Editing Spacetime (section)

=== Time dilation and length contraction ===
[[File:Spacetime diagram of invariant hyperbola.png|thumb|Figure 2–8. The invariant hyperbola comprises the points that can be reached from the origin in a fixed proper time by clocks traveling at different speeds]]
Fig. 2-8 illustrates the invariant hyperbola for all events that can be reached from the origin in a proper time of 5&nbsp;meters (approximately {{val|1.67|e=−8|u=s}}). Different world lines represent clocks moving at different speeds. A clock that is stationary with respect to the observer has a world line that is vertical, and the elapsed time measured by the observer is the same as the proper time. For a clock traveling at 0.3&nbsp;''c'', the elapsed time measured by the observer is 5.24&nbsp;meters ({{val|1.75|e=−8|u=s}}), while for a clock traveling at 0.7&nbsp;''c'', the elapsed time measured by the observer is 7.00&nbsp;meters ({{val|2.34|e=−8|u=s}}).<ref name="Schutz" />{{rp|220–221}}

This illustrates the phenomenon known as [[time dilation]]. Clocks that travel faster take longer (in the observer frame) to tick out the same amount of proper time, and they travel further along the x–axis within that proper time than they would have without time dilation.<ref name="Schutz" />{{rp|220–221}} The measurement of time dilation by two observers in different inertial reference frames is mutual. If observer O measures the clocks of observer O′ as running slower in his frame, observer O′ in turn will measure the clocks of observer O as running slower.

{{anchor|Figure 2-9}}
[[File:Animated Spacetime Diagram - Length Contraction.gif|thumb|Figure 2–9. In this spacetime diagram, the 1&nbsp;m length of the moving rod, as measured in the primed frame, is the foreshortened distance OC when projected onto the unprimed frame.]]
[[Length contraction]], like time dilation, is a manifestation of the relativity of simultaneity. Measurement of length requires measurement of the spacetime interval between two events that are simultaneous in one's frame of reference. But events that are simultaneous in one frame of reference are, in general, not simultaneous in other frames of reference.

Fig. 2-9 illustrates the motions of a 1&nbsp;m rod that is traveling at 0.5&nbsp;''c'' along the ''x'' axis. The edges of the blue band represent the world lines of the rod's two endpoints. The invariant hyperbola illustrates events separated from the origin by a spacelike interval of 1&nbsp;m. The endpoints O and B measured when {{′|''t''}}&nbsp;=&nbsp;0 are simultaneous events in the S′ frame. But to an observer in frame S, events O and B are not simultaneous. To measure length, the observer in frame S measures the endpoints of the rod as projected onto the ''x''-axis along their world lines. The projection of the rod's ''world sheet'' onto the ''x'' axis yields the foreshortened length OC.<ref name="Collier" />{{rp|125}}

(not illustrated) Drawing a vertical line through A so that it intersects the ''x''′ axis demonstrates that, even as OB is foreshortened from the point of view of observer O, OA is likewise foreshortened from the point of view of observer O′. In the same way that each observer measures the other's clocks as running slow, each observer measures the other's rulers as being contracted.

In regards to mutual length contraction, [[#Figure 2-9|'''Fig.&nbsp;2-9''']] illustrates that the primed and unprimed frames are mutually [[Lorentz transformation#Coordinate transformation|rotated]] by a [[hyperbolic angle]] (analogous to ordinary angles in Euclidean geometry).<ref group=note>In a [[Cartesian plane]], ordinary rotation leaves a circle unchanged. In spacetime, hyperbolic rotation preserves the [[hyperbolic metric]].</ref> Because of this rotation, the projection of a primed meter-stick onto the unprimed ''x''-axis is foreshortened, while the projection of an unprimed meter-stick onto the primed x′-axis is likewise foreshortened.