Schild's ladder

For the book, see Schild's Ladder.
Two rungs of Schild's ladder. The segments A1X1 and A2X2 are an approximation to first order of the parallel transport of A0X0 along the curve.

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.

A curve in M with a "vector" X0 at A0, identified here as a geodesic segment.
Select A1 on the original curve. The point P1 is the midpoint of the geodesic segment X0A1.
The point X1 is obtained by following the geodesic A0P1 for twice its parameter length.

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

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

The steps of the Schild's ladder construction are:

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).

Notes

  1. Schild's ladder requires not only geodesics but also relative distance along geodesics. Relative distance may be provided by affine parametrization of geodesics, from which the required midpoints may be determined.
  2. The parallel transport which is constructed by Schild's ladder is necessarily torsion-free.
  3. A Riemannian metric is not required to generate the geodesics. But if the geodesics are generated from a Riemannian metric, the parallel transport which is constructed in the limit by Schild's ladder is the same as the Levi-Civita connection because this connection is defined to be torsion-free.

References

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