I'm taking an undergrad GR course, and our text (Lambourne) mentions covariant and contravariant vectors and tensors ad-nauseum, but never really gives a formal definition for what they are, and how they are unique from each other in any physical sense (other than their difference in transformations). Is there any physical intuition behind these two labels? There should be, right? If they differ in how they transform with transformation of coordinates, doesnt that indicate that there has to be some way of visualizing their difference, since coordinate transformations are easily visualized?


This whole business of covariant vs contravariant is very old school. Some very old texts go into ways of visualizing this. I would suggest instead learning about tangent vectors (contravariant) and 1-forms (covariant) and the equivalence between tangent vectors and directional derivatives.

Associate the vector $\vec{v}$ with the derivative operator $\vec{\frac{d}{d\lambda}}$ by saying that there is a curve parameterized by $\lambda$ that has $\vec{v}$ as it's tangent vector.

Similarly, associate to the function $f$ the 1-form $df$. A 1-form is a linear map from tangent vectors onto real numbers. A 1-form $df$ maps a tangent vector $\vec{\frac{d}{d\lambda}}$ to the real number $df \left( \vec{\frac{d}{d\lambda}} \right) \equiv \frac{df}{d\lambda}$.

Once you are comfortable with this idea, you will notice that we can introduce a coordinate system $x^i$ and tangent vectors $\frac{\partial}{\partial x^i}$ and one-forms $dx^i$. Note that from our rule, $dx^i \left( \vec{\frac{\partial}{\partial x^j} } \right) = \delta^i_j$.

You can then parameterize your curve with the functions $x^i(\lambda)$. Note that from the chain rule

$\vec{ \frac{d}{d\lambda} } = \frac{\partial x^i}{\partial \lambda} \vec{\frac{\partial}{\partial x^i}}$

and you can use what we've produced so far to show that

$df = \frac{\partial f}{\partial x^i} dx^i$.

When all is said and done, you can prove that

$df \left( \vec{\frac{d}{d\lambda}} \right) = \frac{\partial x^i}{\partial \lambda} \frac{\partial f}{\partial x^j} \delta_i^j = \frac{df}{d\lambda}$

is coordinate independent, as it should be.

From there on, you can define arbitrary tensors as multilinear maps taking $n$ 1-forms and $m$ vectors onto real numbers. The utility of this construction is that it is very geometrical and at the same time not tied to coordinates (abstract). You also never have to wonder which way a thing transforms, because it's always the natural way.

I recommend you pick up a good book on differential geometry for physicists. Geometrical Methods of Mathematical Physics by Schutz is OK, his GR book is probably more useful. The bible by Misner, Thorne and Wheeler goes into great depth into this business and has handy visualizations of n-forms if you are so inclined.

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  • $\begingroup$ Lionel and Sigma: I'm not sure that Schutz's GR book is as good a reference for geometry as it used to be. True, it does still include the discussion of the one form as you and Mark's answer describe them, but stuff like the Lie derivative and much of the other geometrical discussion that used to be in his book has had to make way for expanded chapters of GR experimental evidence and issues. I think Schutz even says something about using his geometry book together with a second reading of his "first course on GR" in the latter's preface. $\endgroup$ – Selene Routley Nov 25 '13 at 3:12
  • $\begingroup$ True, I used both simultaneously. $\endgroup$ – lionelbrits Nov 25 '13 at 10:48

Here is a visualization from Geometrical Methods of Mathematical Physics by Schutz. The co-vector is here called a "one form". His notation $\langle \tilde{\omega} , \bar{V} \rangle$ is equivalent to $\omega_\alpha V^\alpha$, which you might be used to seeing.

enter image description here

Note that when the magnitude of $\bar{V}$ increases, the arrow gets longer. When the magnitude of $\tilde{\omega}$ increases, the parallel surfaces get closer together.

Physically, vectors and covectors are not meaningfully differet from each other. Mathematically, either can be defined as the dual space of the original, so they have all the same properties. You could switch which one is the arrow and which one the set of parallel lines in that picture. If you choose displacements to be prototypical vectors, then other quantities will be either vectors or covectors depending on how they're related to displacements. For example, velocities are just derivatives of displacements, so they are also vectors. Gradients act on displacements to produce scalars, so they are co-vectors.

