I'm confused trying to understand what's happening in terms of spacetime geodesics when a ball is thrown and its trajectory plotted, height against time to give a parabola.
I read (from more than one source) that although the curve is a parabola in three (spatial) dimensions “we must recognize that a massive body like the Earth actually curves the coordinate system itself, so that rather than following a curved path in a flat (Cartesian) coordinate system, the ball actually follows a minimum-distance path, or geodesic, in a curved coordinate system.” In other words, the problem here is “mapping curved space-time onto a flat piece of paper.”
OK. But then I also read (again from more than one source) that, as an example of the weakness of the Earth's gravitational field, if the ball is in the air for, say, 1 second, using $ct$ units this is equivalent to $3\times10^{8}\,\mathrm{m.}$ One author then talks about plotting this curve on an ordinary $ct/x$ spacetime diagram saying “in this case the spacetime is practically flat, and thus the geodesic is very close to a straight line.”
OK again, but aren't these two positions contradictory? If you aren't allowed to plot curved spacetime using Cartesian coordinates in the first example, how come you are in the second? And vice versa of course.
Thanks