Gravitational Time Dilation

[Reaper]

As some of you may know, General Relativity does predict that gravitational fields do affect the flow of time. The exact equation for it is:

 

 

And this has got me wondering. What would happen if you were to stand on a Neutron Star or hover near a black hole (assuming you aren't crushed by the overwhelming gravitational force). Could you, in principle, live far longer? Or if you had a second spacecraft far enough from the gravitational field exerted by these bodies would one observe the people on the Neutron Star (or black hole) go through time much slower relative to them, similar to what you would see at near-light speeds?

 

Another thing that I'm wondering is, would you get different results in an experiment depending on the strength of the gravitational field? The experiments I'm referring to is particle accelerators. What I'm wondering is, would it matter if you built a particle accelerator in the sky as opposed to the ground (please ignore the technical challenges involved, this is purely hypothetical). In many noted experiments a Muon decays at a slower rate when accelerated to near light speeds. But since time goes slower on the ground than it does in the sky, would this effect be noticeable. Like, would a muon decay faster in the sky then it would on the ground? Or is the gravitational field of the Earth too weak to be of any significance. If it is, would it, and other affects associated with Time Dilation, have much more of an impact on or near a black hole or neutron star, or other massive body like the Sun or Jupiter?

[swansont]

GPS satellites run 45 microseconds a day fast due to gravitational time dilation effects (and also 7 microseconds slow due to kinematic time dilation), so this is a fractional change of several parts in 10¹⁰