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Although this is saying the same thing as Lionel's answer and Mark's answer from a different standpoint, another idea that I like in describing the tangent space is to think of the one dimensional $C^1$ space curve (or spacetime curve) within the manifold $M$ as a grounding concept. So our fundamental idea is some function ("A Path" or "A Trail") through the manifold $M$ and centred on some point $p\in M$ which is constant for the present:

$$\sigma:(-\epsilon,\epsilon) \subset \mathbb{R}\to M$$

such that ${\rm d}_t \sigma(t)$ also exists in the same interval $(-\epsilon,\epsilon)$ and such that $\sigma(0) = p\in M$ and $\sigma(\epsilon)\neq p$.

After all one dimensional paths, even if very windy, have made sense to us human and related animals ever since we've needed to find water, food and the way back to our cave!

Then, the tangent space $T_p M$ at point $p\in M$ is the set of equivalence classes of such paths, where we define two such paths $\sigma_1: (-\epsilon,\epsilon)\to M$ and $\sigma_2: (-\epsilon,\epsilon)\to M$ as "equivalent" if their "tangents" are the same at $p$, i.e. if : $\left.{\rm d}_t \sigma_1(t)\right|_{t=0} = \left.{\rm d}_t \sigma_2(t)\right|_{t=0}$. We can then readily define scalar multiples of tangents and additions of tangents: here we must be a little careful because what we are doing of course is implicitly labelling $M$ with one of its atlas's charts so that we are implicity thinking about paths as functions $\sigma:(-\epsilon,\epsilon) \to \mathbb{R}^m$ and their "tangents" ${\rm d}_t \sigma:(-\epsilon,\epsilon) \to \mathbb{R}^m$, where $m$ is the manifold's dimension and this is how we compare paths and declare them to be "equivalent" in the above way. Otherwise, in general, there is of course no notion of the linear operations of scaling and addition in the manifold itself $M$.

So now "contravariant" vectors (or simply plain vectors) are objects that live in such tangent spaces.

Okay, all this is long winded, but my point is that I actually thing of wiggly, squirmy "threads" in families (the latter defined by this equivalence) when I think of tangent vectors and not little arrows. This is something I personally find very helpful as one can imagine something "real" within the manifold itself (and, implicitly through a chart, within our homely and wonted friend $\mathbb{R}^m$) and not simply some idea of "arrows" stuck all over the manifold by some graffiti vandal!

So now, with this concept, we take up Mark's Answer to imagine the one form - or what you're calling a covariant vector (or sometimes covector). Actually, I find the idea of a dual vector space pretty neat, so I generally sit with the mathematician's idea of the one-form. In finite dimensional $\mathbb{R}^m$, a dual vector - a linear functional $\mathbb{R}^m\to\mathbb{R}$ is always an inner product as in Mark's answer (this assertion is the same as saying that $\mathbb{R}^m$ is a complete metric space) and indeed can be represented by its "components" - the values of the functional for the basis vectors of $T_p M$, with all values in $T_p M$ then following from linearity. So this (co)vector (one form) uniquely defines the vector orthogonal to it (modulo a multiplicative constant). The spacing between the level planes of this linear functional defines the "length" of the covector.

  • if you want to get heavy handed, this is where the Riesz Representation Theorem comes on stage - although you don't need anything like the full strength of this theorem to discuss the ideas here.

Now, if your background is optics like me, you've got a very strong and concrete example of the one form. Namely, the wave vector $\tilde{k}$. This beast forms inner products $\left<\tilde{k},\,\underset{\sim}{r}\right>$ with position vectors $\underset{\sim}{r}$ to give you the local phase of the plane wave component it represents. The maximum rate of change of phase in radians per metre is the length of the covector $\tilde{k}$.

Indeed, in Minkowsky spacetime, the four-wavevector is a one-form - a covector:

$$\tilde{k} = (\omega,-k_x,-k_t,-k_z);\qquad \omega = \sqrt{k_x^2+k_y^2+k_z^2}\,c$$

Now, to go to arbitrary valence tensors, if you haven't got some of the references in Lionel's or Mark's answers, a great introductory discussion is given in the first chapter of Kip Thorne's physics 136 course which is downloadable from here. He talks about all these ideas in terms linear functionals and "slots" for components, rather like you would go about representing and storing these ideas in computer memory (with a countably infinite word size, of course, to represent real numbers exactly!).

