Editing Special relativity (section)

=== Time dilation ===
{{See also|Time dilation}}
The time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers' reference frames.

Suppose a [[clock]] is at rest in the unprimed system ''S''. The location of the clock on two different ticks is then characterized by {{nowrap|1=Δ''x'' = 0}}. To find the relation between the times between these ticks as measured in both systems, {{EquationNote|3|Equation&nbsp;3}} can be used to find:
: <math>\Delta t' = \gamma\, \Delta t </math>{{pad|4}}for events satisfying{{pad|4}}<math>\Delta x = 0 \ .</math>

This shows that the time (Δ{{prime|''t''}}) between the two ticks as seen in the frame in which the clock is moving ({{prime|''S''}}), is ''longer'' than the time (Δ''t'') between these ticks as measured in the rest frame of the clock (''S''). Time dilation explains a number of physical phenomena; for example, the lifetime of high speed [[muon]]s created by the collision of cosmic rays with particles in the Earth's outer atmosphere and moving towards the surface is greater than the lifetime of slowly moving muons, created and decaying in a laboratory.<ref>{{cite book|author1=Daniel Kleppner|author2=David Kolenkow|name-list-style=amp|title=An Introduction to Mechanics|date=1973|pages=[https://archive.org/details/introductiontome00dani/page/468 468–70]|publisher=McGraw-Hill |isbn=978-0-07-035048-9|url=https://archive.org/details/introductiontome00dani/page/468}}</ref>

[[File:Observer_in_special_relativity.svg|thumb|Figure 4–2. Hypothetical infinite array of synchronized clocks associated with an observer's reference frame]]
Whenever one hears a statement to the effect that "moving clocks run slow", one should envision an inertial reference frame thickly populated with identical, synchronized clocks. As a moving clock travels through this array, its reading at any particular point is compared with a stationary clock at the same point.<ref name="French_1968"/>{{rp|149–152}}

The measurements that we would get if we actually ''looked'' at a moving clock would, in general, not at all be the same thing, because the time that we would see would be delayed by the finite speed of light, i.e. the times that we see would be distorted by the [[Doppler effect]]. Measurements of relativistic effects must always be understood as having been made after finite speed-of-light effects have been factored out.<ref name="French_1968">{{cite book |last1=French |first1=A. P. |title=Special Relativity |date=1968 |publisher=W. W. Norton & Company |location=New York |isbn=0-393-09793-5}}</ref>{{rp|149–152}}

==== Langevin's light-clock ====
{{anchor|Langevin's Light-Clock}}
[[File:Langevin Light Clock.gif|thumb|320px|Figure 4–3. Thought experiment using a light-clock to explain time dilation]]
[[Paul Langevin]], an early proponent of the theory of relativity, did much to popularize the theory in the face of resistance by many physicists to Einstein's revolutionary concepts. Among his numerous contributions to the foundations of special relativity were independent work on the mass–energy relationship, a thorough examination of the twin paradox, and investigations into rotating coordinate systems. His name is frequently attached to a hypothetical construct called a "light-clock" (originally developed by Lewis and Tolman in 1909<ref name="Lewis_Tolman_1909">{{cite journal |last1=Lewis |first1=Gilbert Newton |authorlink1=Gilbert N. Lewis|last2=Tolman |first2=Richard Chase| authorlink2=Richard Chase Tolman| title=The Principle of Relativity, and Non-Newtonian Mechanics |journal=Proceedings of the American Academy of Arts and Sciences |date=1909 |volume=44 |issue=25 |pages=709–726 |doi=10.2307/20022495 |jstor=20022495 |url=https://en.wikisource.org/wiki/The_Principle_of_Relativity,_and_Non-Newtonian_Mechanics |access-date=22 August 2023}}</ref>), which he used to perform a novel derivation of the Lorentz transformation.<ref name="Cuvaj_1971">{{cite journal |last1=Cuvaj |first1=Camillo |title=Paul Langeyin and the Theory of Relativity |journal=Japanese Studies in the History of Science |date=1971 |volume=10 |pages=113–142 |url=http://www.isc.meiji.ac.jp/~sano/hssj/pdf/Cuvaj_C-1972-Langevin_Relativity-JSHS-No_10-pp113-142.pdf |access-date=12 June 2023}}</ref>

