By Newtonian physics if we are above a infinite plane with acceleration of gravity $2$ then falling object reach to plane in $1$ second and this is independent of horizontal speed of falling body. Now in GR... (I prefer to throw objects in $xz$-plane and throw $y$-axis away and use it for $t$)
(About time of falling) Is time of falling is independent of horizontal speed of throwing? By clock of the throwing objects? By clock of the observer who throws objects?
(About curvature of falling) I mean curvature in spacetime curve. I considered simply parabola of Newtonian physics extended to spacetime. As I computed, curvature of parabola in falling by zero horizontal throwing speed I found curvature $2$ . (by $c=1$). if I throw object horizontally by speed $c=1$, I found curvature $\frac{1}{2}$ for parabola of Newtonian mechanics. (MTW in page 32 says this curvatures are almost the same for speeds of a throwing ball and a bullet) Are in GR curvatures of true geodesics the same? or really differ so much? how much?
And... Could you give metric and geodesics for this plane in GR?
