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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Using Galilean relativity, special relativity, and General relativity as an example.
All of them are theories that deal with change of measures with the change of the frame of reference. Galilean relativity is the simplest one: Change the frame of reference, and you get different speeds (by just adding the difference of speed between the frames). This is basically the famous “you walking on a train in movement, your speed relative to the ground is your speed relative to the train plus the speed of the train relative to the ground”.
However this formula doesn’t work well with both Maxwell’s equations and Newtonian mechanics, so either one of the three should be wrong. Well… Then comes special relativity, where the time and space dilation makes the Electromagnetism and Newton’s laws be consistent again with change of frame of reference(Kinda…) In this case, Galilean relativity ends up being an approximation of special relativity when the difference in speed of the frames of reference are small compared to the speed of light (the math behind is the lorentz factor “γ=1/√(1-v²/c²)”). So, in this the older theory approximates because the formula of the new one converges to the old formula for a specific set of values.
But SR has a problem: It doesn’t work for accelerated frames of reference. The math trick is to describe changes in the speed vector (direction and magnitude) when you change to an accelerated frame of reference, as if the coordinates where the particle is moving have been distorted (that’s what curved spacetime is about, it’s like drawing some curve in some white rubber surface, and then stretching it so the curve becames a straight line. GR describes “how and where the rubber is stretched”).
The math of GR is complicated, but it ends up that if we use it to change the frame of reference between two non-accelerated frames, we end up with the formulas of SR (I think the best analogy I can came up with, is how the formula of the length of the diagonal of a rectangular right prism “d² = √(x² + y² + z²)” becomes the formula of the length of the diagonal of a rectangle when one of the dimensions of the prism becomes 0).
So, in this case, it isn’t an approximation, it is that for a specific set of values, the formula of general relativity becomes the formulas of special relativity (or in the opposite way: The General Relativity is a generalization that includes SR).
In case you are curious, in the original paper of General Relativity, that you can find the english translation, there’s a part where Einstein shows how SR is a special case of GR. You won’t be able to fully understand without knowing Tensor Calculus, but you can have a vague idea, by reading the proof, of how some terms are cancelled and SR math appears.
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