My understanding here is that the ball (like everything else) is moving at a constant "speed" through spacetime. From the Earth's reference frame, when the bal

My understanding here is that the ball (like everything else) is moving at a constant "speed" through spacetime. From the Earth's reference frame, when the ball is frozen in space, all of that speed is in the time component (as is yours). The reason its path is parabolic is due to the very slight curvature in spacetime induced by the Earth; if you instead stand in a "flat" reference frame (e.g. freefall) and watch the ball, it will indeed look like it's going in a straight line (but the Earth will follow a curved path in spacetime, allowing it to "catch" the ball). You can verify this using the link to https://timhutton.github.io/GravityIsNotAForce/ https://timhutton.github.io/GravityIsNotAForce/. I won't push my luck by pretending to understand how this changes when your spatial velocity is nontrivial (intuitively I feel like it should not even possible to differentiate between spatial and temporal components except relative to your own reference frame, but I could easily be wrong there). But in the case you describe it seems pretty clear.