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If I hold such an accelerometer in my hand (so that it is attached to
my local reference frame), and suddenly an "ideal" trap-door in the
floor opens (t=0) and me and my accelerometer fall through "in ideal
fashion" (IOW pretend there is no drag, no rotation due to my spastic
twisting when I realize I'm falling, etc), I submit that the
accelerometer will not change its reading, even momentarily. From
t=0, every particle in the accelerometer and me are falling together,
and by looking at the accelerometer alone, it's as if we were still
standing on the floor.