Dear Jules Verne's SA,
Great work on the constant density solution. One thing, though: to travel across a secant, you'd need some sort of frictionless track system, because you'd be pulled toward the center of the earth. Just use the Shell Theorem: the shell containing you cancels itself out, so you just calculate the attraction between yourself and a sphere defined by your radius. It is indeed a cool idea, though, especially the constant time, which makes it sound like a slow wormhole. Thanks.