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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It depends on the situation.
Sometimes it can be that one theory is a special case of another.
Often it can be down to approximations being used. For example, in maths there is a thing called a “Taylor series”, which is a way of taking any nice function (for a specific definition of ‘nice’) and expressing it as a power series. So taking a weird, repeating function like sin x, we could say that in certain situations:
> sin x = x – x^(3)/6 + x^(5)/120 – x^(7)/7! + …
Now that is an infinite series, so we could never calculate it exactly. But if x is small, then x^3 would be really small, and x^5 would be incredibly small and so on. So if we wanted to approximate we could ignore most of the terms and just use the first couple.
Hooke’s Law for springs is a first order approximation – i.e. it ignores any powers of x (or whatever we are calling it) from x^2 upwards. Some of the physical models for friction or air resistance involve low-order approximations (using just x, or x^(2)). The normal formula for Kinetic Energy:
> KE = 1/2 mv^(2)
drops out of the Special Relativity version
> KE = (γ – 1)mc^2
if we say (v/c)^(4) is small enough to ignore (as with any higher powers), in which case γ = 1 + 1/2 (v/c)^(2) + …
This is the same sort of thing that gets us Newtonian Gravity out of General Relativity.
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