That equivalence plays a crucial role. The central idea of General Relativity is that there are many different ways of assigning spacetime coordinates to events, all of which ways can be described by the same laws of physics. These ways include coordinate systems that are accelerating with respect to each other. In one coordinate system, all the objects at rest in the other seem to be accelerating, with nearby ones accelerating at the same rate. In that coordinate system you'd need a gravitational field to explain why, for no other reason, all the objects in some region were accelerating together.
Now lets look at the Newtonian version of this, which not only has inertial masses, mi
, but also gravitational masses, mg
. Since the gravitational force on an object is proportional to its mg
, and the acceleration is given by F/mi
, the acceleration would be proportional to mg
. Unless every object has the same mg
then gravity will cause nearby objects to accelerate differently. That's completely different from the effects of changing coordinate systems.
So General Relativity only makes sense if every object has the same mg
. It's most convenient then to just call both masses the same thing.
(published on 11/16/2009)