Schild's ladder

In the theory of general relativity, and differential geometry more generally, Schild's ladder is a first-order method for approximating parallel transport of a vector along a curve using only affinely parametrized geodesics.

thus corresponds to an approximately parallel translated tangent vector at

Formally, consider a curve γ through a point A0 in a Riemannian manifold M, and let x be a tangent vector at A0.

Then x can be identified with a geodesic segment A0X0 via the exponential map.

This geodesic σ satisfies The steps of the Schild's ladder construction are: This is a discrete approximation of the continuous process of parallel transport.

If the ambient space is flat, this is exactly parallel transport, and the steps define parallelograms, which agree with the Levi-Civita parallelogramoid.

which is equal to the integral of the curvature over the interior of the triangle, by the Ambrose-Singer theorem; this is a form of Green's theorem (integral around a curve related to integral over interior), and in the case of Levi-Civita connections on surfaces, of Gauss–Bonnet theorem.

Two rungs of Schild's ladder. The segments A 1 X 1 and A 2 X 2 are an approximation to first order of the parallel transport of A 0 X 0 along the curve.
A curve in M with a "vector" X 0 at A 0 , identified here as a geodesic segment.
Select A 1 on the original curve. The point P 1 is the midpoint of the geodesic segment X 0 A 1 .
The point X 1 is obtained by following the geodesic A 0 P 1 for twice its parameter length.