The mechanism inside the watch work the same. It is time itself that runs slower.
The clock will still go tick-tock at the same rate it is just that time itself is slowed down or sped up depending on from where you look at it. To the person looking at the clock it will work the same.
It is not even just space.
You can have time pass at different speeds here on earth. High mountains and deep valleys have time pass at different speeds.
The effect is however so small that we can barely measure it.
Once you get away from Earth you can get much greater differences.
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