PeroK said:
Suppose two balls were "dropped" at the same time on opposite sides of the Earth. Which way would the Earth fall?
Answering this requires drawing a key distinction between local and global phenomena. Locally, you can view the ball and the feather as being at rest in an inertial frame, in which physics works exactly like it does in special relativity. The Earth is accelerating upward in this local inertial frame.
Globally, there is no single inertial frame that encompasses the entire Earth, and, as your observation makes clear, local inertial frames on opposite sides of the Earth do not "line up" with each other. This is because of spacetime curvature (which can be ignored in a single local inertial frame). So to properly account for the behavior of objects globally around the Earth, you have to talk about spacetime curvature and how it affects geodesics (the paths of freely falling objects).
However, even on this global view, it's still true that the ball and the feather would follow the same paths (the same geodesics), despite their different masses. So even globally, the motion of freely falling objects can't depend on any property of the objects; it has to depend only on properties of spacetime itself. That is the real point of Einstein's observation.
inertiaforce said:
The video actually said that Einstein's view was that no force was acting on the feather or on the ball.
Which is correct: if you attach accelerometers to the feather and the ball, they will read zero, indicating that no force is acting on them.
This definition of "force" is different than the Newtonian one; but the Newtonian one had the disadvantage that "forces" like gravity (and also centrifugal force and other "fictitious forces") could be acting without there being any possible direct measurement of them with an accelerometer. Einstein's definition is cleaner because it makes "force" correspond exactly with direct accelerometer measurements.
I notice, btw, that you are now
not using the word "falling" in connection with what the video said. Saying that no force is acting on the ball and feather is not the same as saying they are "not falling", because "no force acting" can be measured directly (as I described above), but "falling" vs. "not falling" can depend on how you choose to describe the motion, i.e., it's observer-dependent. Einstein was trying to focus on things that are
not observer-dependent.