The geodesic equation is as follows:
The details aren’t important, but this equation lets you find the straight lines on a curved manifold. I have seen three derivations of this equation and thought I would share them here. Each derivation sheds some new light on the equation.
1. Minimize Distance
If you have some path on your curved manifold from one point to another, you can measure the distance of your path. If you want this path to be a local minimum (meaning all nearby paths are longer) then you can derive the geodesic equation using the Euler Lagrange equation. This derivation shows you that “straight” lines are the ones that minimize distance in a very global sense.
2. Parallel Transport
If you have a curved manifold you usually picture it sitting in some higher dimensional space. A warped piece of paper sits nicely in regular three dimensional space. A warped higher dimensional manifold might also sit nicely in an even higher dimensional space. If you have, say, a little ant moving in some direction on this manifold, embedded in a higher dimensional space, it will take a step forward along it’s velocity vector. Now, note that it’s velocity vector is tangent to the manifold. If the ant moves straight along the velocity vector it will fall off the manifold. Instead, the ant must be constrained to move on the manifold. Therefore it’s velocity vector must change a little bit when it moves. The amount it’s velocity vector changes is given exactly by that upside down L figure in the equation. Through the process of taking a step forward as best the ant can along its velocity vector, “parallel transporting” its velocity vector along this path, then taking another step forward, it then moves again along this new transported velocity vector.
This is a beautiful proof because it gives this dynamical local description of what’s going on. A trajectory is generated by a single initial velocity vector tangent to the manifold and keeps following this vector, with the vector constrained to be tangent to the current point.
3. The Equivalence Principle
This is the physics proof of the above equation. It imagines the trajectory as the path of a falling observer in a space time manifold. Einstein’s equivalence principle states that for any free falling observer there should be a reference frame in which the observer is in regular flat (Minkowski) space time. In this reference frame, the observer moves at a constant velocity along a straight line in space time. Then, doing a coordinate transform from that inertial reference frame into your preferred coordinates, out pops the geodesic equation.
This derivation is beautiful for many reasons, of which I will list a few. For one, it gives beautiful physical intuition. There is some reference frame in which we know thing move along straight lines and the equations are easy. It stands to reason that when we transform coordinates, these equations are still “straight.” Furthermore, it shows how the motion of particles is guided completely by the curves of space time. The particle is really just moving in a constant velocity in a straight line in its inertial frame!
That these three derivations all fit together show how beautiful physics is. All these ideas are completely different ways of getting at the exact same thing.