Editing Spacetime (section)

=== Relativity of simultaneity ===
{{Main|Relativity of simultaneity}}
{{anchor|Figure 2-6}}
[[File:Relativity of Simultaneity Animation.gif|thumb|Figure 2–6. Animation illustrating relativity of simultaneity]]

All observers will agree that for any given event, an event within the given event's future light cone occurs ''after'' the given event. Likewise, for any given event, an event within the given event's past light cone occurs ''before'' the given event. The before–after relationship observed for timelike-separated events remains unchanged no matter what the [[Frame of reference|reference frame]] of the observer, i.e. no matter how the observer may be moving. The situation is quite different for spacelike-separated events. [[#Figure 2-4|'''Fig.&nbsp;2-4''']] was drawn from the reference frame of an observer moving at {{nowrap|1=''v'' = 0.}} From this reference frame, event&nbsp;C is observed to occur after event&nbsp;O, and event&nbsp;B is observed to occur before event&nbsp;O.<ref name="plato.stanford.edu">{{cite web|last1=Savitt|first1=Steven|title=Being and Becoming in Modern Physics. 3. The Special Theory of Relativity|url=https://plato.stanford.edu/entries/spacetime-bebecome/#Spec|website=The Stanford Encyclopedia of Philosophy|publisher=Metaphysics Research Lab, Stanford University|access-date=26 March 2017|archive-date=11 March 2017|archive-url=https://web.archive.org/web/20170311015404/https://plato.stanford.edu/entries/spacetime-bebecome/#Spec|url-status=live}}</ref>

From a different reference frame, the orderings of these non-causally-related events can be reversed. In particular, one notes that if two events are simultaneous in a particular reference frame, they are ''necessarily'' separated by a spacelike interval and thus are noncausally related. The observation that simultaneity is not absolute, but depends on the observer's reference frame, is termed the [[relativity of simultaneity]].<ref name="plato.stanford.edu"/>

Fig. 2-6 illustrates the use of spacetime diagrams in the analysis of the relativity of simultaneity. The events in spacetime are invariant, but the coordinate frames transform as discussed above for Fig.&nbsp;2-3. The three events {{nowrap|1=(A, B, C)}} are simultaneous from the reference frame of an observer moving at {{nowrap|1=''v'' = 0.}} From the reference frame of an observer moving at {{nowrap|1=''v'' = 0.3''c'',}} the events appear to occur in the order {{nowrap|1=C, B, A.}} From the reference frame of an observer moving at {{nowrap|1=''v'' = −0.5''c''}}, the events appear to occur in the order {{nowrap|1=A, B, C}}. The white line represents a ''plane of simultaneity'' being moved from the past of the observer to the future of the observer, highlighting events residing on it. The gray area is the light cone of the observer, which remains invariant.

A spacelike spacetime interval gives the same distance that an observer would measure if the events being measured were simultaneous to the observer. A spacelike spacetime interval hence provides a measure of ''proper distance'', i.e. the true distance = <math>\sqrt{-s^2}.</math> Likewise, a timelike spacetime interval gives the same measure of time as would be presented by the cumulative ticking of a clock that moves along a given world line. A timelike spacetime interval hence provides a measure of the ''proper time'' = <math>\sqrt{s^2}.</math><ref name="Schutz" />{{rp|220–221}}

{{anchor|Invariant hyperbola}}