Visualizing tensors

[Genady]

There are two ways to visualize tensors as coordinate independent, "geometric" objects (that I know of). One way is what I'd call, "dynamic": tensor is a linear function that takes in vectors and 1-forms and puts out numbers. "A tensor is a machine", Gravitation by Misner, Thorne, Wheeler. 

Another way is what I'd call, "static": tensor is an equivalence class of sets of components that transform into each other with a coordinate transformation. "A tensor is something that transforms like a tensor", Einstein Gravity in a Nutshell by Zee.

To me, the "static" definition is easier to visualize and to use. What is your preference, if any?

[joigus]

I, for one, don't feel any need to visualize a tensor. Never have, I must say.

To me, it's an algebraically motivated concept. In some cases it might be useful to picture something geometric going on (example: the energy momentum tensor in GR).

As an example of the converse, the Einstein tensor is very notorious for being essentially the only second-order tensor you can form that's covariantly constant. I'm sure pure mathematicians will tell you that there is no known role that this tensor plays in pure geometry --unless you invoke GR for some reason.

[Genady]

Good. I did not mean "visualizing" each individual tensor, but rather relating to a tensor concept. To you, it as an algebraically motivated concept. I understand that it means, sets of indexed components that transform in certain ways. IOW, the "Zee's definition."

Edited September 8, 2023 by Genady

[Markus Hanke]

13 hours ago, Genady said:

What is your preference, if any?

I’m with MTW - I look at it as a map that maps vectors and forms into real numbers (or other maps). 

[Genady]

Defining tensors as maps is fine, but there is an asymmetry in this view that I don't like much. One starts with something "static", say, vectors. Then, one defines forms and tensors as maps of maps etc. of the vectors into numbers. The numbers are "static", too. I like that in the "static" view, all of them, numbers (scalars), vectors, forms, tensors are entities on equal footing which "interact" through operations.

Moreover, some operations, e.g., contraction, are defined using basis, i.e., components, albeit being basis independent. Frame independent component notation has other advantages, as MTW admits here:

So, why not to start with this view?

The mapping then is just another contraction. And it does not matter if a tensor maps a vector or a vector maps a tensor, they just together contract to make a number or something else.

[Markus Hanke]

3 hours ago, Genady said:

So, why not to start with this view?

I don’t quite understand what you mean here - vectors, forms and scalars are themselves just tensors, so all of these objects are already on equal footing from the beginning. The slots (indices) simply tell you what mappings are possible, and what the rank of the resulting tensor will be.

[studiot]

21 hours ago, Genady said:

There are two ways to visualize tensors as coordinate independent, "geometric" objects (that I know of). One way is what I'd call, "dynamic": tensor is a linear function that takes in vectors and 1-forms and puts out numbers. "A tensor is a machine", Gravitation by Misner, Thorne, Wheeler. 

Another way is what I'd call, "static": tensor is an equivalence class of sets of components that transform into each other with a coordinate transformation. "A tensor is something that transforms like a tensor", Einstein Gravity in a Nutshell by Zee.

To me, the "static" definition is easier to visualize and to use. What is your preference, if any?

I think those disciplines that activeley use tensors have a head start on those which could do but usually avoid them.

I wish I had had a copy of this book when I first met tensors.

Geologists make extensive use of certain tensors so one can get an immediate handle on the subject via rock mechanics and, to a lesser extent, soil mechanics.

 

The build up from one dimension is echoed in the explanation of GR given by Baez

https://math.ucr.edu/home/baez/einstein/einstein.pdf

 

and also by Koleki

https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf

 

Essentially this view runs that you have a coordinate system (which is therefore geometric) and a bunch of (physical) quantities of interest at some or all points spanned by the coordinate system.

This setup is organised into an array of coordinate points, an array of quantities and an array of coefficients connecting the first two arrays.

The coeffeciant array being the tensor array.

Its coefficients may be constants or expressions which depend upon the coordinates.

 

When one starts some form of continuum mechanics it is usual to first work in one dimension, then two, then three and so on.

This offers the opportunity to go back to a lower dimension to see what is happening, although there is always the possibility of loosing some aspect that only occurs in higher dimensions .

An example of such loss is the loss of the difference between contra and co variance in one dimension.

 

Some other books with useful visula models are from Fleisch

 

and Bickley

 

 

 

[joigus]

The best approach to a tensor, I think, should be dealing first with Euclidean tensors. Ie, fixed entities that represent multilinear mappings acting on vectors on a flat space. On a flat space we can identify points with vectors quite directly by means of an affine structure. On a manifold OTOH, points are not vectors, nor can they be identified with such. We must introduce vector structures point to point (the so-called tangent space at point x T_M(x) M standing for "manifold"). So I would introduce tensors in two steps. First: What they do at a point; then considering how what they do changes from point to point. That naturally leads to a parallel transport as a rule to take vectors at one point to vectors at a different point. Flat spaces have no connection (or parallel transport), even though they have tensors.

Another thing that confuses people is the difference co-variant / contra-variant, which occurs long after one finds tensors and has to do with vectors "naturally" having two alternate bases when we are not in an orthonormal frame.

Trying to bring all these aspects together into one pictorial explanation is, perhaps, misguided. I don't know. The root "tens-" in tensor doesn't help either. It seems to suggest something directly physically interpretable having to do with tension. They are multilinear operators; that's what they are. Angular velocity is a 2-tensor on a flat space, but in disguise.

 

x-posted with @studiot

[Genady]

Thank you very much for all the suggestions and thoughts.