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Wikipedia's article on torsion includes the following excerpt:

Suppose that an observer is moving along a geodesic, and carries with herself a system of rigid straight measuring rods (a coordinate system). Each rod is a straight segment; a geodesic. Assume that each rod is parallel transported along the trajectory. The fact that these rods are physically carried along the trajectory means that they are Lie-dragged, or propagated so that the Lie derivative of each rod along the tangent vanishes.

I don't understand the last sentence, which seems to equate Lie dragging with physically dragging an object. That's certainly a nice physical interpretation for something I thought was just a mathematical tool, but I can't see how Lie dragging is even relevant here.

  • Why is Lie dragging needed to describe what happens when you hold a ruler? Isn't that just parallel transport of the vector pointing along the ruler?
  • Along what vector field are we doing the Lie dragging? Isn't there only one vector around, the observer's velocity?
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To start with, note that the measuring rods determine a coordinate system. This means that they are a congruence of observers rather than a single observer with vectors. The velocity vectors define a local vector field $u^i$, and the tangent vector along one of the measuring rods define another local vector field $t^i$. Since, as you note, $t^i$ is parallel transported along $u^i$ we have $t^i{}_{;j}u^j \equiv 0$, but since the rods define a coordinate system along with the parametrization of the paths we can take $u^i = \partial_{0}$ and say $t^i = \partial_{1}$. Thus $[u^i,t^i] = 0$, i.e. $t^i$ is Lie-dragged along $u^i$. It is really just the fact that they define a coordinate system. From this you can see how their conclusion about torsion and twisting connections follow, because non-zero torsion then implies $u^i{}_{;j}t^j \neq 0$, so that the rods are not moving parallel to the central observer.

EDIT: By a congruence we simply mean that we do not treat it as a single path, but rather consider the continuum to follow different paths that do not intersect (mathemically it is a set of integral curves that defines a vector field). Your rulers define coordinates if you can use them as such physically, so pretty much if you stick it out and it is long enough to not be approximated to a single point. That's why dragging a reasonably rigid physical object (because a "ruler" must not be an actual ruler of course) makes it Lie dragged.

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    $\begingroup$ Thanks for the answer! That makes sense for my second question, but I still don't understand the physical meaning. If I were floating in space and I had a large amount of rulers with me, how exactly would I hold them to form a 'congruence of observers'? What does it mean, in the excerpt, to 'carry [with me] a coordinate system'? $\endgroup$ – knzhou Aug 24 '16 at 3:10
  • $\begingroup$ @knzhou I added some additional explanation. Does that help? $\endgroup$ – Erik Jörgenfelt Aug 24 '16 at 10:22

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