I really really need to get past my hang-ups in calculus, so I can start to understand Special and General Relativity better. I keep getting hung up on this one "thought experiment" that is supposed to illustrate the equivalence of "Inertial Mass" and "Gravitational Mass". Because, except at the very limit... it doesn't show that at all.

Ok, the thought experiment is the classic one of the person in an enclosed room. Using a pendulum, timing the swings, said person is in theory unable to distinguish whether the room is resting on a planet, or if the room is aboard a spaceship undergoing the equivalent acceleration.

However, unless the gravitational source is very distant, all it takes is two, non-swinging pendulums to make the determination. On a ship under thrust, the two pendulums will be parallel. On a planet, the two pendulums will will each point towards the center of mass, meaning that they will make - admitted, in most circumstances extremely tiny - angles with each other.

So, since this thought experiment is supposed to illustrate the equivalence of inertial and gravitational mass - I'm left with the conclusion that in fact, except at the limit where the gravitational field is indistinguishable from flat, the two masses are *NOT* the same.

And that makes my head hurt. Because I'm pretty certain I'm not smarter than Einstein, Feynman, Guth, et al., who have been all over this territory. But, damn, that's some scary math, too.