Two rungs of Schild's ladder. Thesegments A1X1 and A2X2 are anapproximation to first order of theparallel transport of A0X0 along thecurve.

A curve in M with a "vector" X0 at A0,identified here as a geodesic segment.

Select A1 on the original curve. The point P1is the midpoint of the geodesic segmentX0A1.

The point X1 is obtaingeodesic A0P1 for twlength.

From Wikipedia, the free encyclopedia

In the theory of general relativity, and differential geometry moregenerally, Schild's ladder is a first-order method for approximatingparallel transport of a vector along a curve using only affinelyparametrized geodesics. The method is named for Alfred Schild, whointroduced the method during lectures at Princeton University.

1 Construction2 Approximation3 Notes4 References

The idea is to identify a tangent vector x at a point with a geodesic segment of unit length , and toconstruct an approximate parallelogram with approximately parallel sides and as anapproximation of the Levi-Civita parallelogramoid; the new segment thus corresponds to anapproximately 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:

Let X0 = σ(1), so the geodesic segment has unit length.Now let A1 be a point on γ 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 affineparameter to traverse.Construct the geodesic A0P1, and extend it to a point X1 so that the parameter length of A0X1 is doublethat of A0P1.Finally construct the geodesic A1X1. The tangent to this geodesic x1 is then the parallel transport of X0 toA1, at least to first order.

This is a discrete approximation of the continuous process of parallel transport. If the ambient space is flat, thisis exactly parallel transport, and the steps define parallelograms, which agree with the Levi-Civitaparallelogramoid.

In a curved space, the error is given by holonomy around the triangle which is equal to the integralof the curvature over the interior of the triangle, by the Ambrose-Singer theorem; this is a form of Green'stheorem (integral around a curve related to integral over interior).

Schild's ladder requires not only geodesics but also relative distance along geodesics. Relative distancemay be provided by affine parametrization of geodesics, from which the required midpoints may bedetermined.

1.

The parallel transport which is constructed by Schild's ladder is necessarily torsion-free.2.A Riemannian metric is not required to generate the geodesics. But if the geodesics are generated from aRiemannian metric, the parallel transport which is constructed in the limit by Schild's ladder is the same asthe Levi-Civita connection because this connection is defined to be torsion-free.

3.

Kheyfets, Arkady; Miller, Warner A.; Newton, Gregory A., Schild's ladder parallel transport procedurefor an arbitrary connection.Misner, Charles W.; Thorne, Kip S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman,ISBN 0-7167-0344-0

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