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I'm a secondary student who never studied any general relativity level maths (yet!), but I heard that in general relativity objects with mass and energy deform space-time, and therefore bodies move in the shortest path in this deformed space. At least this is what I think this article means.

Wikipedia - General Relativity

There is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow. The curvature is, in turn, caused by the energy–momentum of matter.

Here's my question: If this is the case how come objects moving initially in the same direction but at different speeds, have end up having different trajectories? I mean, the length of a path doesn't depend on the speed you traverse it with? Unless some relativistic phenomena is at work?

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The trajectories are worldlines in 4D spacetime, not just paths in 3D space. They maximise proper time along the trajectory. They do not minimise distance along the curve traced through space. An object tracing a fixed path in space at different speeds would trace different worldlines in space-time.

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Philip Gibbs: "The trajectories are worldlines in 4D spacetime, not just paths in 3D space. They maximise proper time along the trajectory." -- If a given worldline happened to "maximize proper time" (between any two events which were visited by that worldline) then such a worldline is called "free" (or: "the worldline of a free participant"). However, this was not really required by the OP. –  user12262 Oct 27 '13 at 19:22
    
@user12262 Of course this is assuming no (other) forces, just gravity. The description implied by the OP did not seem to suggest that other forces would be involved but it is worth clarifying. –  Philip Gibbs Oct 27 '13 at 22:22

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