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An object moves at a constant speed relative to the local spacetime but due to the warped spacetime it appears as if accelerating. How does spacetime curvature exactly causes this effect?

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The relevant part of the mathematics here goes by the name "geodesic deviation". This is the fact that lines which are as straight as possible, near any given event, do not behave the same way as straight lines in a Euclidean space. The worldlines of Earth and a nearby object such as a dropped apple are both geodesic, but the two geodesics approach one another more and more rapidly. A nice analogy is to draw geodesic lines on a sphere. If you start with two lines at the equator, both heading in the direction "north" and setting out parallel to one another, and then continuing each as straight as possible, then they will nevertheless meet at the north pole. Something similar happens to worldlines in spacetime, only now you have 4 dimensions and the temporal dimension is not quite like the spatial ones.

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