Gravity Train

The gravity train thought experiment

The idea of a gravity-powered train through the Earth first appears in a 1679 letter from Robert Hooke to Isaac Newton, though it gained wider fame through later popularizations by Lewis Carroll and, in the twentieth century, by physicist Paul Cooper in a 1966 American Journal of Physics article. The concept is beguilingly simple: drill a straight, frictionless, evacuated tunnel between any two points on Earth’s surface, drop a vehicle in at one end, and gravity alone will carry it to the other end — no engine required. The vehicle accelerates toward the midpoint of the tunnel (the point of maximum depth), then decelerates symmetrically until it arrives at rest at the far city. What makes this remarkable is that the transit time is independent of the tunnel’s length.

Why 42 minutes, always

The mathematics follows from a beautiful result about gravity inside a uniform-density sphere. At radius r inside the Earth, only the mass interior to r contributes a net gravitational pull (Newton’s shell theorem eliminates the outer layers). For uniform density, the gravitational acceleration at distance r from the center is g(r) = gₚ(r/R), proportional to r. The component of this acceleration along the tunnel chord turns out to be −ω²x, where x is displacement from the midpoint and ω = √(g/R). This is exactly the equation for simple harmonic motion, identical in form to a mass on a spring. The half-period (one-way trip time) is T = π/ω = π√(R/g) ≈ 2530 seconds ≈ 42.2 minutes. Because ω depends only on Earth’s radius and surface gravity — not on the chord length — every tunnel gives the same transit time, whether from London to Paris (340 km apart) or from pole to pole through the center (12,742 km). The maximum speed, reached at the tunnel’s midpoint, does depend on the chord length: for a tunnel through Earth’s center it reaches about 7.9 km/s (orbital velocity), while a short tunnel produces only a gentle ride.

Connections and impossibilities

The 42-minute result conceals a beautiful coincidence. The orbital period of a satellite skimming Earth’s surface is Tₒₛₚ = 2π√(R/g) ≈ 84.4 minutes — exactly twice the gravity train’s one-way trip time, because both systems share the same angular frequency ω. This also connects to the brachistochrone problem: the fastest frictionless slide between two points on Earth’s surface (through a curved tunnel shaped as a hypocycloid) takes even less time, but the straight-tunnel version has the elegant property that all trips are synchronous. In practice, the gravity train is thoroughly impossible: Earth’s interior reaches over 5,000°C at the core, pressures exceed 360 GPa, the mantle is viscous rock, and maintaining a vacuum in a tunnel thousands of kilometers long presents engineering challenges that make even Elon Musk’s Hyperloop look modest. Air resistance alone would rob the train of nearly all its energy. Yet as a thought experiment, it illuminates how simple harmonic motion, shell theorems, and orbital mechanics are all faces of the same inverse-square law — the deep unity beneath Newton’s gravity.