Geodesics

In the specialized language of "differential geometry", a geodesic is path of extremal length betwen two points. But the two points have to be in some well-defined kind of space equipped with a means of defining paths between points and measuring their distance. This kind of space (or spacetime) is called a "manifold with a metric".

Here is a familiar example: the X-Y plane with the rule for computing differential arc length along as arbitrary path being ds^2 = dx^2 + dy^2. The geodesic paths on this manifold with this metric are just straight lines. A straight line gives the shortest distance between two points in this space with this metric.

On a two-sphere of radius a with the metric rule ds^2 = a^2(dtheta^2 + Sin[theta]^2 dfi^2), the geodesic paths are arcs of great circles.

When we're dealing with spacetime rather than just space, things get more complicated. A spacetime metric contains minus signs, whereas a space metric has all positive coefficients, so that we encounter paths that have zero spacetime arc length.

The Einstein equations tell us that particles on a given spacetime manifold with a metric that solves the Einstein equations will travel on the geodesic paths for that spacetime and metric. Particles on geodesic paths are called "freely falling" particles. This means the particle is travelling only under the influence of gravity -- without rocket ships or wheels or feet or any other means of guiding its motion.

Geodesic paths with the zero arc length property described above turn out to represent the paths of massless particles, and this in turn winds up enforcing causality in the motions of particles in that spacetime.

Now physics can be expressed in the language of differential geometry.

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