Samiksha Jaiswal (Editor)

Schild's ladder

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit
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
    (Text) CC BY-SA


    Similar Topics