Re: if gravity slows time does radioactivity increase in planet's center?

In the words of my cosmologist friend Amy: "Gravity doesn't slow time."

It seem like this this is contrary to your understanding of relativity
and no doubt you'll want an explanation, so here it is:

A phenomenon called time dilation does indeed occur when two inertial
reference frames (non-accelerating points of view in which Newton's laws
apply)are accelerating with respect to one another. In plain language: if
you put a person in spaceship and and watch it accelerate away from Earth,
time will appear to slow down for the person on the ship. But this time
dilation is only apparent to you. The person on the ship will experience
time on the ship passing normally.

So time dilation does occur, *but* it only happens if one frame of
reference is accelerating. If the spaceship in the previous example were
sitting still or coasting at a constant speed with respect to the earth, we
wouldn't see a time-stretch for the person on the ship.

So it's not gravity that stretches time, it's the acceleration of two
intertial reference frames with respect to one another. Any observers have
to be in these separate frames of reference for them to see any effect. 

Now let's get back to your question. If you were standing on the surface of
the earth with a Geiger counter and a radioactive sample (a lump of 
cesium 137, let's say) then you and the cesium would be in the same
inertial reference frame. If you went to the moon and did your
radioactivity experiment there, your Geiger counter still wouldn't ping any
faster because you'd still be in the same reference frame. Even if you and
the cesium were floating in the icy depths of space where there was a
miniscule gravitational field, it wouldn't make the decay happen any
faster. 

But if you put the cesium on the Space Shuttle, fired it away from
you, and recorded the results with a super-duper mono-directional
Geiger counter that could work over long distances, then you WOULD see 
the decay rate slow down!