Editing Time dilation (section)

===Simple inference===
[[Image:Time-dilation-002-mod.svg|thumb|center|upright=3|'''Left''': Observer at rest measures time 2''L''/''c'' between co-local events of light signal generation at A and arrival at A.<br />'''Right''': Events according to an observer moving to the left of the setup: bottom mirror A when signal is generated at time ''t'=''0, top mirror B when signal gets reflected at time ''t'=D/c'', bottom mirror A when signal returns at time ''t'=2D/c'']]
Time dilation can be inferred from the observed constancy of the speed of light in all reference frames dictated by the [[Special relativity#Postulates|second postulate of special relativity]]. This constancy of the speed of light means that, counter to intuition, the speeds of material objects and light are not additive. It is not possible to make the speed of light appear greater by moving towards or away from the light source.<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 |last=Lerner |first=Lawrence S. |url=https://books.google.com/books?id=B8K_ym9rS6UC&pg=PA1051 |title=Physics for Scientists and Engineers, Volume 2 |publisher=[[Jones and Bartlett]] |year=1996 |isbn=978-0-7637-0460-5 |pages=1051–1052}}</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=2nd |pages=28–29}}</ref>

Consider then, a simple vertical clock consisting of two mirrors {{math|A}} and {{math|B}}, between which a light pulse is bouncing. The separation of the mirrors is {{math|''L''}} and the clock ticks once each time the light pulse hits mirror {{math|A}}.

In the frame in which the clock is at rest (see left part of the diagram), the light pulse traces out a path of length {{math|2''L''}} and the time period between the ticks of the clock <math>\Delta t</math> is equal to {{math|2''L''}} divided by the speed of light {{math|''c''}}: 
:<math>\Delta t = \frac{2 L}{c}</math>

From the frame of reference of a moving observer traveling at the speed {{math|''v''}} relative to the resting frame of the clock (right part of diagram), the light pulse is seen as tracing out a longer, angled path {{math|2''D''}}. Keeping the speed of light constant for all inertial observers requires a lengthening (that is dilation) of the time period between the ticks of this clock <math>\Delta t'</math> from the moving observer's perspective. That is to say, as measured in a frame moving relative to the local clock, this clock will be running (that is ticking) more slowly, since tick rate equals one over the time period between ticks 1/<math>\Delta t'</math>. 

Straightforward application of the [[Pythagorean theorem]] leads to the well-known prediction of special relativity:

The total time for the light pulse to trace its path is given by:
:<math>\Delta t' = \frac{2 D}{c}</math>
The length of the half path can be calculated as a function of known quantities as:
:<math>D = \sqrt{\left (\frac{1}{2}v \Delta t'\right )^2 + L^2}</math>
Elimination of the variables {{math|''D''}} and {{math|''L''}} from these three equations results in:
{{Equation box 1
|indent= :
|title='''Time dilation equation'''
|equation=<math> \Delta t' = \frac{\Delta t}{\sqrt{1 - \frac{v^2}{c^2}}} = {\gamma}{\Delta t} </math>
|cellpadding =10
|border
|border colour = #50C878
|background colour = #ECFCF4}}
which expresses the fact that the moving observer's period of the clock <math>\Delta t'</math> is longer than the period <math>\Delta t</math> in the frame of the clock itself. The [[Lorentz factor]] gamma ({{math|''γ''}}) is defined as<ref name="Forshaw 2014">{{cite book |last1=Forshaw |first1=Jeffrey |last2=Smith |first2=Gavin |title=Dynamics and Relativity |publisher=[[John Wiley & Sons]] |date=2014 |isbn=978-1-118-93329-9 |url=https://books.google.com/books?id=5TaiAwAAQBAJ }}</ref>
:<math>\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}</math>
Because all clocks that have a common period in the resting frame should have a common period when observed from the moving frame, all other clocks{{em dash}}mechanical, electronic, optical (such as an identical horizontal version of the clock in the example){{em dash}}should exhibit the same velocity-dependent time dilation.<ref>{{Cite journal |last1=Galli |first1=J. Ronald |last2=Amiri |first2=Farhang |date=Apr 2012 |title=The Square Light Clock and Special Relativity |journal=[[The Physics Teacher]] |publisher=[[American Association of Physics Teachers]] |volume=50 |issue=4 |page=212 |bibcode=2012PhTea..50..212G |doi=10.1119/1.3694069 |s2cid=120089462}}</ref>