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I'm reading Geroch's lecture notes on GR, and in the first chapter he makes the following assertion:

People inside an elevator freely falling in a uniform gravitational field cannot distinguish their situation from that of an elevator at rest in no gravitational field [...] by no physics experiment inside the elevator can the two situations be distinguished.

I've always seen the equivalence principle expressed as the fact that uniform gravitational field = uniform acceleration. Is Geroch asserting something that isn't true, or is there something subtle that I'm missing?

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    $\begingroup$ I don't understand your source of confusion. $\endgroup$ – Myridium Aug 5 '17 at 3:07
  • $\begingroup$ That should be read as "at rest in a uniform gravitational field = uniform acceleration", not "freely falling in a uniform gravitational field" which would be locally equivalent to "no acceleration, no gravitational field". $\endgroup$ – doetoe Aug 5 '17 at 16:45
  • $\begingroup$ It sounds like you're asking about two different formulations of the equivalence principle, one of which is about zero apparent field and one which is about nonzero apparent field. Is your question asking for an explanation of the relationship between them? $\endgroup$ – user4552 Aug 5 '17 at 19:39
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I think the answer is as follows:

The people inside the elevator move along the same geodesics as anything they could use to test if they were "accelerating" (since the field is uniform), and hence they cannot tell that they are "accelerating". Hence their frame is locally inertial (i.e. for however far the field is uniform).

A person at rest in no gravitational field is also in a (globally) inertial frame. Hence, locally, both frames are "the same" (or, really, analogous).

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I share these paragraphs from a book of general relativity that explains very simply everything: The inertial force that appears in an accelerated system produces the same effects as a "real" gravitational field. Gravity can be anulled locally by choosing a system in free fall or created by choosing an accelerated equivalent system. That is, there is an equivalence between the effects of a real gravitational field and those of a field of fictitious inertial forces. Thus, there is no physical experiment that can differentiate between gravity and the effects of using an agreeably accelerated system.

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I've always seen the equivalence principle expressed as the fact that uniform gravitational field = uniform acceleration.

This is a misleading statement. An observer following a geodesic path does not feel any force nor acceleration, the observer is free falling. This is true for the free fall to the center of a gravitational source - for example the earth - as well as for any orbital movement around the earth. To distinguish between the increase of velocity from a "mechanical" source (a car or a train) and from a gravitational source the first is called an acceleration and the second is called the following of its geodesic path.

BTW in the absence of a gravitational field the rotation of a bucket with water shouldn't bring the effect of the elipsoid of the water surface. Without gravitational field don't exist g odesic paths nor any acceleration.

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