msumm said:
... If is the distance changes at a constant rate then the 2 statements are equivalent, right?
We are talking about something that is at an intuitive level. To make math'ly rigorous sense of relative motion as a concept you need the two objects to be in the same reference frame, which is typically not the case. So this is purely intuitive---how you think about it.
Personally I think of an object as moving if there is some destination that it is approaching. It has to be "going somewhere". If there is no definable place that it is getting nearer, then it isn't moving (for me, intuitively).
You are welcome to have different intuitive notions about it. Intuitive perspective on things is not hard science that we have to argue about
So take the balloon example. All existence is concentrated on the 2D surface, there is no surrounding 3D space, no inside or outside of the balloon surface. And there are two dots painted on the surface. The distance between them is increasing but
neither dot is going anywhere. For me, in that 2D universe, neither is moving.
Or take two widely separated galaxies in a curved expanding universe. In GR you can't get one (flat, non-expanding) reference frame to fit the whole thing---it makes too bad a fit. Like trying to dress in clothes made of wooden boards.
So the two galaxies are in two different reference frames. Each one has a locally defined frame that approximates its immediate neighborhood. But there is no global shared frame. Relative motion is math'ly undefined.
Perhaps each galaxy is actually moving slightly in its own frame. Galaxies do typically have their own small "peculiar" velocity of a few hundred km/s. But this is negligible compared with typical distance expansion rates which can be several times the speed of light. So let's ignore the socalled "peculiar" velocities and consider them as both stationary.
Neither one is getting closer to anything else, just like the dots on the balloon. But the distance between is increasing.
With distance expansion, the rate has changed greatly over the U history, and it is changing now, but
nobody ever feels any acceleration. In what direction would the acceleration be?
Distance expansion is the same in all directions, for everybody. Nobody is going anywhere so there is no acceleration in any direction.
So anyway, my intuition seems to work OK for me and it says that this is not ordinary motion (and would not be ordinary motion even if it were at constant rate as you suggested). You have to decide what your feeling about it is.
Mathematically speaking it is rigorously NOT relative motion, as I said, because no global frame. Could be that it is advisable to align intuition with the math, and with the rough astro community consensus in this case.