Lie Dragging and Lie Derivatives

[Zag]

Dear friends,

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 $M$ into itself by using the congruence curves of a given vector field $\bar{V} = \frac{\mathrm{d}}{\mathrm{d}\lambda}$ parametrized by $\lambda$. Therefore, a point $P$ of the manifold can be mapped to a point $Q$ by this method - both points belonging to the same congruence curve and separated by a $\Delta\lambda$ in the curve's parameter. (See Figure 3.3 of Page 1)

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

$f^{*}_{\Delta\lambda}(Q) = f(P)$

for the same $P$ and $Q$ mentioned above. Schutz also adds that in case $f^{*}_{\Delta\lambda}(Q) = f(Q)$ the function is said to be invariant under the given map and, furthermore, in case this same equality holds for all $\Delta\lambda$, the function $f$ 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 $M$: if the field $\bar{W} = \frac{\mathrm{d}}{\mathrm{d}\mu}$ is dragged along the congruence of $\bar{V} = \frac{\mathrm{d}}{\mathrm{d}\lambda}$ by a change $\Delta\lambda$ in the curve's parameter, we will obtain a new field defined by $\frac{\mathrm{d}}{\mathrm{d}\mu^{*}_{\Delta\lambda}}$ and described by congruence with parameter $\mu^{*}_{\Delta\lambda}$. 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 $\frac{\mathrm{d}}{\mathrm{d}\mu^{*}_{\Delta\lambda}} = \frac{\mathrm{d}}{\mathrm{d}\mu}$ for all $\Delta\lambda$. Also, for fields which are Lie dragged, the Lie commutator is intuitively zero: $[\frac{\mathrm{d}}{\mathrm{d}\lambda}, \frac{\mathrm{d}}{\mathrm{d}\mu}] = 0$.

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 $f^{*}$ is defined by $\frac{\mathrm{d}f^{*}}{\mathrm{d}\lambda} = 0$ 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 $\bar{U}$ that has been dragged through $\bar{V} = \frac{\mathrm{d}}{\mathrm{d}\lambda}$ (thus generating a new field $\bar{U}^{*}$), the following Lie bracket holds: $[\bar{U}^{*}, \bar{V}] = 0$. 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 $0$ anyway... Any thoughts and comments on this issue would be greatly appciated!

Thank you very much!
Zag

 

[Ben Niehoff]

The asterisk is usually used to denote the pullback ($\varphi^*$) or pushforward ($\varphi_*$) of a map $\varphi$. If your book is for some reason using it to denote the "new" quantities, that is quite unfortunate.

Has your book talked about pullbacks and pushforwards? You need them to define Lie derivatives and Lie dragging of tensors.

Still reading your post, I'll add more in a bit.

 

[Ben Niehoff]

Frankly, I would toss this book and find another. The author has introduced notation that directly contradicts standard notation (using * to denote "new" quantities and apparently not bothering to discuss pullbacks), and moreover has introduced terminology that directly contradicts standard terminology: saying a quantity is "Lie dragged" when he means "invariant under Lie dragging". This is the most bizarre and confusing introduction to Lie derivatives I have ever read. (Note: "bizarre and confusing" is often the result when authors try to "simplify" a topic by neglecting to introduce important concepts).

Lie dragging is a very simple concept. Imagine you have a flow on your manifold (exactly like a fluid flow). The streamlines of this flow are the congruence you have been talking about. The vector field V is the velocity field of this flow, and the parameter $\lambda$ is like "time". Lie dragging is then the process of carrying quantities along this flow.

Lie dragging of functions is pretty obvious: the new function's value at the end of the flow is the old function's value at the start of the flow.

To define Lie dragging of tensor quantities, use the fact that, at a given value of $\lambda$, the flow is a diffeomorphism from your manifold to itself. The pushforward of this diffeomorphism then gives you a natural map between the tangent spaces at the old and new points. From there you can build maps for tensors.

