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What is the meaning of gravitational acceleration $g$ in the General Theory of Relativity, given that objects in free fall experience weightlessness?

Can gravitational acceleration be thought as a measure of how much the spacetime is curved?

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g is the acceleration with which objects move through space-time, from the point of view of an inertial reference frame. Gravity is just the curvature of space-time created from any mass (according to GR it's not a "real force", but a fictitious force).

Sure, from the reference frame of the object in free-fall, it's not accelerating. From the point of view from an inertial reference frame, it is accelerating through space-time due to how gravity curves space-time in "valleys".

Here's a link to a good question for more on this:

Does GR imply a fundamental difference between gravitational and non-gravitational acceleration?

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