The Principle of Reciprocal Time Dilation

by Wolfgang G. Gasser

There are two principally different kinds of time-dilation, which can be combined in different ways. In case of two clocks, one next to the other, only one clock can run slower than the other. Reciprocal time-dilation is impossible. This is the kind of time dilation, occurring in particle accelerators and cyclotrons. The high-speed particle, considered as a clock, runs slower than any clock placed in the accelerator. This time-dilation is rather asymmetric and absolute than reciprocal, mutual, symmetric and relative.

Only a second kind of time dilation allows for reciprocal time-dilation. In order to realize it in the simplest form, one must compare one clock with a number of other clocks, all running at the same pace, yet each with a different time offset. One possible realization: Clocks are fixed on a long tape and adjusted in such a way that a clock (despite running at the same pace) is the more in advance (i.e. in the future), the more the clock is situated in front (positive x-axis), e.g. advancing each 0.2 sec per meter in front. An observer moving at constant speeds of 1 m/s relative to the tape gets the impression that his own time runs slower than tape-time (1 sec per 1.2 sec).

This principle of time-dilation entails two important consequences: The tape cannot be closed (resp. made cyclic), as in one direction all is 'future' and in the other direction all is 'past'.

The second consequence: If the moving observer changes direction (moving at same speed of 1 m/s) and the clocks on the tape are not readjusted with new time offsets, then observer time runs faster than tape-time (1 sec per 0.8 sec). The analogous of the (normal) twin-paradox resolution is this: The tape-clock where the moving observer is situated at direction-change is bound by the presence of the observer, so it cannot be changed. Yet all other clocks receive new time-offsets:

` 0.0`
`--+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+--`
` 0.0   0.2   0.4   0.6   0.8   1.0   1.2   1.4   1.6   1.8   2.0`
` `
`                               5.0`
`--+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+--`
` 5.0   5.2   5.4   5.6   5.8   6.0   6.2   6.4   6.6   6.8   7.0`
` `
`                                                            10.0`
`--+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+--`
`10.0  10.2  10.4  10.6  10.8  11.0  11.2  11.4  11.6  11.8  12.0`
` `
`                                                            10.0`
`--+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+--`
`14.0  13.8  13.6  13.4  13.2  13.0  12.8  12.6  12.4  12.2  12.0`
` `
`20.0`
`--+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+--`
`24.0  23.8  23.6  23.4  23.2  23.0  22.8  22.6  22.4  22.2  22.0`

We also recognize that the direction change does not work properly, if the moving observer is expanded, as more than one clock is bound:

`                                                      10.0  10.0`
`--+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+--`
`10.0  10.2  10.4  10.6  10.8  11.0  11.2  11.4  11.6  11.8  12.0`
` `
`                                                      10.0  10.0`
`--+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+--`
`14.0  13.8  13.6  13.4  13.2  13.0  12.8  12.6  12.4  12.2  12.0`
` `

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