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... There may be some insights into this buried in general
relativity, but since GR is based on the postulate that gravitational
and inertial mass are the same thing, any insights might well end up
being tautological. David Bowman will probably have something
relevant to say about this.
The PoE says that there is no way to distinguish the effects of a
local gravitational field from the effects of being in an absolutely
accelerating reference frame.
...
To first order, general relativity theory
predicts equivalence between gravitational
mass and inertial mass, but
a) GR might be wrong, and
b) even within GR, you have to be a bit fussy
about how you state the equivalence principle,
or you'll get a "principle" that isn't even
theoretically true. ...