it is commonly argued that the equivalence principle implies that everything must fall locally in the same direction, because any local variation of accelerations in a small enough neighbourhood is equivalent to an inertial force.
Is the purpose of this question is to understand, what are the bounds for a frame to be considered local, when the space might be compactified, and some of the macroscopic asymmetries of fields we see are explainable by symmetries in the inner dimensions.
For instance, consider the following argument: antimatter cannot fall away from matter, otherwise a local frame falling toward matter will not see local antimatter falling inertially
don't get me wrong, there might be good reasons why in the particular case of anti-matter, it will fall exactly like regular matter (some of those might be found on this answer), however, i'm concerned about this particular argument.
In a compactified space, what we mean with locally in the context of macroscopic observations needs to be refined; for instance consider the scenario where antimatter (or for the purpose of avoiding controversy, lets call it $matter \dagger$ ) falls gravitationally away from regular matter; it seems to me that the principle of equivalence still holds if what we mean as a local neighbourhood means actually:
$\lbrace$ usual local neighbourhood in macroscopic spacetime $\rbrace$ $\times$ $\lbrace$ the subset of the compactification space where regular matter lives $\rbrace$
in this scenario, $matter \dagger$ falls inertially in its own neighbourhood given by removing the regular matter sector from the local neighbourhood in the whole space (macroscopic plus compactified)
I even made a ugly picture of the so called scenario:
Question 1: Are there known reasons why such reinterpretations of the equivalence principle, taking into account the additional dimensions, are flawed?
Question 2: What does the original Kaluza-Klein theory has to say about how and under what forces the electric negative and positive charges move?
I'm sorry for the rather crude arguments, hope the intent is rather clear though