People often have trouble visualizing how time dilation can be "mutual" in special relativity. That is, if you have two clocks moving with respect to each other, in each clock's rest frame the other clock runs slower. The following example illustrates how this can be possible without contradictions.(adsbygoogle = window.adsbygoogle || []).push({});

Imagine a set of space stations, labeled A through E. They are at rest with respect to each other and spaced equally along a line, [itex]\Delta x = \sqrt{3} = 1.732[/itex] light-years (ly) apart. In their rest frame, their clocks are synchronized: they run at the same rate, and in step with each other.

Also imagine a set of space ships, labeled A through E. They are traveling along the line connecting the space stations. In the stations' rest frame the ships have the same spacing as the stations, and are traveling at a speed of [itex]v = \sqrt{3}/2 = 0.866[/itex] light-years per year (ly/yr). Note the speed of light is c = 1 ly/yr, exactly. The ships carry clocks that are constructed identically to the stations' clocks.

In the stations' rest frame, at time 0.00 according to the stations' clocks, each ship passes the corresponding station simultaneously. At this instant, set each of the ships' clocks to 0.00 also. We assume that the ships pass the stations closely enough that there is no problem with this procedure, e.g. from time delays in sending signals between them.

In the stations' rest frame, each ship takes [itex]\Delta x / v = \sqrt{3} / (\sqrt{3} / 2) = 2.00[/itex] yr to travel the distance from one station to the next, so 2.00 yr elapses on each station's clock as ship A travels from station A to station B, etc.

Meanwhile, because of time dilation, exactly half as much time, 1.00 yr, elapses on each ship's clock. You should be able to verify this from the given speed and the time-dilation equation.

The attached diagram shows a series of snapshots showing the positions of the ships and stations, and the the readings on their clocks (the "flags" attached to each object), at 2.00-yr intervals in the stations' rest frame.

In the next post I will start to consider what this looks like in the ships' rest frame.

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# How two clocks can each run slow in the other's reference frame

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