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It is the equivalence principle that provides the bridge between the ideal $SR$ model and the real world. According to it, we can find at each event a set of local inertial frames (LIFS), which may be small or large depending on

I.) Distribution of nearby masses

ii.) Accuracy we require

Rindler: Relativity, Special, General and Cosmological, page-36

What does Rindler mean here by the smallness or largeness of an inertial frame?

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The Equivalence Principle (EP) is valid locally: if you perform a local experiment in an inertial frame of reference, you can't say if you're freefalling or not falling, you can't say if you're attracted downwards or accelerated upwards. If you perform a nonlocal experiment, you can say what kind of motion you're following.
Of course, no physics experiment is perfectly local: a perfectly local experiment would perfectly follow the EP, and all nonlocal experiments violate the EP to some degree.
The locality of the equivalence principle is then to be meant as "if your experiment room (alias frame of reference) is small enough compared to the precision of the measurements you want to make and the intensity of outside forces, then the EP is valid".
This means that if the elevator in Einstein's gedankenexperiment was a very big elevator, big enough to perform an experiment about Earth's curvature inside it, then the observer inside could have said whether he was freefalling or not falling at all.

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  • $\begingroup$ What constitutes an expxeriment? $\endgroup$ May 20 at 17:19
  • $\begingroup$ @Aplateofmomos everything that you do with your eyes open, really! Jokes aside, if you can measure an observable (the mass of a particle, a scattering angle, etc) then you're conducting an experiment. $\endgroup$ May 20 at 17:53
  • $\begingroup$ Could you please see this question as well? $\endgroup$ May 21 at 8:15
  • $\begingroup$ @Aplateofmomos I have no idea about that actually, I'm sorry $\endgroup$ May 21 at 11:07
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I've never seen it phrased like that, but I understand it as the following.

Strictly speaking, the frame is defined locally, on a point. So the further you move from that point, the less inertial the frame becomes.

It's reasonable to assume that, if the mass distribution is too dense, the approximation will become bad very quickly, which would explain point 1.

As for any approximation, deciding when it becomes too bad to be useful is a matter of "taste". So it's up to the scientist to decide how big the neighborhood (in the mathematical sense) of the given point can be before the frame is too far from being inertial. That would be point 2.

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I don't have the book but I think he means the following. At any point in spacetime you can assign an inertial frame, similar to how you can assign a tangent line to a point on a curve. The size of this local inertial frame is basically the largest region of spacetime that can be reasonably approximated by this inertial frame. To refer back to the curve example: for the curve we would say the largest interval that can be reasonably approximated by a straight line. The larger the curvature at that point, the smaller this region will be.

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