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Zag

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I am currently studying some concepts on Differential Geometry using the book

*"Geometrical Methods of Mathematical Physics"*by Bernard F. Schutz and have so far read up to the beginning of Chapter 3 entitled

*"Lie Derivatives and Lie Groups"*. Even though Chapters 1 and 2 are very clear, Chapter 3 introduces a very a suspicious definition for Lie Derivatives which seems to contradict all the discussion on Lie Dragging that precedes it. I would be very grateful if anyone could help me understand what the author is trying to communicate since it sounds completely obscure to me. Below I write my thoughts on what I understood regarding Lie Dragging and Lie Derivatives, and also attach to this topic links to photocopied pages of the book as references.

Regarding Lie Dragging as described by Schutz, it is defined as a natural way of mapping a manifold [itex]M[/itex] into itself by using the congruence curves of a given vector field [itex]\bar{V} = \frac{\mathrm{d}}{\mathrm{d}\lambda}[/itex] parametrized by [itex]\lambda[/itex]. Therefore, a point [itex]P[/itex] of the manifold can be mapped to a point [itex]Q[/itex] by this method - both points belonging to the same congruence curve and separated by a [itex]\Delta\lambda[/itex] in the curve's parameter. (See Figure 3.3 of Page 1)

Using this idea Schutz proceeds to defined how a function [itex]f[/itex] can be dragged along the congruence curves of [itex]\bar{V}[/itex]. The definition is that the new function, [itex]f^{*}_{\Delta\lambda}[/itex], dragged by a change [itex]\Delta\lambda[/itex] in the curve's parameter is such that:

[itex]f^{*}_{\Delta\lambda}(Q) = f(P)[/itex]

for the same [itex]P[/itex] and [itex]Q[/itex] mentioned above. Schutz also adds that in case [itex]f^{*}_{\Delta\lambda}(Q) = f(Q)[/itex] the function is said to be

*invariant*under the given map and, furthermore, in case this same equality holds for all [itex]\Delta\lambda[/itex], the function [itex]f[/itex] is said to be

*Lie dragged*.

A similar discussion on how vector fields can be dragged in this fashion is also presented in §3.3 (See Pages 2 and 3). The operation of Lie dragging clearly generates a new vector field on [itex]M[/itex]: if the field [itex]\bar{W} = \frac{\mathrm{d}}{\mathrm{d}\mu}[/itex] is dragged along the congruence of [itex]\bar{V} = \frac{\mathrm{d}}{\mathrm{d}\lambda}[/itex] by a change [itex]\Delta\lambda[/itex] in the curve's parameter, we will obtain a new field defined by [itex]\frac{\mathrm{d}}{\mathrm{d}\mu^{*}_{\Delta\lambda}}[/itex] and described by congruence with parameter [itex]\mu^{*}_{\Delta\lambda}[/itex]. Here Schutz also defines

*invariant*and

*Lie dragged*fields using definitions analogous to those for usual functions. In particular, a field is said to be

*Lie dragged*when [itex]\frac{\mathrm{d}}{\mathrm{d}\mu^{*}_{\Delta\lambda}} = \frac{\mathrm{d}}{\mathrm{d}\mu}[/itex] for all [itex]\Delta\lambda[/itex]. Also, for fields which are

*Lie dragged*, the Lie commutator is intuitively zero: [itex] [\frac{\mathrm{d}}{\mathrm{d}\lambda}, \frac{\mathrm{d}}{\mathrm{d}\mu}] = 0[/itex].

So far so good, but my issue with Schutz's definitions begin when Lie Derivatives are defined (See Pages 4 and 5). The motivation presented on Page 4 is very clear and helps the physicist gain some intuition about what is going on, and motivates the definition of Lie derivatives for functions. However, Schutz apparently goes beyond what was stated before and says that a dragged function [itex]f^{*}[/itex] is defined by [itex]\frac{\mathrm{d}f^{*}}{\mathrm{d}\lambda} = 0[/itex] which doesn't seem to make any sense! That would be true for

*Lie dragged*functions which are totally invariant under the dragging, but in general a function which goes through Lie dragging is different at every point!

The same weird reasoning appears on Page 5 when the Lie drivative for vector fields is discussed. Schutz insists that for a field [itex]\bar{U}[/itex] that has been dragged through [itex]\bar{V} = \frac{\mathrm{d}}{\mathrm{d}\lambda}[/itex] (thus generating a new field [itex]\bar{U}^{*}[/itex]), the following Lie bracket holds: [itex] [\bar{U}^{*}, \bar{V}] = 0[/itex]. However, once again, this would be true only for

*Lie dragged*fields which are totally invariant under the dragging, but in general a vector field which goes through Lie dragging is different at every point!

So it is really hard forme to grasp what the author had in mind. Why go through all the Lie dragging dscussion if none of that is used? It appears that in the end functions and fields are just assumedto be totally invariant (

*Lie dragged*) under the map, but in that case the Lie derivatives would all be [itex]0[/itex] anyway... Any thoughts and comments on this issue would be greatly appciated!

Thank you very much!

Zag