Geometrical Vectors

Doing Faraday proud

The concept of a field got its first geometrical incarnation thanks to Michael Faraday?s line drawings. Since, its treatment and interpretations have been progressively analytical as the sophistication of physical description has necessitated its abstraction. Gabriel Weinreich has convinced me through his wonderful little monograph that there is more than meets the eye if one cared to look and extend geometrical reasoning to those vector concepts that can be understood by our everyday intuition.

The primary strength of his method, at least initially, lies in a description of vector relationships in rubbery space. The raison d?etre is twofold, as the author points out quite early. One appeals to the pedagogical advantage of exploiting the capacity of our brain to extrapolate 3D data from what is essentially a 2D image on the retina (in conjunction with the stereoscopic nature of vision, of course), and hence lending itself to think in topologically invariant terms. The other leaves the forms of relationships invariant in this rubber space geometry, which potentially saves some nightmare calculations from their tedium.

Weinreich hints, however, that this formulation might not represent ?interesting? physical laws, but that there is a lot to be gained from this perspective. Like a well-told story, what exactly is gained emerges with an elegant lucidity only toward the end of the book, where he motivates the definition of the metric through topological reasoning. He does this by first demonstrating the problems faced when representing physical laws, like Maxwell?s equations, using his invariant formulation, then introduces a coordinate representation that is necessitated by the need for this metric. Ultimately, he provides an alternative to his invariant formulation by introducing a scaling factor for orthogonal basis vectors which is represented in terms of a Cartesian system. This is but one of the infinite possibilities, making the definition of the metric independent of a Cartesian or orthogonal basis.

In the present development, from the earliest arguments about invariance, one thing I did find missing was any mention about fundamental and derived quantities, which would have provided support for arguments about space distortions. A case in point being the example of the capacitor field. At the same time, I cannot help but wonder if the omission was deliberate, so the author could emphasize that particular physical systems did not affect topological invariance. If one were oblivious to dimensional analysis, this would be understandable, but given that the audience of this book is primarily students of physics, I don?t think it would be easy to turn that blind an eye.

Weinreich builds his invariant scaffolding by creating four types of vectors to facilitate his intuitive outlook, dividing them into ?plane? and ?line? types. In addition, he introduces three types of scalars. If that didn?t seem fanciful enough, he draws out his hat both a polar and an axial sense to each. That leaves us to juggle fourteen objects and their relationships!

However, he offers an accurate picture for each, and goes on to define how they satisfy closure properties, and then extends his arguments and constructions to vector operations. One could get lost in the myriad permutations if he hadn?t brought it all together, reasoning why his exotic vectors don?t spawn an unending array of objects of increasing complexity. The finiteness of the flavors, both for scalars and vectors, turns out a be a consequence of the simplicity of their representation. Weinreich is quite straightforward in his justification: either constructed objects can be drawn, or they need to make sense from a invariance viewpoint, which is to say that the response to space transformations is linear. This is where a remarkably consistent description emerges, and it comes as quite a surprise to see how he even blends the three flavors of scalars he introduces to the coherent set of rules between vector and scalar quantities. Newer objects mushroom but can be appropriately classified and unruly objects are politely but firmly discarded, keeping the balance - nothing unnecessary, and nothing insufficient.

The introduction of the gradient, divergence and curl, in addition to the vector integral relations, Stokes? and the Gauss?s theorem, have a wonderfully elegant geometrical construction and interpretation, and I think this where the simplicity of the approach reaps its benefits. If one were to visualize fields in terms of lines or surfaces, gleaning these limiting relations in so direct a manner would have possibly moved the man who taught us how to think in terms of fields at the first place - Michael Faraday. If one were to reminisce about a middle school essay involving a hypothetical meeting with a historical character, it doesn?t get better than this.

After fabricating and floating around in rubber space, and having defined field operations, Weinreich comfortably descends into a coordinate system, albeit general, where he grounds his vectors in a basis. The interesting thing is that he derives the basis from the idea of a scalar field, and defines it so. When I thought back about this approach in retrospect, I realized that he had introduced the idea of a reciprocal lattice through this definition. This was the most pleasant shock! Maybe it is my sheer naivete, but I hadn?t come across this concept except through the conventional condensed matter treatment - the wave vector approach. In fact, I had to rub my eyes again to see that if the basis was defined in terms of the field, and its reciprocal basis was defined in terms of the gradient, this was symbolically obvious even in the differential notation! So here was a new perspective to reciprocity.

That said, I felt there was an element lacking in the treatment of the abstract vector quantities introduced for constructions. What I would have liked to see was a discussion of dependent and independent variables threaded through the topics. It would have provided some of the geometric transformations a physical context, especially when differential operations and formation of the basis are discussed. Again, I understand that this point is debatable, and any discussion of dependent quantities would have to be associated with the physical system at hand, and may not have kept the discussion as general as the author had intended it to be.

The development of vector differential relations in his terms of the generalized basis follows, and we see how, true to its promise of invariance, its forms are exactly the ubiquitous and familiar Cartesian differentials. After this, Weinreich demonstrates how his house of cards almost collapses the moment Maxwell?s equations are introduced. This underlines the need for a metric and we are back to our conventional Cartesian system, but now with the knowledge that it represents just one basis and metric, and we can express other bases in terms of it. The idea of orthogonal bases is then motivated from a proportionality relation between reciprocal bases and we see how this leads to an alternative representation of a basis in terms of unit vector and position-dependent scale factors. Finally, comfortable switching bases, and representing the differential operators in these, we realize there isn?t anything special about Rene Descarte, and we close with a generic definition of a metric, as a stepping stone to understanding more abstract geometries.

Summarizing, the topological route that Weinreich follows ultimately frees the representation of basis vectors in Cartesian terms, and defines the metric in terms of invariant basis vectors. This approach motivates the idea of a metric from a geometrical viewpoint but inspired by the form of a physical law. We get to see how the means itself can lead to a somewhat different end.

The author succinctly points out at the end of this book how his approach could be a victim o...