I believe that I understand the basic concept that time flows slower inside a gravity well (*). But in my brief internet searches, the only examples/discussions that I’ve found discuss time dilation in terms of a spot well outside the surface of a sun/planet/etc versus another spot on its surface. What about, say, a spot at the surface of the earth relative to another spot at/near the center of the earth? Does time continue to go slower as we approach the center of the earth, does it speed up again, or does it stay constant once you’re inside the surface? (**)
The gravity would decrease as you burrowed down towards the center of the earth since only the mass below you would exert a gravitational field on you. (In both Newtonian and Einsteinian gravity, there is no gravity inside a perfectly symmetrical hollow sphere.) Therefore, time would run slightly faster inside the Earth than it would at the surface of the Earth.
Incorrect. The gravitational field decreases as you go down (well, eventually, at least-- It’s roughly constant through most of the mantle, due to density variation), but time dilation depends on the gravitational potential, not on the field. So you would get more time dilation the deeper you go, and at the center of the Earth, the amount of time dilation would be at a local maximum.
Chronos has it right. I’ll just add that actually calculating the gravitational potential at the center of the earth (assumed to be a uniform sphere) is a standard physics problem and not all that hard. If I remember correctly, it works out that the difference in clock rates between the surface and the center of the Earth is exactly one-half the difference in the rates between the surface and “infinity”.
Ah, of course. Now I get it. Even if you have a hollow spherical shell, any radio signal sent from the inside to the outside is subjected to the shell’s gravity. The photons lose energy, which means they’re redshifted, which means that time is slower inside the shell than outside.