Schild's ladder

In the theory of general relativity, and differential geometry more generally, Schild's ladder is a method for parallel-transporting a vector along a curve using only geodesics. The method is named for Alfred Schild, who introduced the method during lectures at Princeton University.

Consider a curve γ through a point "A"₀ in a Riemannian manifold "M", and "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. In detail, consider the geodesic σ in "M" such that

:$sigma(0)=A\_0,$ :$sigma\text{'}(0)\; =\; x.,$

Let "X"₀ = σ(1). Now let "A"₁ be a point on γ close to "A"₀, and construct the geodesic "X"₀"A"₁. Let "P"₀ be the midpoint of "X"₀"A"₁ in the sense that the segments "X"₀"P"₀ and "P"₀"A"₁ take an equal affine parameter to traverse. Construct the geodesic "A"₁"P"₀, 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"₀. 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.

ee also

* Levi-Civita parallelogramoid‎

References

*citation|first1=Arkady|last1=Kheyfets|first2=Warner A.|last2=Miller|first3=Gregory A.|last3=Newton|title=Schild's ladder parallel transport procedure for an arbitrary connection.
* 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