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. The method is named for Alfred Schild, who introduced the method during lectures at Princeton University.

Construction
The idea is to identify a tangent vector x at a point $$A_0$$ with a geodesic segment of unit length $$A_0X_0$$, and to construct an approximate parallelogram with approximately parallel sides $$A_0X_0$$ and $$A_1X_1$$ as an approximation of the Levi-Civita parallelogramoid; the new segment $$A_1X_1$$ thus corresponds to an approximately parallel translated tangent vector at $$A_1.$$

Formally, consider a curve &gamma; 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 &sigma; satisfies


 * $$\sigma(0)=A_0\,$$
 * $$\sigma'(0) = x.\,$$

The steps of the Schild's ladder construction are:
 * Let X0 = &sigma;(1), so the geodesic segment $$A_0X_0$$ has unit length.
 * Now let A1 be a point on &gamma; close to A0, and construct the geodesic X0A1.
 * Let P1 be the midpoint of X0A1 in the sense that the segments X0P1 and P1A1 take an equal affine parameter to traverse.
 * Construct the geodesic A0P1, and extend it to a point X1 so that the parameter length of A0X1 is double that of A0P1.
 * Finally construct the geodesic A1X1. The tangent to this geodesic x1 is then the parallel transport of X0 to A1, at least to first order.

Approximation
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.

In a curved space, the error is given by holonomy around the triangle $$A_1A_0X_0,$$ 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.