I have a related question which may broaden the image for conceptual clarity. Imagine an object (me) moving through "flat" outer space, far from any gravitational bodies. We can say that the geodesic I am travelling along is essentially straight or flat, as is its worldline, correct? So me moving at a constant velocity simply moves through space (time) in a straight line. Now we introduce a gravitational body into the picture, say a moon-sized object. Ok, so now the spacetime geodesic is no longer flat, it is curved in the vicinity of the moon. I am cruising along in my flat spacetime straight geodesic when all of a sudden I start to feel this curved space develop in front of me. My geodesic begins to become more and more curved as I slingshot past the moon in a curved path. Ok, that's all fine and good, but my question is why do I not maintain a constant velocity while I am following this curved geodesic? Why is there acceleration involved? Where does that come from? Is there some sort of physical scenario where one would follow a curved geodesic where they maintained constant velocity? I'm guessing no, but why? Similarly, does the rate of change of the geodesic, or simply the degree of curvature, determine the value of the acceleration? For instance, say instead of the moon I go cruising by an Earth-sized object. The curvature of the spacetime geodesic/worldline is more pronounced or "steep" there, if you will, therefore there is more gravity and a greater acceleration impelled upon a passing object. My question is why does a greater curvature compel a greater acceleration? Why isn't just the path of the moving body affected by the curvature while it continues to travel at a constant velocity. Edit: Now I'm going for "dumb question" broke so forgive me: It doesn't have anything to do with centripetal acceleration caused by the curved path, does it?