How does Curvature Relate to 'Gravitation'?

I have searche many genral relativity texts and have not found an answer to the following question: How does curvature translate into the Newtonian idea of gravitation? For example, how is Newton's law of gravitation, where all matter attracts all matter, an approximation to the idea of curvature? I vaguely know that the idea is related to the equation of geodesic deviation, but could somebody explain this more clearly?

Thanks!
 

atyy

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The handwavy idea is to define a new sort of acceleration. In Newtonian physics, a force produces an acceleration. This can be measured by an accelerometer. The funny thing is that when an object is acted on by gravity alone, ie. when it is falling freely, an accelerometer attached to it reads zero, even though the acceleration is not zero. If we define a new sort of "G-for-geometry-acceleration" that is what the accelerometer reads, then we end up with a geometric theory of gravity. In this view, gravity curves space, and freely falling particles move with zero G-acceleration which means they move on "straight lines" in curved spaces called geodesics.
 
Thanks! Unfortunately, all GR yests I have pored over do not emphasize this aspect of GR enough, in my opinion.
 
Sorry! Substitute 'texts' for 'yests'. To say my keyboard is antiquated would be quite an understatement.
 

A.T.

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Sorry! Substitute 'texts' for 'yests'. To say my keyboard is antiquated would be quite an understatement.
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