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If the gravity effect we witness is due to objects travelling along geodesics, why is the geodesic different for objects with different velocities if there is no gravitational force as such?

For example, a fast asteroid may veer toward a planet and miss it whereas a slower asteroid may get closer to the planet and possibly hit it.

What would cause an object moving in a continuous direction (as it must) to take a different path because it is slower, unless there is another force acting on it?

Does space curve more dramatically for slower objects? If so, is this related to time dilation?

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    $\begingroup$ Since asking I realised that I wasn't thinking about space-time, meaning that the trajectories of fast and slow objects are actually different, therefore would be on different geodesics! Apologies. $\endgroup$ – CompanyDroneFromSector7G Oct 16 '14 at 12:36
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    $\begingroup$ You can answer your own question... $\endgroup$ – innisfree Oct 16 '14 at 12:42
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    $\begingroup$ Indeed, if you realized the answer to your question, I would encourage you to post an answer. We love to see when people learn and answer questions themselves. I personally get this warm and fuzzy feeling to see learning in action. Plus you can get rep and badges if the satisfaction of others isn't enough motivation for you $\endgroup$ – Jim Oct 16 '14 at 13:07
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I realised why this happens.

Objects moving at different speeds have different trajectories in spacetime, so they naturally follow different geodesics.

This is explained very well here: http://curious.astro.cornell.edu/physics/the-theory-of-relativity/140-physics/the-theory-of-relativity/general-relativity/1023-if-gravity-is-a-curvature-of-space-rather-than-a-force-why-do-a-ball-and-bullet-follow-different-paths-intermediate

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    $\begingroup$ I remember making this point here. $\endgroup$ – Nikolaj-K Oct 16 '14 at 13:46

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