Editing Time dilation (section)

===Proper time and Minkowski diagram===
{{Multiple images
|align=right
|width=200
|direction=horizontal
|background-color = white
|image1=Zeitdilatation3.svg
|caption1=Clock C in relative motion between two synchronized clocks A and B. C meets A at ''d'', and B at ''f''.
|image2=EigenzeitZwill.svg
|caption2=[[Twin paradox]]. One twin has to change frames, leading to different [[proper time]]s in the twin's world lines.
|header=Minkowski diagram and twin paradox
}}

In the [[Minkowski diagram]] from the first image on the right, clock C resting in inertial frame S′ meets clock A at ''d'' and clock B at ''f'' (both resting in S). All three clocks simultaneously start to tick in S. The worldline of A is the ct-axis, the worldline of B intersecting ''f'' is parallel to the ct-axis, and the worldline of C is the ct′-axis. All events simultaneous with ''d'' in S are on the x-axis, in S′ on the x′-axis.

The [[proper time]] between two events is indicated by a clock present at both events.<ref name="taylor">{{Cite book |last1=Taylor |first1=Edwin F. |url=https://archive.org/details/spacetimephysics00edwi_0 |title=Spacetime Physics: Introduction to Special Relativity |last2=Wheeler |first2=John Archibald |publisher=W. H. Freeman |year=1992 |isbn=978-0-7167-2327-1 |location=New York |url-access=registration}}</ref> It is invariant, i.e., in all inertial frames it is agreed that this time is indicated by that clock. Interval ''df'' is, therefore, the proper time of clock C, and is shorter with respect to the coordinate times ''ef=dg'' of clocks B and A in S. Conversely, also proper time ''ef'' of B is shorter with respect to time ''if'' in S′, because event ''e'' was measured in S′ already at time ''i'' due to relativity of simultaneity, long before C started to tick.

From that it can be seen, that the proper time between two events indicated by an unaccelerated clock present at both events, compared with the synchronized coordinate time measured in all other inertial frames, is always the ''minimal'' time interval between those events. However, the interval between two events can also correspond to the proper time of accelerated clocks present at both events. Under all possible proper times between two events, the proper time of the unaccelerated clock is ''maximal'', which is the solution to the [[twin paradox]].<ref name=taylor />