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There are two foundative Equivalence Principles in General relativity:

Weak Equivalence Principle (WEP): the dynamics of a test particle in a gravitational field is independent of its mass and internal composition. (WEP is equivalent to say that the ratio between the gravitational mass $m_g$ and the inertial mass $m_i$ has a universal value $k$. This value can be considered as the scalar $1$).

Einstein equivalence principle (EEP): A frame linearly accelerated relative to an inertial frame in special relativity is LOCALLY identical to a frame at rest in a gravitational field.

Most textbooks say that "obviously" (EEP) implies (WEP) but the converse is not true.

I don't understand why the implication (EEP) $\Rightarrow$ (WEP) is true, and moreover I'd like a counterexample showing that (WEP) $\not\Rightarrow$ (EEP).

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EEP has something more than what WEP has. WEP states that in a small reign of space-time,there is no difference for a particle to free fall (to move in a gravitational field) or to move in a box with acceleration $g$ in the inverse direction.

Then Einstein stated more and said that not only for free fall but also for any non-gravitational experiment, this equivalence exists.

For example, we consider an electron and a proton.They fall with the same acceleration in the gravitational field and independent of their mass.If we then combine them and make an atom,the free fall acceleration of that atom would be the same.We know that the mass of atom is less than the total mass of electron and proton and negative potential is added to the system. Here we see that in addition to mass,gravitation is also coupled with energy.

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I agree that in the textbook you considered, explanation is not clear. First, please check my answer here: graviton and principle of equivalence

Personally, I prefer to unify WEP and EEP, but let's distinguish them as in your textbook. The example you provided for WEP is definitely mechanical, so not-gravitational experiment, hence the implication (EEP) ⇒ (WEP) indeed is obvious. As counter example I would suggest an experiment with photon. For instance the Doppler effect, being a consequence of special relativity is a part of general relativity. Being an experiment on photons, it does not tell you anything about mass of particles.

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EEP implies WEP:

Take two test particles of different masses/compositions in a gravitational field. Give them the same initial conditions (i.e. start them at the same place with the same initial velocity). We want to know if the WEP holds.

Analyze the experiment according to a frame at rest in the gravitational field. We can use the EEP to figure out how the particles will behave. Consider what takes place over a split second in the region surrounding the test particles ("locally"). The EEP says the physics governing the situation is the same as if we were in an accelerated frame relative to some inertial frame in the absence of gravity. We know what happens in that inertial frame. There are no forces on the objects. They travel in straight lines at constant velocity. Since they start at the same place and have the same initial velocity they are still together a short time later. Now viewing the situation back to our original frame (the one at rest in the gravitational field) we know that the two particles travel along the same trajectory over every short time interval. The dynamics are therefore independent of their mass and composition. That's the WEP.

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A counterexample where WEP holds but not EEP in some hypothetical gravity theory:

Imagine this new gravity obeys the WEP so every particle falls the same way in a gravitational field regardless of its composition. But under this new law of gravity test particles over 1 gram turn green when they are in a gravitational field. Put a 0.5g and a 2g particle next to each other in a gravitational field. One of them turns green. This would not happen if we were in an accelerated frame in the absence of gravity. So EEP does not hold in this model of gravity.

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