page_title: Schild's ladder
section_title: Construction
hierarchical_section_title: Schild's ladder / Construction
context_page_description: 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.
context_section_description: The idea is to identify a tangent vector x at a point  with a geodesic segment of unit length , and to construct an approximate parallelogram with approximately parallel sides  and  as an approximation of the Levi-Civita parallelogramoid; the new segment  thus corresponds to an approximately parallel translated tangent vector at 
  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
  The steps of the Schild's ladder construction are:
  Let X₀ = σ(1), so the geodesic segment  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.
caption: Select A1 on the original curve.  The point P1 is the midpoint of the geodesic segment X0A1.