Contravariant - covariant notations

Question

Working in curvilinear coordinates, one can define basis vectors corresponding to those coordinates. In the figure below (taken from here), $\{\mathbf{g_i}\}$ are base vectors corresponding to the change in $\{\Theta_i\}$. These are called covariant base vectors. One can also define another set of base vectors, $\{\mathbf{g^i}\}$, such that $\mathbf{g^i} \ \mathbf{g_j} = \delta_{i}^j$. These are called the contravariant basis and a vector $\mathbf{v}$ can be written as $\mathbf{v} = v^i \mathbf{g}_i = v_i \mathbf{g}^i$.

Now I understand that the summation convention says that if an index appears as a subscript and a superscript, the summation is implicit. My question is whether there is any (other) practical reason for denoting the components in the covariant basis with upper indices and in the contravariant basis with lower indices. For instance, it would be much more natural to me to write $\mathbf{v} = v_i \mathbf{g}_i = v^i \mathbf{g}^i$ (where now $v_i$ denotes the component $v^i$ from the figure). This way the indices in the components would match the indices in their basis: in the basis with the upper index, components would be with upper index and vice versa. The norm and scalar product would work out to be the same since $\mathbf{u} \mathbf{v} = u_i \mathbf{g}_i v^j \mathbf{g}^j = u_i v^i$. Obviously, in this case indices appearing twice at the same level have to be summed.

I also understand that the co- and contravariant components transform differently. However, here I'm interested in the benefits of the upper-lower notation as opposed to a matching, upper-upper and lower-lower notation.

Answer 1

I also understand that the co- and contravariant components transform differently. However, here I'm interested in the benefits of the upper-lower notation as opposed to a matching, upper-upper and lower-lower notation.

But that's precisely the point - objects with a lower index (whether they be basis vectors or components) transform in one way, while objects with an upper index transform the other way, such that their contractions remain invariant. In your suggested convention, index placement does not determine transformation behavior, whereas in the standard convention it does.

Of course, this is all conventional. If you want, you can write all indices upstairs, but write "covariant" indices with red ink and "contravariant" indices with blue ink. But there is good reasoning behind the existing convention, and that (along with the fact that it is essentially universal at this point) is enough for me not to worry about rocking the boat.

Answer 2

The components of a vector and the basis vectors transform differently. The components of a vector transform contravariantly and the basis vectors transform covariantly, so writing

$$\vec V=v_i\hat e_i \tag{1}$$

is not just different notation, it is incorrect. The choice to label upstairs indices as contravariant and downstairs as covariant is arbitrary (as far as I know), much like defining positive and negative charge.

The reason contravariant and covariant indices have to be contracted together and not contravariant-contravariant or covariant-covariant contraction is because in the context of physics they produce "frame invariant" quantities. Different observers will disagree on some measurements, but they will always agree on quantities defined by tensor contraction. I have actually asked this question before and G.Smith was nice enough to write out this answer with an explicit example.