seems to me that this should imply that a local observer standing on the earth (so not free falling at all) should be considered as an accelerating, non inertial frame.
Yes, an observer standing on the earth is not inertial in relativity. The definitive test is to have the observer carry a good accelerometer. In this case it will indicate an acceleration of 1 g upwards, conclusively showing that the observer is non-inertial.
Just a nitpick on language: an observer isnโt a reference frame, he or she has a reference frame, or even better there is a reference frame where he or she is at rest.
there is another, more geometrical, equivalent formulation of EEP:
Locally spacetime looks like ๐4
This is not the precise formulation of the geometrical formulation, but it's good enough.
Agreed, it is good enough for present purposes.
This means that in every sufficiently small region of spacetime it's like being into a inertial special relativity frame, so no accelerating, no gravity, no shenanigans.
It does not mean that at all. You can certainly have accelerating reference frames with pseudo-gravitational forces in ๐4. All ๐4 means is that you cannot have any tidal effects.
๐4 is a flat spacetime manifold and can be equipped with an endless number of coordinate systems, including non-inertial ones. What โlocally spacetime looks like ๐4โ means is that there exist local coordinates where the metric is the Minkowski metric (to first order), but it does not restrict you to using those coordinate systems.
More physically it means that tidal effects become negligible at small scales. The measurable effects from curvature, or tidal effects, are second order so they go away to first order at small enough scales.
But the geometrical formulation states that every sufficiently small reference frame, myself included, should be like an inertial SR frame!
No, the observer is unambiguously non-inertial. The geometrical formulation does not contradict that at all. The geometrical formulation merely says that in a small region spacetime is flat, not that an observer is inertial. It is perfectly consistent to have non-inertial observers and reference frames in flat spacetime. Only tidal effects are forbidden.
Within the frame of the free falling elevator the reading on an accelerometer always matches the acceleration with respect to the frame. Therefore the free falling frame is inertial.
In the frame of the ground an accelerometer at rest reads an upward acceleration of $g$ despite having no acceleration with respect to the ground. Therefore the groundโs frame is non-inertial.
Each frame can determine if they are inertial or not by looking at their own accelerometers and their own frame, without reference to any other frame. But the inertial vs non-inertial designation is exactly backwards from what you had indicated. A free falling frame is inertial and the ground frame is accelerating upwards at $g$.
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Your argument is true classically also. You see, the effect of Gravitation close to earth and that of acceleration in flat spacetime is same. But this equivalence goes way past just the elevator experience, where all the observer feeling is a state of weightlessness or equal weight. While introducing GR Einstein took this equivalence farther and said that there are no laws of physics that can distinguish between an accelerated frame and a stationary frame in gravitational field. This goes for all laws including Electro-magnetism. So yes, if you see a charged particle in accelerated frame it should radiate. This can then be extrapolated and we can say that a charged particle which is stationary in a gravitational field should also radiate.