A light-clock is imagined to be a box of perfectly reflecting walls wherein a light signal reflects back and forth from opposite faces. The concept of time dilation is frequently taught using a  light-clock that is traveling in uniform inertial motion perpendicular to a line connecting the two mirrors.<ref>{{Cite book |last1=Cassidy |first1=David C. |url=https://books.google.com/books?id=rpQo7f9F1xUC&pg=PA422 |title=Understanding Physics |last2=Holton |first2=Gerald James |last3=Rutherford |first3=Floyd James |publisher=[[Springer-Verlag]] |year=2002 |isbn=978-0-387-98756-9 |pages=422}}</ref><ref>{{Cite book |last=Cutner |first=Mark Leslie |url=https://books.google.com/books?id=2QVmiMW0O0MC&pg=PA128 |title=Astronomy, A Physical Perspective |publisher=[[Cambridge University Press]] |year=2003 |isbn=978-0-521-82196-4 |page=128}}</ref><ref>{{Cite book |last1=Ellis |first1=George F. R. |url=https://books.google.com/books?id=Hos31wty5WIC&pg=PA28 |title=Flat and Curved Space-times |last2=Williams |first2=Ruth M. |publisher=[[Oxford University Press]] |year=2000 |isbn=978-0-19-850657-7 |edition=2n |pages=28–29}}</ref><ref name="Feynman_Lectures_1">{{cite book |last1=Feynman |first1=Richard P. |last2=Leighton |first2=Robert B. |last3=Sands |first3=Matthew |title=The feynman lectures on physics; vol I: The new millennium edition |date=2011 |publisher=Basic Books |isbn=978-0-465-02414-8 |page=15-5 |url=https://www.feynmanlectures.caltech.edu/I_15.html |access-date=12 June 2023}}</ref> (Langevin himself made use of a light-clock oriented parallel to its line of motion.<ref name="Cuvaj_1971"/>)

Consider the scenario illustrated in {{nowrap|Fig. 4-3A.}} Observer A holds a light-clock of length <math>L</math> as well as an electronic timer with which she measures how long it takes a pulse to make a round trip up and down along the light-clock. Although observer A is traveling rapidly along a train, from her point of view the emission and receipt of the pulse occur at the same place, and she measures the interval using a single clock located at the precise position of these two events. For the interval between these two events, observer A finds {{tmath|1= t_\text{A} = 2 L/c }}. A time interval measured using a single clock that is motionless in a particular reference frame is called a ''[[proper time interval]]''.<ref name="Halliday_1988">{{cite book |last1=Halliday |first1=David |last2=Resnick |first2=Robert |title=Fundamental Physics: Extended Third Edition |date=1988 |publisher=John Wiley & sons |location=New York |isbn=0-471-81995-6 |pages=958–959}}</ref>

Fig. 4-3B illustrates these same two events from the standpoint of observer B, who is parked by the tracks as the train goes by at a speed of {{tmath|1= v }}. Instead of making straight up-and-down motions, observer B sees the pulses moving along a zig-zag line. However, because of the postulate of the constancy of the speed of light, the speed of the pulses along these diagonal lines is the same <math>c</math> that observer A saw for her up-and-down pulses. B measures the speed of the vertical component of these pulses as <math display=inline>\pm \sqrt{c^2 - v^2},</math> so that the total round-trip time of the pulses is <math display=inline>t_\text{B} = 2L \big/ \sqrt{ c^2 - v^2 } = {}</math>{{tmath|1= \textstyle t_\text{A} \big/ \sqrt {1 - v^2 / c^2} }}. Note that for observer B, the emission and receipt of the light pulse occurred at different places, and he measured the interval using two stationary and synchronized clocks located at two different positions in his reference frame. The interval that B measured was therefore ''not'' a proper time interval because he did not measure it with a single resting clock.<ref name="Halliday_1988"/>

==== Reciprocal time dilation ====
In the above description of the Langevin light-clock, the labeling of one observer as stationary and the other as in motion was completely arbitrary. One could just as well have observer B carrying the light-clock and moving at a speed of <math>v</math> to the left, in which case observer A would perceive B's clock as running slower than her local clock.

There is no paradox here, because there is no independent observer C who will agree with both A and B. Observer C necessarily makes his measurements from his own reference frame. If that reference frame coincides with A's reference frame, then C will agree with A's measurement of time. If C's reference frame coincides with B's reference frame, then C will agree with B's measurement of time. If C's reference frame coincides with neither A's frame nor B's frame, then C's measurement of time will disagree with ''both'' A's and B's measurement of time.<ref>{{Cite book |last=Adams |first=Steve |url=https://books.google.com/books?id=1RV0AysEN4oC&pg=PA54 |title=Relativity: An introduction to space-time physics |publisher=[[CRC Press]] |year=1997 |isbn=978-0-7484-0621-0 |page=54}}</ref>