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.

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

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