When is something covariant or contravariant?

[omega_minus]

I'm pretty comfortable with special relativity, and at least familiar with the principles of the general theory, but recently I've tried to learn SR using tensors. It is my first foray into this branch of mathematics. I understand they're handy because they represent invariant objects, but the learning is otherwise slow going. One basic question I can't seem to be able to find a direct answer to is this:

What does it mean for something to be covariant or contravariant? In other words, what's the difference between the two?

When I see equations in my books they seem to randomly choose which quantities are represented by each one. Could anyone give me some examples of each and tell me how they were decided to be what they are?

Thanks.

 

[dx]

The gradient df (local linear behavior) of a function f : M → R on the manifold is the typical example of a covariant vector, and the tangent vector d/dt of a parametrized curve t → g(t) on the manifold is the typical example of a contravariant vector. These are geometric objects independent of the cooridnate representation of these objects. The tangent space (vector space of tangent vectors at a point) is dual to the cotangent space. For instance, the action of d/dt on df would give df/dt.

On a manifold with a metric tensor, there is a natural isomorphism between the tangent and cotangent spaces, i.e. a covector g(v, ) is associated with the vector v. Thus people used to speak about the 'covariant components' and the 'contravariant components' of v as if they were referring to the same object. However, on general manifolds there is no such natural mapping, and you cannot decide randomly which type of vector you choose to represent something. Moreover, even when such a mapping is available, the objects are naturally one or the other. For example, in the old vector analysis, the gradient of a function was represented by a vector perpendicular to the level surfaces of the gradient, even though it is naturally a 1-form. This is an unnatural representation because 'perpendicularity' is not preserved by linear transformations, but only Euclidean transformations. In mechanics, people thought of force and momentum as vectors, but now they are understood to be naturally 1-forms.

 

[Fredrik]

This post might be useful. (It explains the terms, but doesn't give any examples).

 

[omega_minus]

Fredrik, thanks for the links. You and dx have done a lot of writing on the subject, most of which is new territory for me. I like your position of using vector and covector in place of contra- and covariant.
I have read elsewhere that one can define which of these two a given quantity can be described by. For example, a paper I'm reading now says you choose which EM tensor to use depending on whether you want to describe the electric and magnetic fields by vectors or one-forms. In this view, is it even correct to ask the question "Is a magnetic field a vector or covector?" This seems contrary to dx's post above, however.
Thanks for all of your help.

 

[Fredrik]

The terms I like the best are "tangent vector" and "cotangent vector".

I should probably mention one detail that I didn't think of when I wrote that post I linked to. The "something that transforms as" definition of tensors assigns the terms "covariant vector" and "contravariant vector" to things that are neither tangent vectors nor cotangent vectors. For example, the n-tuple (\partial_1,\dots,\partial_n). Each member of this tuple is a tangent vector, but the whole tuple "transforms" as the components of a cotangent vector, so by the "something that transforms as" definition, this would be a covariant vector even though its components aren't even numbers.

I don't know much about the specific things you asked about now, but I think you're right. I'll leave it for someone else to provide more details, or perhaps tell us that we're both wrong. (I don't have time to think about it right now).

 

[dx]

I have read elsewhere that one can define which of these two a given quantity can be described by. For example, a paper I'm reading now says you choose which EM tensor to use depending on whether you want to describe the electric and magnetic fields by vectors or one-forms. In this view, is it even correct to ask the question "Is a magnetic field a vector or covector?" This seems contrary to dx's post above, however. Thanks for all of your help.

You are of course free to choose any representation as long as it is used properly and consistently, but I think one can say that forces are naturally 1-forms and not vectors. The work done by a force F on a particle that moves a distance s in space was calculated as the dot product of the vectors F and s. The notion of dot product only exists in a Euclidean space, but the idea of work done by a force on system displaced by some amount makes sense even in a general configuration space which is not the Euclidean space in which a point particle moves. What force is really doing is assigning numbers to small displacements, i.e. it is a linear function on the tangent spaces of configuration space. Conservative forces are derived from potentials and are naturally expressed as 1-forms: F = -dV, where V is a function on configuration space.

 

[atyy]

http://sophia.dtp.fmph.uniba.sk/~fecko/regensburg_2007.html

Fecko says that if the dynamics of a field are defined by an action, then the the quantities in the action are forms, since those are designed to be integrated.

 

[omega_minus]

Well, I certainly appreciate all of the input. I am convinced you all know exactly what you're talking about :) I think my biggest problem is that in all of my searching, nowhere (and I mean nowhere) have I ever found an example of tensor calculations that use numbers (i.e. a worked example of a problem solvable with tensors). The descriptions are all pretty much the same, starting with the definition of tensors, or how they transform, or a formula to calculate a given element of a tensor, etc. Then they stick to variables and abstract ideas but never actually calculate anything. They write things like x_ixⁱ...x_nxⁿ... and so on. As an electrical engineer and only a recreational physicist my ability to abstract reality away is not as good as I would hope. I am too used to learning the theory, being given a set of conditions, then calculating a number or function that corresponds to a physical quantity.
If anyone has a link to some real-world (preferably SR) worked-out examples of tensor analysis I would appreciate it. Again, thank you all for your time and expertise. I did learn a good bit from this thread but I know I have a long way to go.

\Omega^-