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[[Image:Schild's ladder step 4.svg|thumb|right|Two rungs of Schild's ladder. The segments ''A''<sub>1</sub>''X''<sub>1</sub> and ''A''<sub>2</sub>''X''<sub>2</sub> are an approximation to first order of the [[parallel transport]] of ''A''<sub>0</sub>''X''<sub>0</sub> along the curve.]]
In the theory of [[general relativity]], and [[differential geometry]] more generally, '''Schild's ladder''' is a [[First-order approximation|first-order]] method for ''approximating'' [[parallel transport]] of a vector along a curve using only [[Torsion of connection#Geodesics and the absorption of torsion|affinely parametrized]] [[geodesic]]s. 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 <math>A_0</math> with a geodesic segment of unit length <math>A_0X_0</math>, and to construct an approximate parallelogram with approximately parallel sides <math>A_0X_0</math> and <math>A_1X_1</math> as an approximation of the [[Levi-Civita parallelogramoid]]; the new segment <math>A_1X_1</math> thus corresponds to an approximately parallel translated tangent vector at <math>A_1.</math>
 
{|
|[[Image:Schild's ladder step 1.svg|thumb|250px|right|A curve in ''M'' with a "vector" ''X''<sub>0</sub> at ''A''<sub>0</sub>, identified here as a geodesic segment.]]
|[[Image:Schild's ladder step 2.svg|thumb|250px|right|Select ''A''<sub>1</sub> on the original curve. The point ''P''<sub>1</sub> is the midpoint of the geodesic segment ''X''<sub>0</sub>''A''<sub>1</sub>.]]
|[[Image:Schild's ladder step 3.svg|thumb|250px|right|The point ''X''<sub>1</sub> is obtained by following the geodesic ''A''<sub>0</sub>''P''<sub>1</sub> for twice its parameter length.]]
|}
 
Formally, consider a curve &gamma; through a point ''A''<sub>0</sub> in a [[Riemannian manifold]] ''M'', and let ''x'' be a [[tangent vector]] at ''A''<sub>0</sub>. Then ''x'' can be identified with a geodesic segment ''A''<sub>0</sub>''X''<sub>0</sub> via the [[exponential map]]. This geodesic &sigma; satisfies
 
:<math>\sigma(0)=A_0\,</math>
:<math>\sigma'(0) = x.\,</math>
 
The steps of the Schild's ladder construction are:
* Let ''X''<sub>0</sub> = &sigma;(1), so the geodesic segment <math>A_0X_0</math> has unit length.
* Now let ''A''<sub>1</sub> be a point on &gamma; close to ''A''<sub>0</sub>, and construct the geodesic ''X''<sub>0</sub>''A''<sub>1</sub>.
* Let ''P''<sub>1</sub> be the midpoint  of ''X''<sub>0</sub>''A''<sub>1</sub> in the sense that the segments ''X''<sub>0</sub>''P''<sub>1</sub> and ''P''<sub>1</sub>''A''<sub>1</sub> take an equal affine parameter to traverse.  
* Construct the geodesic ''A''<sub>0</sub>''P''<sub>1</sub>, and extend it to a point ''X''<sub>1</sub> so that the parameter length of ''A''<sub>0</sub>''X''<sub>1</sub> is double that of ''A''<sub>0</sub>''P''<sub>1</sub>.
* Finally construct the geodesic ''A''<sub>1</sub>''X''<sub>1</sub>. The tangent to this geodesic ''x''<sub>1</sub> is then the parallel transport of ''X''<sub>0</sub> to ''A''<sub>1</sub>, 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 <math>A_1A_0X_0,</math> 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).
 
== 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 of connection|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=Arkady|last1=Kheyfets|first2=Warner A.|last2=Miller|first3=Gregory A.|last3=Newton|title=Schild's ladder parallel transport procedure for an arbitrary connection. International Journal of Theoretical Physics
20001200, Volume 39, Issue 12, pp 2891-2898 }}.
* {{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
}}
 
[[Category:Connection (mathematics)]]
[[Category:First order methods]]

Latest revision as of 11:44, 10 December 2014

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