Editing Lorentz transformation (section)

===Physical implications===

A critical requirement of the Lorentz transformations is the invariance of the speed of light, a fact used in their derivation, and contained in the transformations themselves. If in {{mvar|F}} the equation for a pulse of light along the {{mvar|x}} direction is {{math|1=''x'' = ''ct''}}, then in {{mvar|F′}} the Lorentz transformations give {{math|1=''x′'' = ''ct′''}}, and vice versa, for any {{math|−''c'' < ''v'' < ''c''}}.

For relative speeds much less than the speed of light, the Lorentz transformations reduce to the [[Galilean transformation]]:<ref>{{cite book |title=International Edition University Physics |author1=George Arfken |edition= |publisher=Elsevier |year=2012 |isbn=978-0-323-14203-8 |page=367 |url=https://books.google.com/books?id=XzQK42x6uaEC}} [https://books.google.com/books?id=XzQK42x6uaEC&pg=PA367 Extract of page 367]</ref><ref>{{cite book |title=Basic Electromagnetism |author1=E.R. Dobbs |edition=illustrated |publisher=Springer Science & Business Media |year=2013 |isbn=978-94-011-2112-5 |page=113 |url=https://books.google.com/books?id=v3fsCAAAQBAJ}} [https://books.google.com/books?id=v3fsCAAAQBAJ&pg=PA113 Extract of page 113]</ref>
<math display="block">\begin{align}
t' &\approx t \\
x' &\approx x - vt
\end{align}</math>
in accordance with the [[correspondence principle]]. It is sometimes said that nonrelativistic physics is a physics of "instantaneous action at a distance".<ref>{{harvnb|Einstein|1916}}</ref>


Three counterintuitive, but correct, predictions of the transformations are:
;[[Relativity of simultaneity]]
: Suppose two events occur along the x axis simultaneously ({{math|1=Δ''t'' = 0}}) in {{mvar|F}}, but separated by a nonzero displacement {{math|Δ''x''}}.  Then in {{mvar|F′}}, we find that <math>\Delta t' = \gamma \frac{-v\,\Delta x}{c^2} </math>, so the events are no longer simultaneous according to a moving observer. 
;[[Time dilation]]
: Suppose there is a clock at rest in {{mvar|F}}. If a time interval is measured at the same point in that frame, so that {{math|1=Δ''x'' = 0}}, then the transformations give this interval in {{mvar|F′}} by {{math|1=Δ''t′'' = ''γ''Δ''t''}}. Conversely, suppose there is a clock at rest in {{mvar|F′}}. If an interval is measured at the same point in that frame, so that {{math|1=Δ''x′'' = 0}}, then the transformations give this interval in {{mvar|F}} by {{math|1=Δ''t'' = ''γ''Δ''t′''}}. Either way, each observer measures the time interval between ticks of a moving clock to be longer by a factor {{mvar|γ}} than the time interval between ticks of his own clock.
;[[Length contraction]]
: Suppose there is a rod at rest in {{mvar|F}} aligned along the {{mvar|x}} axis, with length {{math|Δ''x''}}.  In {{mvar|F′}}, the rod moves with velocity {{math|-''v''}}, so its length must be measured by taking two simultaneous ({{math|1=Δ''t′'' = 0}}) measurements at opposite ends.  Under these conditions, the inverse Lorentz transform shows that {{math|1=Δ''x'' = ''γ''Δ''x′''}}. In {{mvar|F}} the two measurements are no longer simultaneous, but this does not matter because the rod is at rest in {{mvar|F}}. So each observer measures the distance between the end points of a moving rod to be shorter by a factor {{math|1/''γ''}} than the end points of an identical rod at rest in his own frame. Length contraction affects any geometric quantity related to lengths, so from the perspective of a moving observer, areas and volumes will also appear to shrink along the direction of motion.