dalehileman wrote:... I had also assumed that a clock in a spaceship circumnavigating our planet sends back clock signals at regular intervals sequentially behind, which would seem to constitute a pretty reliable indication that it's apparently running slower throughout its trip
The GPS system has a network of satellites in high orbits. Each one is an altitude of about 20,000 km and orbits at about 14,000 km/hour. Each has an atomic clock that "ticks" with an accuracy of 1 nanosecond (1 billionth of a second). Because an observer on the ground sees the satellites in motion relative to them, Special Relativity predicts:
(1) We should see their clocks ticking more slowly by around 7 microseconds (millionths of a second) per day because of the time dilation effect of their relative motion.
Effect (1): GPS clocks would lose 7 microseconds a day.
(2) 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 further away from a massive object such as the Earth (e.g. in orbit) will seem to tick faster than those located on its surface. In the case of the GPS system, the effect is to speed them up by 45 microseconds a day.
Effect (2): GPS clocks would gain 45 microseconds a day.
The combination of effects (1) and (2) means that if nothing was done the orbiting clocks would gain about 38 microseconds per day (45-7) and the system would be useless.
The engineers who designed the GPS system included these relativistic effects when they designed and deployed the system. For example, to counteract the General Relativistic effect once on orbit, they slowed down the ticking frequency of the atomic clocks before they were launched so that once they were in their proper orbit stations their clocks would appear to tick at the correct rate as compared to the reference atomic clocks at the GPS ground stations. Further, each GPS receiver has built into it a microcomputer that (among other things) performs the necessary relativistic calculations when determining the user's location.
Incidentally, this shows that General Relativity is very much more than a theory. Everybody who uses Google Maps on their phone, or a sat-nav system in a car or plane or bus or train relies on it every day.