Then to define the Lie derivative, you look at your flow for an infinitesimal $\lambda$, and compare your quantities' actual values (at the new point) to their Lie-dragged values. The Lie derivative measures how much your quantity is resisting motion along the flow, per unit parameter.

This has a pretty direct analogy with fluid mechanics. A function is like a collection of test particles, and the Lie derivative measures whether they move along with the fluid or not. A vector field is like a collection of little arrows; the Lie derivative measures whether the little arrows need to apply any forces to maintain their length and orientation under the fluid flow.

 

[Zag]

Thank you for your reply Ben Niehoff. Indeed I've noticed that the book is not as rigorous as one would expect - at first that seemed like an advantage, but I guess now I'm feeling the difficulties due to the oversimplification.

Could you suggest a book for a physicist who's interested in learning Differential Geometry?

Again, thank you very much!
Zag

 

[Fredrik]

I don't think there's a better choice than Lee. ("Introduction to smooth manifolds" and "Riemannian geometry: an introduction to curvature"). Unfortunately they're pretty heavy on topology right at the start (and not so much later). A physics student will probably want to skip the difficult proofs that involve a lot of topology, and just study the stuff that isn't too heavy. That's how I learned the basics of differential geometry myself (except I was using Spivak, not Lee).

 

[Zag]

Thank you all for your replies! I decided to start from ground zero and read "Introduction to Topological Manifolds" by Lee and hopefully in the future I will start "Introduction to Smooth Manifolds" by the same author.Zag

 

[Matterwave]

For physics, I would think smooth manifolds would be much more important and pertinent than topological manifolds. Certainly topological manifolds are the broader idea, but they have a much more limited use in physics than smooth manifolds.

Of course, I have not read either text by Lee, so it's possible that his book on topological manifolds has a lot of pertinent material, but I just thought I'd give my 2 cents.

 

[Fredrik]

The point of studying "topological" first is that it covers the theorems of point-set topology, which are used heavily in the early parts of "smooth". So if you want to understand everything you're doing the first time you're doing it, then it's the logical order. The problem is that if you're interested in some topic from "smooth", e.g. tensors, you will now have to spend several months studying topology before you can even begin to study tensors.

The alternative is to study "smooth" first, skipping everything that involves topology, and then later, if you feel that you want to or need to, fill the gaps in your knowledge by studying topology and the heavy proofs in "smooth". I prefer the latter approach, because the main ideas behind most important concepts in differential geometry don't involve much topology.

 

[Matterwave]

The point of studying "topological" first is that it covers the theorems of point-set topology, which are used heavily in the early parts of "smooth". So if you want to understand everything you're doing the first time you're doing it, then it's the logical order. The problem is that if you're interested in some topic from "smooth", e.g. tensors, you will now have to spend several months studying topology before you can even begin to study tensors.

The alternative is to study "smooth" first, skipping everything that involves topology, and then later, if you feel that you want to or need to, fill the gaps in your knowledge by studying topology and the heavy proofs in "smooth". I prefer the latter approach, because the main ideas behind most important concepts in differential geometry don't involve much topology.

One can get a quick and dirty overview of point-set topology in e.g. Wald appendix A. I don't know about the relative usefulness of studying a huge mathematical topic like topology just so one can rigorously define a few of the items that are pertinent. As such, I would second using the latter approach.

 

[JezuzStardust]

A late response. I am currently studying Schutz as well and find it equally confusing. I often find that I have to look up things in more rigorous books in order to understand what is meant, which makes the Schutz's book kind of useless.

However, I think I can actually answer the questions you posed. The function $f^*$ is obtained from $f$ by requiring that it coincides at one point $f^* (\lambda_0) = f( \lambda_0 + \Delta \lambda)$ and then Lie dragging determine the value of $f^*$ everywhere else. This means that, by definition, we have $d f^* / d\lambda = 0$. Hence the functions $f$ and $f^*$ are only equal at one point in general. It is done in the same way for $U^*$.

I hope this was some help. Btw, wait until you get to the proof of the Frobenius theorem. This completely threw me off using the book.