How free-fall paths reveal curvature

Release two balls side by side above an ideal spherical Earth, initially at rest relative to one another. Each falls towards Earth’s centre. Their paths draw together because the direction towards the centre differs slightly at their two starting positions.

What matters is how their separation changes. Two objects in flat space can approach each other simply because they were sent in different directions. Here, the balls begin with no motion relative to one another, then start drawing together as they fall. That change is their relative acceleration.

Straight paths on a curved surface

A globe provides a useful analogy. Start two walkers at different points on the equator and send both due north along lines of longitude. They follow great-circle paths, which are the surface’s equivalent of straight lines, yet their separation decreases towards the pole.

The straightest available paths in a geometry are called geodesics. In general relativity, freely falling objects follow geodesics through spacetime. An observer drawing position against time may see a curved line, even though no supporting force acts on the object.

The globe helps us picture how geometry affects neighbouring paths. It represents a curved surface; the ball’s path involves time as well as space, so the analogy covers only part of the idea.

Tidal effects

Two balls falling one above the other behave differently from two falling side by side. The lower ball experiences a stronger downward acceleration in the ground-based description, so the vertical separation tends to increase.

This stretching in one direction and squeezing in another is a tidal effect. It also explains the limit of our falling-room example: objects can float together approximately, while careful measurements still reveal small changes in their separation.

Remaining at a fixed height

A stone resting on the ground would fall if its support disappeared. The ground keeps pushing it away from that free-fall path, holding it at the same height. Around Earth, the material beneath the surface provides this support without the planet having to expand.

We have used the word “geodesic” for a free path through spacetime. The next lesson makes that idea more concrete by following a stone that is held, released and thrown. It will also explain how a path can be straight in the spacetime sense while looking curved from the ground.