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Because the Earth spins higher altitude locations that are farther away from the Earth's center are moving faster in relation to low altitude locations. This means that the hermit sitting on top of the huge mountain is actually living in a location where time moves slightly slower than other places, according to relativity.
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No it doesn't. The hermit moves faster, but he travels farther and ends up there at the same time as people at lower altitudes. A giant clock with perfect time and a watch with perfect time will always show the same point in time.