[Physics] Is the covariance or contravariance of vectors/tensors something that can be “visualized”


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?

Best Answer

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.

Related Question