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.
To use your example, both Newtonian gravity and GR are describing the same physical phenomenon. The difference between them is about 400 years of additional work done by several generations of physicists. People like Hamilton (not Alexander) and Lagrange and Mach expanded on, revised, and criticized Newton’s theories. Einstein was the beneficiary of all this work which helped him develop GR.
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)
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.
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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