Editing Spacetime (section)

==== Mutual time dilation ====
Mutual time dilation and length contraction tend to strike beginners as inherently self-contradictory concepts. If an observer in frame S measures a clock, at rest in frame S', as running slower than his', while S' is moving at speed ''v'' in S, then the principle of relativity requires that an observer in frame S' likewise measures a clock in frame S, moving at speed −''v'' in S', as running slower than hers. How two clocks can run ''both slower'' than the other, is an important question that "goes to the heart of understanding special relativity."<ref name="Schutz" />{{rp|198}}

This apparent contradiction stems from not correctly taking into account the different settings of the necessary, related measurements. These settings allow for a consistent explanation of the ''only apparent'' contradiction. It is not about the abstract ticking of two identical clocks, but about how to measure in one frame the temporal distance of two ticks of a moving clock. It turns out that in mutually observing the duration between ticks of clocks, each moving in the respective frame, different sets of clocks must be involved. In order to measure in frame S the tick duration of a moving clock W′ (at rest in S′), one uses ''two'' additional, synchronized clocks W<sub>1</sub> and W<sub>2</sub> at rest in two arbitrarily fixed points in S with the spatial distance ''d''.
: <small>Two events can be defined by the condition "two clocks are simultaneously at one place", i.e., when W′ passes each W<sub>1</sub> and W<sub>2</sub>. For both events the two readings of the collocated clocks are recorded. The difference of the two readings of W<sub>1</sub> and W<sub>2</sub> is the temporal distance of the two events in S, and their spatial distance is ''d''. The difference of the two readings of W′ is the temporal distance of the two events in S′. In S′ these events are only separated in time, they happen at the same place in S′. Because of the invariance of the spacetime interval spanned by these two events, and the nonzero spatial separation ''d'' in S, the temporal distance in S′ must be smaller than the one in S: the ''smaller'' temporal distance between the two events, resulting from the readings of the moving clock W′, belongs to the ''slower'' running clock W′.</small>

Conversely, for judging in frame S′ the temporal distance of two events on a moving clock W (at rest in S), one needs two clocks at rest in S′.
: <small>In this comparison the clock W is moving by with velocity −''v''. Recording again the four readings for the events, defined by "two clocks simultaneously at one place", results in the analogous temporal distances of the two events, now temporally and spatially separated in S′, and only temporally separated but collocated in S. To keep the spacetime interval invariant, the temporal distance in S must be smaller than in S′, because of the spatial separation of the events in S′: now clock W is observed to run slower.</small>

The necessary recordings for the two judgements, with "one moving clock" and "two clocks at rest" in respectively S or S′, involves two different sets, each with three clocks. Since there are different sets of clocks involved in the measurements, there is no inherent necessity that the measurements be reciprocally "consistent" such that, if one observer measures the moving clock to be slow, the other observer measures the one's clock to be fast.<ref name="Schutz" />{{rp|198–199}}

{{multiple image|perrow = 1|total_width=250
| image2 = Spacetime Diagrams of Mutual Time Dilation B.png  |width2=300|height2=300
| image4 = Spacetime Diagrams of Mutual Time Dilation D.png  |width4=300|height4=300
| footer = Figure 2-10. Mutual time dilation }}
Fig. 2-10 illustrates the previous discussion of mutual time dilation with [[Minkowski diagram]]s. The upper picture reflects the measurements as seen from frame S "at rest" with unprimed, rectangular axes, and frame S′ "moving with ''v''&nbsp;>&nbsp;0", coordinatized by primed, oblique axes, slanted to the right; the lower picture shows frame S′ "at rest" with primed, rectangular coordinates, and frame S "moving with −''v''&nbsp;<&nbsp;0", with unprimed, oblique axes, slanted to the left.

Each line drawn parallel to a spatial axis (''x'', ''x''′) represents a line of simultaneity. All events on such a line have the same time value (''ct'', ''ct''′). Likewise, each line drawn parallel to a temporal axis (''ct'', ''ct′'') represents a line of equal spatial coordinate values (''x'', ''x''′).
: <small>One may designate in both pictures the origin ''O'' (= {{′|''O''}}) as the event, where the respective "moving clock" is collocated with the "first clock at rest" in both comparisons. Obviously, for this event the readings on both clocks in both comparisons are zero. As a consequence, the worldlines of the moving clocks are the slanted to the right ''ct''′-axis (upper pictures, clock W′) and the slanted to the left ''ct''-axes (lower pictures, clock W). The worldlines of W<sub>1</sub> and W′<sub>1</sub> are the corresponding vertical time axes (''ct'' in the upper pictures, and ''ct''′ in the lower pictures).</small>
: <small>In the upper picture the place for W<sub>2</sub> is taken to be ''A<sub>x</sub>'' > 0, and thus the worldline (not shown in the pictures) of this clock intersects the worldline of the moving clock (the ''ct''′-axis) in the event labelled ''A'', where "two clocks are simultaneously at one place". In the lower picture the place for W′<sub>2</sub> is taken to be ''C''<sub>''x''′</sub>&nbsp;<&nbsp;0, and so in this measurement the moving clock W passes W′<sub>2</sub> in the event ''C''.</small>
: <small>In the upper picture the ''ct''-coordinate ''A<sub>t</sub>'' of the event ''A'' (the reading of W<sub>2</sub>) is labeled ''B'', thus giving the elapsed time between the two events, measured with W<sub>1</sub> and W<sub>2</sub>, as ''OB''. For a comparison, the length of the time interval ''OA'', measured with W′, must be transformed to the scale of the ''ct''-axis. This is done by the invariant hyperbola (see also Fig. 2-8) through ''A'', connecting all events with the same spacetime interval from the origin as ''A''. This yields the event ''C'' on the ''ct''-axis, and obviously: ''OC''&nbsp;<&nbsp;''OB'', the "moving" clock W′ runs slower.</small>

To show the mutual time dilation immediately in the upper picture, the event ''D'' may be constructed as the event at ''x''′&nbsp;=&nbsp;0 (the location of clock W′ in S′), that is simultaneous to ''C'' (''OC'' has equal spacetime interval as ''OA'') in S′. This shows that the time interval ''OD'' is longer than ''OA'', showing that the "moving" clock runs slower.<ref name="Collier" />{{rp|124}}

In the lower picture the frame S is moving with velocity −''v'' in the frame S′ at rest. The worldline of clock W is the ''ct''-axis (slanted to the left), the worldline of W′<sub>1</sub> is the vertical ''ct''′-axis, and the worldline of W′<sub>2</sub> is the vertical through event ''C'', with ''ct''′-coordinate ''D''. The invariant hyperbola through event ''C'' scales the time interval ''OC'' to ''OA'', which is shorter than ''OD''; also, ''B'' is constructed (similar to ''D'' in the upper pictures) as simultaneous to ''A'' in S, at ''x''&nbsp;=&nbsp;0. The result ''OB''&nbsp;>&nbsp;''OC'' corresponds again to above.

The word "measure" is important. In classical physics an observer cannot affect an observed object, but the object's state of motion ''can'' affect the observer's ''observations'' of the object.