For example, how does Newtonian gravity happen to work just the same as GR on low energy scales? Is it simply a simplified special case version of GR, or is there something deeper mathematically going on?
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Basically it comes down to are you measuring it accurately enough for the difference to matter?
For example. If we measure the passage of time on the ground vs that of the passage of time in a satellite in space it’s not going to seem like much of a difference. An astronaut isn’t going to notice any difference.
However, if you want your GPS system to work it has to be taken into account. Newtonian gravity calculations aren’t accurate enough. You have to use GR (and also SR) to get the system to work otherwise the system will think you’re many km away from where you really are. The difference of the passage of time between the two locations isn’t much, but it’s enough to throw out the calculation of where you are when using radio signals going at the speed of light. A difference in just tiny fractions of a second will matter a lot.
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