Any Way You Fall

Drill a straight tunnel clean through the Earth, step in, and let go — no air, no friction, nothing but gravity. You reach the far side in about forty-two minutes. Drill a different tunnel, one that misses the centre entirely, and drop again. Same forty-two minutes. Every straight tunnel keeps the same clock. Drag the mouths of the tunnel below and see.

The forty-two minutes is not a coincidence and it is not a heavy calculation once you see it. Inside a uniform ball of rock, only the mass below your feet pulls on you; the shell above cancels out (Newton proved this). That inner mass shrinks as you fall, so gravity weakens in exact proportion to how deep you are — pulling hardest at the surface, vanishing at the very centre. A force that pulls you back in proportion to your distance from the middle is the definition of simple harmonic motion: the same law as a mass on a spring, or a pendulum in a small swing. You don't fly past the centre and escape; you slow, stop exactly at the far mouth, and would fall back forever. And the period of that oscillation depends only on the planet — not on where you started, not on how long the tunnel is.

That last clause is the strange one, and it is worth dwelling on. A tunnel from pole to pole is nearly 12,700 km long. A tunnel between two cities a short hop apart is a stubby chord near the surface. The stone in the long tunnel has vastly farther to travel — but it also falls through a much steeper drop and reaches a much higher speed at the middle, and the two effects cancel exactly. Set the spread to anything you like above and read the transit time: it does not move. This is a cousin of the tautochrone — the curve on which a bead takes the same time to reach the bottom no matter where you release it. Here the equal-time property lives not in a special curve but in gravity's own shape inside a uniform world.

The real Earth is faster — and breaks the spell

Switch the model above to Realistic. The Earth is not uniform: it has a small, ferociously dense iron core. Because so much mass is packed near the middle, gravity does not fade away as you descend through the mantle — it actually holds near 9.8 m/s² almost the whole way down, and only collapses once you enter the core. Stronger pull for most of the trip means a quicker fall: the pole-to-pole tunnel now takes about 38 minutes, not 42. And the beautiful equal-time coincidence quietly dies — a shallow tunnel and a deep one no longer share a clock. The tidy answer was a gift of the uniform assumption; the real planet keeps its own, messier time. Both numbers below are integrated live in your browser from the same physics, and both are re-checked offline against the standard Earth model.