Editing Special relativity (section)

=== 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>