Schild's ladder
# 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 $A_{0}X_{0}$, 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 $A_{1}.$

| A curve in M with a "vector" $X_0$ at $A_0,$ identified here as a geodesic segment. | Select $A_1$ on the original curve. The point $P_1$ is the midpoint of the geodesic segment $X_0A_1.$ | The point $X_1$ is obtained by following the geodesic $A_0P_1$ for twice its parameter length. |
| - | - | - |

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 $A_0X_0$ 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 $A_{0}X_{0}$ has unit length.
* Now let $A_1$ be a point on γ close to $A_0,$ and construct the geodesic $X_0A_1.$
* Let $P_1$ be the midpoint of $X_0A_1$ in the sense that the segments $X_0P_1$ and $P_1A_1$ take an equal affine parameter to traverse.
* Construct the geodesic $A_0P_1,$ and extend it to a point $X_1$ so that the parameter length of $A_0X_1$ is double that of $A_0P_1.$
* Finally construct the geodesic $A_1X_1.$ 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 $A_{1}A_{0}X_{0},$ 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), and in the case of Levi-Civita connections on surfaces, of Gauss–Bonnet theorem.

