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Schild's ladder

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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.

Contents

Construction

The idea is to identify a tangent vector x at a point A 0 with a geodesic segment of unit length A 0 X 0 , and to construct an approximate parallelogram with approximately parallel sides A 0 X 0 and A 1 X 1 as an approximation of the Levi-Civita parallelogramoid; the new segment A 1 X 1 thus corresponds to an approximately parallel translated tangent vector at A 1 .

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

σ ( 0 ) = A 0 σ ′ ( 0 ) = x .

The steps of the Schild's ladder construction are:

  • Let X0 = σ(1), so the geodesic segment A 0 X 0 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 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 1 A 0 X 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).

    References

    Schild's ladder Wikipedia
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