What you are thinking would be true if the Earth's mass was concentrated to a point. Since it isn't, when you go down, you are nearing the center but you are al

What you are thinking would be true if the Earth's mass was concentrated to a point. Since it isn't, when you go down, you are nearing the center but you are also having more and more mass above you (which doesn't contribute, see http://en.wikipedia.org/wiki/Shell_theorem http://en.wikipedia.org/wiki/Shell_theorem) so the net effect is that the force of gravity goes down, like westbywest said. However, unless I've totally forgotten what little I ever knew about GR, what matters for clocks is the gravitational potential, in such a way that clocks run slower in lower potentials. And the potential is definitely the lowest at the center of the Earth. Otherwise clocks would run the slowest far out in intergalactic space where there's practically no gravity, but that's not the case.