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Kleppner and Kolenkow Pg 347 say

The point becomes even more apparent in the case of the elevator freely falling in the gravitational field. The elevator and all its contents accelerate downward at rate $g$. If the man releases the apple, it will float as if the elevator were motionless in free space. Einstein pointed out that the downward acceleration of the elevator exactly cancels the local gravitational field. From the point of view of an observer in the elevator, there is no way to determine whether the elevator is in free space or whether it is falling in a gravitational field.

And then

We summarize the principle of equivalence as follows: there is no way to distinguish locally between a uniform gravitational acceleration $\mathrm{g}$ and an accelera tion of the coordinate system $A=-g$.

I can't understand how the summarised definition applies on the case discussed first or how the summarised definition has a link with the above example.

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I will try to explain what's happening using some examples here.

Case (1) Assume you are in an elevator on the surface of earth. If you drop the apple, it will fall to the ground at the rate $g \ ms^{-2}$. The gravitational force on the object caused it to fall and not its inertia.

Case (2) Picture yourself in the same elevator, but in space and away from any gravitational field. Assume now that it is accelerating upwards at a rate $-g \ ms^{-2}$. If you drop an object, it will fall to the ground at the rate $g \ ms^{-2}$. No gravitational force caused the object to fall this time, but instead its inertia did.

Now assume that in these two examples, you have no way of knowing where you are. You cannot see what is outside and there is no way that any measurements you make locally (inside the elevator) can indicate to you if you are on earth or in space accelerating upwards. That is, its inertial mass and its gravitational mass appear to have the same affect in both cases.

That is essentially what the second paragraph is telling you. The principle of equivalence, ie., there is no way to distinguish locally between a uniform gravitational field (acceleration) g and an acceleration of the coordinate system $A=−g$ and that gravitational mass and inertial mass are equivalent.

For this to be true, then in outer space far away from any gravitational forces, and your elevator is not in an accelerated motion, you cannot have any of same things mentioned in the above examples happen. In the conditions described by case (1) you had an acceleration caused by gravity. In the conditions described by case (2) you also had a uniform acceleration caused by the apple's inertia.

Therefore, a freely falling elevator in an outside gravitational field cancels the local gravitational field inside on the apple because of its inertial motion. Both the downward acceleration and the apple's inertial motion are equivalent and one exactly cancels the other.

And that is what we see, which is what the first paragraph in your question is explaining. A uniformly accelerated reference frame and a uniform gravitational field are equivalent. And in the absence of both an acceleration in space away from gravity, and gravity in the absence of acceleration, or the case with one cancelling the other (elevator in free fall), there should be no effect on the apple. Hence the apple would just float. A good example is satellites in orbit which are in a gravitational field but the satellite is also actually in free fall. It has a constant acceleration towards the centre of the earth and the astronauts and other objects float - downward acceleration exactly cancelling the local gravitational field as mentioned above.

The fact that gravitational mass and inertial mass are identical is a remarkable result, and the foundation of general relativity.

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If you can cancel a gravitational field by falling downward with an acceleration g, then you can also create a fictitious "gravitational" field by accelerating upward at g. If you're in a small completely closed elevator then you can't determine from experiments or observations in the elevator whether you are in an accelerating rocket in outer space or on earth and not moving.

But, the word "locally" is very important here. You can detect a real gravitational field with sensitive enough equipment. Because you can detect the slight variation in the field. "Locally" means small enough that you can't measure that curvature of spacetime.

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The link is really not explicit, but as the observer can jump from some place and be in free fall, he can also jump from the accelerated ship in the outer space.

In that last case, it is obvious that he becomes an inertial frame just after jump.

Then the accelerated frame is being compared with gravitational field in the sense that it is possible to get rid of it by losing contact with any material stuff of the frame.

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