On a flat plane, the shortest path between two points is a straight line. On a sphere, it’s a great circle. On curved spacetime, it’s a geodesic — the path that a freely falling object follows when no forces act on it except gravity.

General relativity’s central insight: gravity isn’t a force. It’s geometry. Objects in free fall follow the straightest possible paths through curved spacetime. The Earth orbits the Sun not because a force pulls it, but because the Sun’s mass curves spacetime and the Earth follows a geodesic through that curvature.

The geodesic equation

A geodesic extremises the spacetime interval between two events. The resulting equation of motion is:

$$\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu{}_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0$$

where $\tau$ is proper time and $\Gamma^\mu{}_{\alpha\beta}$ are the Christoffel symbols — functions of the metric and its first derivatives.

In flat spacetime, $\Gamma^\mu{}_{\alpha\beta} = 0$ everywhere, and the geodesic equation reduces to $d^2 x^\mu / d\tau^2 = 0$ — constant velocity in a straight line. Newton’s first law is a special case of the geodesic equation.

In curved spacetime, the Christoffel symbols act like a “gravitational correction” that bends the path. But there’s no force — the path is straight; it’s the space that’s curved.

Types of geodesics

In spacetime, geodesics come in three flavours, determined by the metric:

Massive objects follow timelike geodesics. Light follows null geodesics. The distinction is built into the metric signature.

The equivalence principle

Einstein’s equivalence principle: in a small enough region, the effects of gravity are indistinguishable from acceleration. A freely falling elevator feels weightless — locally, spacetime looks flat.

Mathematically: at any point, you can always choose coordinates (called Riemann normal coordinates) in which $g_{\mu\nu} = \eta_{\mu\nu}$ and $\Gamma^\mu{}_{\alpha\beta} = 0$ — flat spacetime, no gravity. But you can’t do this globally if spacetime is curved. The curvature shows up as the failure of this trick to work over extended regions.

This is why geodesics matter: they’re the local definition of “straight” that works in curved space. And free fall — the most natural state of motion — follows them.

Geodesic deviation

Two nearby geodesics on a curved surface don’t stay parallel. On a sphere, two great circles starting “parallel” at the equator converge and meet at the poles. This geodesic deviation is the physical manifestation of curvature — and it’s what tidal forces are. The rate at which neighbouring geodesics converge or diverge is governed by the Riemann curvature tensor.