Geodesics

Free-fall trajectories determined by curved spacetime geometry. Theory, formulas, worked examples and interactive graphs.

Conceptual context

In general relativity, a freely falling particle is not pushed by gravity in the Newtonian sense. It follows the straightest available path through curved spacetime. That path is a geodesic. Geometry determines how its coordinates change, how $r$ evolves, and how the traveler's $\tau$ relates to the time used by external observers. The central challenge is to separate coordinate-dependent behavior from quantities that can be measured locally. A geodesic may therefore have nonzero $d2x^\mu/dlambda2$ while an accelerometer carried by the particle records no proper push.

🟢 Essential level

A geodesic is the natural path of an object moving without engines, drag, or other nongravitational forces.

In flat spacetime, that path would be straight and uniform. Near a mass, geometry changes, so the path appears curved when written in coordinates. The $r$ labels the size of symmetry spheres, while $\tau$ is the reading of the traveler's own clock. The slope of $r$ against $\tau$ is the $\frac{dr}{dtau}$. A negative slope means motion toward smaller radii. The $d2x^\mu/dlambda2$ describes how that coordinate slope changes, but it is not automatically a felt force. Geometry sets the available track, whereas initial conditions select the particular geodesic. This distinction lets us describe free fall without treating gravity as an ordinary force acting inside spacetime.

In November 1915, Einstein used exactly this idea to calculate the precession of Mercury's perihelion: the planet's orbit, understood as a geodesic in the spacetime curved by the Sun, advances by 43 arcseconds per century beyond the Newtonian prediction, precisely the value that had gone unexplained for decades.

🔵 Formal level

Basic formulas

The fundamental relation states that the tangent vector is parallel transported along the path:

Affine geodesic equation

The $d2x^\mu/dlambda2$ is balanced by connection terms built from the metric. These terms are not an extra force; they encode how a locally straight path can look curved in a chosen coordinate system. For a massive particle parametrized by $\tau$, timelike causality is controlled by:

Four-velocity normalization

The $g_mu_{\nu}*u_{\mu}*u_{\nu}$ remains constant and relates temporal and spatial components.

Real student questions

Is a geodesic always the shortest distance? No. A timelike geodesic generally extremizes $\tau$ and can locally maximize it.

Why can $d2x^\mu/dlambda2$ be nonzero in free fall? Because coordinates can bend while proper acceleration remains zero.

Does the horizon stop the object? No. The divergence belongs to Schwarzschild $\frac{dt}{dtau}$, not the local $\tau$.

Can large $\frac{dr}{dtau}$ violate relativity? Not automatically, because it is not necessarily a locally measured speed.

What decides between orbit and plunge? The combination of $E$, $L$, and geometry.

Final synthesis

Geometry supplies the rules of motion, while initial data select one path. Correct interpretation requires separating invariants from coordinate effects and reading $r$ versus $\tau$ through its slope, turning points, and physical limits.