive-date=12 May 2017 |url-status=dead }}</ref><ref name="Boas_1961">{{cite journal |last1=Boas |first1=Mary L. |title=Apparent Shape of Large Objects at Relativistic Speeds |journal=American Journal of Physics |date=1961 |volume=29 |issue=5 |page=283 |doi=10.1119/1.1937751|bibcode=1961AmJPh..29..283B }}</ref>

[[File:M87 jet (1).jpg|thumb|Figure 5–6. Galaxy [[Messier 87|M87]] sends out a black-hole-powered jet of electrons and other sub-atomic particles traveling at nearly the speed of light.]]
Both Fig.&nbsp;5-4 and Fig.&nbsp;5-5 illustrate objects moving transversely to the line of sight. In Fig.&nbsp;5-4, a cube is viewed from a distance of four times the length of its sides. At high speeds, the sides of the cube that are perpendicular to the direction of motion appear hyperbolic in shape. The cube is actually not rotated. Rather, light from the rear of the cube takes longer to reach one's eyes compared with light from the front, during which time the cube has moved to the right. At high speeds, the sphere in Fig.&nbsp;5-5 takes on the appearance of a flattened disk tilted up to 45° from the line of sight. If the objects' motions are not strictly transverse but instead include a longitudinal component, exaggerated distortions in perspective may be seen.<ref name="Muller_2014">{{cite journal |last1=Müller |first1=Thomas |last2=Boblest |first2=Sebastian |title=Visual appearance of wireframe objects in special relativity |journal=European Journal of Physics |date=2014 |volume=35 |issue=6 |page=065025 |doi=10.1088/0143-0807/35/6/065025|arxiv=1410.4583 |bibcode=2014EJPh...35f5025M |s2cid=118498333 }}</ref> This illusion has come to be known as ''[[Terrell rotation]]'' or the ''Terrell–Penrose effect''.<ref group=note>Even though it has been many decades since Terrell and Penrose published their observations, popular writings continue to conflate measurement versus appearance. For example, Michio Kaku wrote in ''Einstein's Cosmos'' (W. W. Norton & Company, 2004. p. 65): "...&nbsp;imagine that the speed of light is only 20 miles per hour. If a car were to go down the street, it might look compressed in the direction of motion, being squeezed like an accordion down to perhaps 1 inch in length."</ref>

Another example where visual appearance is at odds with measurement comes from the observation of apparent [[superluminal motion]] in various [[radio galaxies]], [[BL Lac objects]], [[quasars]], and other astronomical objects that eject [[astrophysical jet|relativistic-speed jets]] of matter at narrow angles with respect to the viewer. An apparent optical illusion results giving the appearance of faster than light travel.<ref>{{cite book|last1=Zensus|first1=J. Anton|last2=Pearson|first2=Timothy J.|title=Superluminal Radio Sources|date=1987|publisher=Cambridge University Press|location=Cambridge, New York|isbn=9780521345606|page=3|edition=1st}}</ref><ref>{{cite web|last1=Chase|first1=Scott I.|title=Apparent Superluminal Velocity of Galaxies|url=http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/Superluminal/superluminal.html|website=The Original Usenet Physics FAQ|publisher=Department of Mathematics, University of California, Riverside|access-date=12 April 2017}}</ref><ref>{{cite web|last1=Richmond|first1=Michael|title="Superluminal" motions in astronomical sources|url=http://spiff.rit.edu/classes/phys200/lectures/superlum/superlum.html|website=Physics 200 Lecture Notes|publisher=School of Physics and Astronomy, Rochester Institute of Technology|access-date=20 April 2017|archive-url=https://web.archive.org/web/20170216155045/http://spiff.rit.edu/classes/phys200/lectures/superlum/superlum.html|archive-date=16 February 2017|url-status=dead}}</ref> In Fig.&nbsp;5-6, galaxy [[Messier 87|M87]] streams out a high-speed jet of subatomic particles almost directly towards us, but Penrose–Terrell rotation causes the jet to appear to be moving laterally in the same manner that the appearance of the cube in Fig.&nbsp;5-4 has been stretched out.<ref>{{cite web|last1=Keel|first1=Bill|title=Jets, Superluminal Motion, and Gamma-Ray Bursts|url=http://pages.astronomy.ua.edu/keel/galaxies/jets.html|website=Galaxies and the Universe – WWW Course Notes|publisher=Department of Physics and Astronomy, University of Alabama|access-date=29 April 2017|archive-url=https://web.archive.org/web/20170301030027/http://pages.astronomy.ua.edu/keel/galaxies/jets.html|archive-date=1 March 2017|url-status=dead}}</ref>