In a previous thread about black holes and their event horizons, the discussion turned to how the equation for time dilation....

Gravitational time dilation - Wikipedia, the free encyclopedia

.....only relates coordinate time measured by a distant observer to local proper time at the event horizon. It doesn't relate local proper time at the event horizon to local proper time at a location further away. Meaning that, even though the formula predicts that a distant observer would observe time to be standing still at the event horizon, it isn't exactly standing still. Things can still fall into the black hole during that observer's lifespan.

Now, looking for a more practical real world example, we have the global GPS system.

Because an observer on the ground sees the satellites in motion relative to them, Special Relativity predicts that we should see their clocks ticking more slowly (see the Special Relativity lecture). Special Relativity predicts that the on-board atomic clocks on the satellites should fall behind clocks on the ground by about 7 microseconds per day because of the slower ticking rate due to the time dilation effect of their relative motion.

Further, the satellites are in orbits high above the Earth, where the curvature of spacetime due to the Earth's mass is less than it is at the Earth's surface. A prediction of General Relativity is that clocks closer to a massive object will seem to tick more slowly than those located further away (see the Black Holes lecture). As such, when viewed from the surface of the Earth, the clocks on the satellites appear to be tickingfasterthan identical clocks on the ground. A calculation using General Relativity predicts that the clocks in each GPS satellite should get ahead of ground-based clocks by 45 microseconds per day.

The combination of these two relativitic effects means that the clocks on-board each satellite should tick faster than identical clocks on the ground by about 38 microseconds per day (45-7=38)! This sounds small, but the high-precision required of the GPS system requires nanosecond accuracy, and 38 microseconds is 38,000 nanoseconds. If these effects were not properly taken into account, a navigational fix based on the GPS constellation would be false after only 2 minutes, and errors in global positions would continue to accumulate at a rate of about 10 kilometers each day! The whole system would be utterly worthless for navigation in a very short time. This kind of accumulated error is akin to measuring my location while standing on my front porch in Columbus, Ohio one day, and then making the same measurement a week later and having my GPS receiver tell me that my porch and I are currently about 5000 meters in the air somewhere over Detroit.

GPS and Relativity

I'm not sure what formula they used to calculate the 45 microseconds per day from GR. Without knowing the satellites' altitude and etc, it would be hard to try and test it against the time dilation formula above.

What I'm wondering is: clearly the effect measured by the satellites is not just an artefact of our choice of coordinate system. If we didn't correct for the daily shift of 38 microseconds, then let the satellite stay up there for a long time, and then brought the satellite back down to Earth, we would surely find that in the course of it's round trip up there and back, the satellite had quite literally "aged" more than we have.

Wouldn't an observer who spent a long time at or near the event horizon of a black hole also age slower than we do on Earth? What formula describes that difference if the formula above doesn't?