How does a scientific theory “approximate” a more fundamental one despite having completely different math?

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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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If you’re fine only with precision of what you can see or feel for what we experience in everyday life, then the equations for GR give the same answers as those for Newtonian mechanics, but the Newtonian ones are much simpler and more intuitive: gravity is a force you feel, your perception of length matches mine, and the time we experience is the exact same

Take special relativity, for example: most equations that relate the two reference frames are just multiplying by γ=1/√(1-v²/c²), unless you’re travelling over 100 million mph, then this term is basically 1 and we get Newtonian mechanics back

For GR, just swap out “speed you’re travelling at” with “escape velocity of the gravity you’re feeling” and you’ll get the same approximation

Once you need to be ridiculously precise (like GPS satellites which need nanosecond-level precision), then you need to use GR instead, but until then you can get along just fine with classical physics (as humans did for 100s of years)

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