An aside about references: I'm not altogether sure that Schutz's GR book is as good a reference for geometry as it used to be (as Lionel makes out). True, it does still include the discussion of the one form as you and Mark's answer describe them, but stuff like the Lie derivative and much of the other geometrical discussion that used to be in his book has had to make way for expanded chapters of GR experimental evidence and issues. I think Schutz even says something about using his geometry book together with a second reading of his "first course on GR" in the latter's preface. So have a browse carefully at the contents of any book you might be thinking of buying - an older copy of Schutz may fit better with you.

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Tensors (or rather tensor fields in case of differential geometry) are very generic and not particularly intuitive objects that can fill a lot of roles - volume elements, endomorphisms, Riemannian metrics are just a few things you can describe with tensors.

However, to get an intuition about co- and contravariance, it's enough to look at tangent vectors and covectors, which can be visualized and are the building blocks of higher-rank tensors.

In differential geometry as traditionally taught in physics courses (mathematicians stopped doing it this way some decades ago), we always work in charts (ie local coordinates).

A vector would be a column vector $$ \begin{pmatrix} v^1\\\vdots\\v^n \end{pmatrix} = (v^i)_{i=1}^n=v^i $$ and a covector a row vector $$ \begin{pmatrix} w_1&\cdots&w_n \end{pmatrix} = (w_i)_{i=1}^n=w_i $$ with duality pairing $$ \begin{pmatrix} w^1&\cdots&w^n \end{pmatrix} \begin{pmatrix} v^1\\\vdots\\v^n \end{pmatrix} =\sum_{i=1}^n w_iv^i = w_iv^i $$ As there is in general no global chart, we need to specify transformation laws and make our vectors and covectors equivalence classes with respect to these transformations.

The tranformations are given by the Jacobi matrix of the coordinate switchover and its inverse, which is obviously necessary to keep pairings invariant.

Now, the coordinates of vectors transform opposite to basis vectors - they are contravariant - whereas the components of covectors transform the same way as basis vectors - they are covariant.

You may encounter the opinion that the difference between vectors and covectors doesn't really matter as in general relativity, you can raise and lower indices as you see fit, which culminates in the idea that there's only a single geometric object - the vector - with co- and contravariant components.

That is in my opinion a pretty bad idea: It only works if there is a distinguished non-degenerate bilinear form available and normally, there's a 'natural' placement of indices due to geometry which might even be relevant to model physical concepts (eg velocity vs momentum in context of Lagrangian and Hamiltonian mechanics).

Besides these technical definitions of vectors and covectors (which do make some sense geometrically once you introduce principal bundles and associated vector bundles, but that's not something typically done in physics courses), there are of course more geometric ones:

We can consider a vector an equivalence class of curves through the same point with first order contact. Similarly, a covector would an equivalence class of real-valued functions at a given point with first order contact.

If we compose representatives of a covector and a vector, we get a function $\mathbb R\to\mathbb R$ and evaluating its derivative gives us the natural pairing.

There are also abstract definitions available: We can identify vectors with their directional derivatives, ie a vector is a linear functional on the space of real-valued functions that respects the Leibniz rule. For covectors, there's an algebraic definition as the ideal of real-valued functions vanishing at a point factored by the ideal generated by product of such functions.

Instead of coming up with definitions for both vectors and covectors, it's enough to define one of those manually (typically vectors) and define the other one by duality, ie as real-valued linear maps.

Now, if you actually want to visualize these objects (as in draw meaningful pictures), the obvious representation for vectors is as little arrows in coordinate space. Now, if there's a distinguished non-degenerate bilinear form available, you can represent covectors via their associated vectors (raising its index) and pairing will just be the Euclidean scalar product.

A second way to visualize covectors would be as (oriented) hyperplane fields, with the pairing as described in Mark's answer. This also does not work for arbitrary manifolds - you need a volume form to do so (this is basically a variant of the Hodge dual with the last step from the Wikipedia explanation omitted).

(If you do not have a bilinear or volume form available, you could of course chose an arbitray local one).

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