Help Me! Question About Tensors and Projections

I need some help. I am working hard to find ways to bring more geometry into introductory calculus-based physics (and conceptual physics as well). By geometry, I mean specifically the geometry associated with vectors and tensors, and the information encoded therein. Yes, I said tensors.

I have been heavily influenced by these notes by Kip Thorne and Roger Blandford that form the basis of their forthcoming textbook. In particular, chapter 1 has deeply affected my view of what introductory physics could, and perhaps should, be like when an emphasis on geometry is provided. The discussion of tensors in these notes presents them as machines that take vectors as inputs and give a real number as output. This is the same approach taken in the famous general relativity textbook coauthored by Thorne (aka MTW) and in a fantastic new book by Jeevanjee. I won’t go into details here because I don’t think it’s necessary for my questions, but I look forward to exploring the approach in more depth in future posts.

Geometric entities, like vectors, have an existence independent of any coordinate system. For example, I can state that an object or particle has 35 units of momentum in a particular direction represented by an arrow and I can do so without choosing a coordinate system. I can project this momentum onto any arbitrary basis and resolve it into components if I want to. Fine. No problem.

But what about a quantity represented by a second rank tensor? What about moment of inertia (MoI0? MoI is a geometric quantity that has an existence independent of any coordinate system, but it isn’t a vector; it’s a second rank tensor that can be represented by a symmetric matrix.

So here is my question. How can I specificy a MoI in a coordinate-independent way analogous to doing so for a vector? For a vector, I can specificy a magnitude and draw a direction. What must I specify for MoI? I think I know the answer. I think I must specify the eigenvalues of the matrix representation of the MoI tensor and the principal axes to which these eigenvalues apply. These can then be projected onto a coordinate basis. Is that correct? If not, could you tell me if what I’m asking is even possible? I don’t see how it can’t be.


2 Comments on “Help Me! Question About Tensors and Projections”

  1. Since the moment of inertia is a symmetric second-rank tensor, you can certainly specify it by giving its eigenvalues and eigenvectors. Physically, you can think of that as giving the ellipsoidal body with the same moment of inertia. The symmetry of the tensor is crucial here; otherwise it’s not guaranteed to have a basis of eigenvectors, or have real eigenvalues. (Although a singular value decomposition might provide similar intuition in the more general case.) And for higher-rank tensors, the theory of eigenvectors is more subtle, and quite recent (only a decade or so old). I certainly don’t think of it as intuitive for rank >= 3.

    Also note that this description uses the metric. To identify eigenvalues and eigenvectors of a tensor, it has to be interpreted as having “one up and one down index”, or in more invariant terms, it has to be a map from a vector space to itself, not a map sending two vectors to a scalar. It seems to me that the most natural version of the moment of inertia is as a fully covariant tensor (both indices down), as in Energy = (1/2) I(omega,omega), but maybe that’s not the most fundamental way to think of it. In any case, if you have a metric, you can identify these different tensor types at will (“raise or lower indices”).

    —David Metzler

    • Joe Heafner says:

      Hi David, and thanks for the insight. I had not considered the ellipsoid description before now! Regarding the metric, I certainly plan on including it as I see it as a conceptual expansion of the Pythagorean theorem.

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