Subject: Is time actually warped by heavenly bodies?

I was once told that the nearer you come to a planet, or any heavenly 
body for that matter, the more time begins to slow down.  Einstein I 
believe stated that the closer you get to the heavenly bodies the more 
"spacetime" is warped and therefore the more time slows down.  An 
experiment was done that had an atomic clock on the Earth's surface 
and one in an airplane, and the one in the airplane was slightly 
faster when checked.  But I have a hard time with this.  Common sense 
says that gravity affects anything anyway, especially an oscillating 
crystal.  But that does not mean that it affects time.  If a clock 
could be designed that was not affected by gravity, and was accurate 
enough to measure time to a small enough increment, I would say that 
time would read no different anywhere in the universe.  Do we have the 
technology to make said clock? Maybe not now, but possibly in the 
future. Does anyone see my point, or has this been thought of before 
and I am insanely wrong?  

Re: Is time actually warped by heavenly bodies?