[obi1kenobi]

The frame of reference in which the mountain peak is stationary is not inertial -- due to the Earth's rotation, it's actually constantly accelerating toward the center of the Earth. Hence, the quote on special relativity you posted doesn't (directly) apply.

However, time passes slower the deeper one gets into a gravity well. The center of the Earth is the center of Earth's gravity well, so the closer to it you are, the slower your clock ticks.

More info: https://en.wikipedia.org/wiki/Gravitational_time_dilation

[sebtoast]

Thank you to all those who replied.

At the top of the article you linked there's the: "This article is about time dilation due to relative gravity. For time dilation due to relative velocity, see Relative velocity time dilation."

I didn't know there were "two forms" of time dilation, knowing what to search for helps a lot.

Thanks again.

[dredmorbius]

On the "centre of Earth's gravity well" bit: what is the net effect of the fact that inside a sphere, the net gravitational attraction depends on the amount, distance, and vector of the gravitational mass in question, and that whilst inside a sphere at any point beneath its surface the net gravitational attraction varies.

In the case of the Earth, it actually increases to a certain depth, then falls as one approaches the centre (or would if one could; imagine, say, the neutrino version of yourself). My first thought was that the associated time dilation would follow a similar trend, though thinking this through, I'm not certain, though I'm leaning toward your answer above as not being accurate.