Editing Special relativity (section)

=== Relativity of simultaneity ===
{{See also|Relativity of simultaneity|Ladder paradox}}
[[File:Relativity of Simultaneity Animation.gif|thumb|Figure 4–1. The three events (A, B, C) are simultaneous in the reference frame of some observer '''O'''. In a reference frame moving at ''v'' = 0.3''c'', as measured by '''O''', the events occur in the order C, B, A. In a reference frame moving at {{nowrap|1=''v'' = −0.5''c''}} with respect to '''O''', the events occur in the order A, B, C. The white lines, the ''lines of simultaneity'', move from the past to the future in the respective frames (green coordinate axes), highlighting events residing on them. They are the locus of all events occurring at the same time in the respective frame. The gray area is the [[light cone]] with respect to the origin of all considered frames.]]

Consider two events happening in two different locations that occur simultaneously in the reference frame of one inertial observer. They may occur non-simultaneously in the reference frame of another inertial observer (lack of [[absolute simultaneity]]).

From {{EquationNote|3|Equation&nbsp;3}} (the forward Lorentz transformation in terms of coordinate differences)
<math display="block">\Delta t' = \gamma \left(\Delta t - \frac{v \,\Delta x}{c^{2}} \right)</math>

It is clear that the two events that are simultaneous in frame ''S'' (satisfying {{nowrap|1=Δ''t'' = 0}}), are not necessarily simultaneous in another inertial frame {{prime|''S''}} (satisfying {{nowrap|1=Δ{{prime|''t''}} = 0}}). Only if these events are additionally co-local in frame ''S'' (satisfying {{nowrap|1=Δ''x'' = 0}}), will they be simultaneous in another frame {{prime|''S''}}.

The [[Sagnac effect]] can be considered a manifestation of the relativity of simultaneity.<ref name="Ashby2003">{{cite journal |last1=Ashby |first1=Neil |title=Relativity in the Global Positioning System |journal=Living Reviews in Relativity |volume=6 |issue=1 |pages=1 |doi=10.12942/lrr-2003-1 |pmid=28163638 |pmc=5253894 |year=2003 |doi-access=free |bibcode=2003LRR.....6....1A }}</ref> Since relativity of simultaneity is a first order effect in {{tmath|1= v }},<ref name=Morin2007/> instruments based on the Sagnac effect for their operation, such as [[ring laser gyroscope]]s and [[fiber optic gyroscope]]s, are capable of extreme levels of sensitivity.<ref name="Lin1979" group="p">{{cite journal |last1=Lin |first1=Shih-Chun |last2=Giallorenzi |first2=Thomas G. |title=Sensitivity analysis of the Sagnac-effect optical-fiber ring interferometer |journal=Applied Optics |date=1979 |volume=18 |issue=6 |pages=915–931 |doi=10.1364/AO.18.000915|pmid=20208844 |bibcode=1979ApOpt..18..915L |s2cid=5343180 }}</ref>