Comment by youzicha on Stupid Questions October 2020 · 2020-10-23T03:32:05.291Z · LW · GW

There is a paper here which does something like this, and draws pretty pictures. The metric has been "absolutized" by replacing the negative coeffecients with their absolute values, so it becomes Riemannian instead of Lorentizian, but the diagram is then annotated with a bunch of yellow triangles showing "the direction of time", and together these two things apparently contain all the information of the original spacetime. For the spacetime around an ordinary planet all the triangles point in the same direction, so this Riemannian version seems like a valid representation, I guess.

Anyway, the red line in Figure 5 in the pdf shows something like what OP was asking about: a ball is thrown straight up, turns around, and falls down along the same path again.

I think maybe the key point is that although the ball is retracing its path in space, in spacetime it's just a long line which never loops back on itself, so it may be easier to believe that it's going "straight ahead".