If you could dig a tunnel in the right shape through the Earth, you could travel from New York to Los Angeles in 42 minutes–by just using gravity! My solution, illustrated using Mathematica, is top ten in a Google search of “gravity train.”
Hypocycloid curves. Digging a hypocycloid curve between any two points on the Earth will allow you to use a gravity train to travel between them. These tunnels are too deep to be practical.
Amanda Maxham 