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Sorry for the long post, for the bad and confusing english, and feel free to correct the errors when you find them. Thank you in advance.

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Sorry for the long post, for the bad and confusing english, and feel free to correct the errors when you find them. Thank you in advance.

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Yes, from the standpoint of General Relativity (GR), the Earth does not feel any "force of gravity" from the Sun, and it moves on the straightest possible path that it can through the spacetime around the Sun. The Earth's path looks curved to us because the spacetime itself is curved by the Sun's mass.the Earth moves around the Sun because of the deformation caused by the Sun on space causing the idea of a gravitic force rather than an actual force.

The same way. A satellite orbiting the Earth moves on the straightest possible path that it can through the spacetime around the Earth, just as the Earth does around the Sun. The satellite's path looks curved because the spacetime around the Earth is curved by the Earth's mass.So, if gravity is a product of the space-time disturbances, how does the Earth's gravity appears?

I'm not sure why you would see a problem with this. The spacetime around the Sun has room for multiple bodies in it. If you're wondering how the deformations due to multiple bodies interact, see below.And if the sun disturbs space like a bowling ball on a matress how can there be several bodies orbiting the sun?

Because they have sideways motion as well, just as a satellite orbiting the Earth does. Objects that don't have enough sideways motion, like a rock dropped from a rooftop, do fall into the object at the center (the Earth, in the case of the rock). Put another way, these bodies have angular momentum, and that affects their trajectories.And why dont these bodies fall to the sun

They do, but which deformations are significant enough to matter depends on the distance scale you are looking at. On the scale of the Solar System, the only significant curvature of spacetime is due to the Sun; the planets are all too small to make much difference. But on the scale of the Earth, the curvature of spacetime due to the Earth is the only one that's significant; the curvature due to the Sun, at the radius of the Earth's orbit, is much too small on the size scale of the Earth.is it because they deform space the same way the sun does

None of the deformations "reverse" any of the others; they all add up (but which ones are large enough to be significant depends on the distance scale, as above). It's worth noting, though, that the effects due to multiple objects do not, in general, add linearly; GR is a nonlinear theory. (In the Solar System, the nonlinearities are extremely small, so for virtually all purposes we can add effects from multiple bodies linearly and get a good enough approximation; but the exact theory is nonlinear.)reversing what the sun does?

Because their sideways motion is different than that of planets, relative to the size of their orbits. Put another way, their angular momentum relative to their orbital energy is very different from that of planets. The orbital parameters of a particular object don't just depend on the properties of the underlying spacetime; they also depend on the specific motion of the object itself.Also, why do comets have much more eliptical orbits than the planets if the sun disturbance on space is even?

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The apple doesn't have any sideways motion (at least, not enough to be significant), so the straightest possible path for it to follow, in the curved spacetime around the Earth, is the path that free-falls straight down. So as soon as the apple is no longer connected to the tree, it will follow that path. (While it's connected to the tree, the tree is keeping it from falling; and the tree itself is held up by its own structure and by the Earth.)

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Ok, thank you very much.

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