General Relativity Tutorial - Geodesics

Keith Ramsay

John has avoided defining geodesics independently of parallel transport, but I think there may be some merit in stating how one can do so. Geodesics are locally extremes of length. In ordinary geometry, paths which take the shortest route between points which are close together on the path. Take a strong cord and stretch it tight.... In general relativity, it is sometimes a maximum of elapsed time. Feynman has a cute illustration in a book of his. Suppose you want to arrive back where you are now in one hour of local time, but with a maximum of time having elapsed for you. Note that going uphill takes you to a place where, informally speaking, time goes faster. But moving fast causes time to go "slower" (informally speaking). What is the tradeoff between the two which leads to an optimum of wasted time?

Geodesics in space-time are the *free-fall* paths of objects. So the right thing to do is to shoot yourself out of a cannon so that in free fall, you return to the same spot on the ground.

If you transport the "forward" direction on a geodesic along the geodesic, it really ought to remain the "forward" direction. If our Roman turns out to be a clutz, we make him follow a taut cord, and point his spear along it to keep it from wobbling. The more difficult problem is to deal with carrying vectors which are pointing off to one side.

One does also require that parallel transport preserve the metric. So the spear shouldn't lengthen, say, as one goes. And the angles between different vectors should remain the same.

As you've pointed out, on a two-dimensional manifold, this is enough. But in 3 or more, spears pointed to the side can wobble around while keeping the same angle to the taut cord.

Let the Roman take not just a spear, but a set of three which are at right angles. One is pointing along the cord; the others remain at right angles to it and to each other. So far, our instructions don't prevent the spears pointed perpendicular to the path from turning around in the plane perpendicular to the path. "Torsion", as it were.

Someone should correct me if I'm mistaken, but I believe that this can be corrected if we require that the tips of the spears take the shortest path compatible with our prior requirements. Spinning them around makes them take a longer path. This is less than absolutely precise, since in fact we want to consider the spears infinitesimal arrows, and for the distance to be an issue, they have to have some small finite length. But I believe that if we make them very small, in the limit the way to carry them which moves them the least will be to parallel transport them as tangent vectors. And this is enough to show how to transport any other tangent vector.