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 γ through a point ''A''₀ in a Riemannian manifold ''M'', and let ''x'' be a tangent vector at ''A''₀. Then ''x'' can be identified with a geodesic segment ''A''₀''X''₀ via the exponential map. This geodesic σ satisfies

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

The steps of the Schild's ladder construction are:
* Let ''X''₀ = σ(1), so the geodesic segment $A_0X_0$ has unit length.
* Now let ''A''₁ be a point on γ close to ''A''₀, and construct the geodesic ''X''₀''A''₁.
* Let ''P''₁ be the midpoint of ''X''₀''A''₁ in the sense that the segments ''X''₀''P''₁ and ''P''₁''A''₁ take an equal affine parameter to traverse.
* Construct the geodesic ''A''₀''P''₁, and extend it to a point ''X''₁ so that the parameter length of ''A''₀''X''₁ is double that of ''A''₀''P''₁.
* Finally construct the geodesic ''A''₁''X''₁. The tangent to this geodesic ''x''₁ is then the parallel transport of ''X''₀ to ''A''₁, 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.

Notes

# 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.
# The parallel transport which is constructed by Schild's ladder is necessarily torsion-free.
# 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

*.
* {{Citation
, first1=Charles W.
, last1=Misner
, authorlink1=Charles W. Misner
, first2=Kip S.
, last2=Thorne
, authorlink2=Kip S. Thorne
, first3=John A.
, last3=Wheeler
, authorlink3=John Archibald Wheeler
, title=Gravitation
, publisher= W. H. Freeman
, year=1973
, isbn=0-7167-0344-0

Connection (mathematics)
First order methods