This is one of those old ideas that’s just too interesting to let go. Tunnel through the Earth to link any two points. A vehicle (say, a train) released at one end will fall through and pop up at the other end about 42 minutes later, regardless of the distance and regardless of the weight of the train. The only energy added is to compensate for friction (incl. drag). Everything else is free.
Unfortunately, tunneling through the Earth is not only impractical, given our understanding of the core, it’s downright impossible. The heat and pressure are only the first problems to encounter. Radiation and magnetic flux are another. The linked article doesn’t go into the current best understanding that the inner and outer cores are molten, moving, and incredibly hazardous. Imagine trying to support a rigid tunnel in a swirling field of viscous magma. And keeping a vacuum in the tunnel would be key, because the air pressure and therefore drag would be much higher than on the surface, even though gravity near the core would go to zero.
However, the principle could work without such drastic tunnels. A tunnel from NY to Washington wouldn’t need to go very deep, perhaps not even past the outer crust. And a gravity tunnel from 14th St. to 42nd St. in NY would be extremely shallow. Of course, taking 42 minutes for that last trip would be a bit of a setback, so a steeper slope would be advised. Our subways could use only the energy they need to overcome friction. They’d work just like a roller-coaster, without the loops.
In wondering why they didn’t do that in the first place, I only came up with two reasons: one is that many subway tunnels are actually just sub-streets, not bored into rock and earth (Broadway, for example, was excavated and then re-covered at street level); and two, putting two trains on the same gravity well is dangerous, since if you lose power and don’t make up for friction to reach the next level station, both trains would swing back and forth like a pendulum, resulting in a potential collision. I think backup brakes could prevent that, but it’s a risk. Of course, I guess the third issue is that the tracks are so bad in NY, that a constant-but-low speed is better than a sprint to the middle of the gravity well.