Why would geodesics depend on velocity in general relativity?

In general relativity, it's the curvature of spacetime that gives the effect of gravity, due to objects following geodesics. What I learnt about geodesics is that they are like straight lines in Euclidean space. But the Wikipedia article on Geodesics in general relativity says that geodesics depend on objects velocity, which contradicts what I originally thought:

I originally thought that, since there's no gravitational force, an object with zero net force of could still experience effect gravity. Now (at least in Euclidean space) if you aren't experiencing force, hence no acceleration, your path (that you'll eventually gone through) shouldn't change as velocity varies, as velocity in this case should merely determine how fast you gone through a certain length of the path.

Wouldn't this be the same for non-Euclidean space? Like if a geodesic is a straight line, wouldn't the path that I'll be taking also be independent from velocity? Am I wrong with my statement above? Or is there something that I've ignored (maybe the curvature due to velocity...as kinetic energy?).

• If I understand correctly (correct me if I don't), does this mean since what I refer as changes in velocity, when substitute it into relativity and requiring it to satisfy $ds^2=dt^2-dx^2$, variation in x direction also effects in t direction. So as the result, what I refer as "changing velocity (in magnitude)" actually corresponds to varying the velocity's angle. Aug 20 '21 at 7:38