How to understand 'covariance' and 'contravariance'

[chaksome]

TL;DR

covariance covariance

When I come into contact with these two concept in the book of Landau, I gradually know how to use $A^i or A_i$ to simplify the calculation in special relativity.
But I found it hard to give an explicit explanation for them(including gauge matrix) in the aspect of physics.
Could you please give me some illumination? Thanks~

 

[Orodruin]

Covariant components transform in the same way as the tangent vector basis, hence "co-". Contravariant components transform in the "opposite" way as the tangent vector basis, hence "contra-".

A vector is a geometical object and does not depend on the basis you use to describe it. Hence, the components need to transform opposite to how the basis they are used with transforms. Therefore, if you use the tangent basis $E_\mu$, you need to use the contravariant components such that the vector $V = V^\mu E_\mu$ is invariant. Conversely, if you use the dual basis $E^\mu$, you need to use the covariant components $V = V_\mu E^\mu$ to have a coordinate independent object.

Things become slightly more complicated in a general manifold where there is no direct link between tangent and dual vectors, but generally tangent vectors (with coordinate bases) then have contravariant components and dual vector covariant components.

 

[Ibix]

An example I found useful is the length of a running track. I can measure it by taking a standard ruler and laying it end-to-end N times. The track length is therefore N units.

But now I redefine my standard unit, and the new standard length is exactly twice my old unit. So I need new standard rulers, and the length of the track becomes N/2 new units.

So, under the length redefinition by a factor of two, ruler lengths covary (they increase by a factor of two) while the counts of rulers needed contravary (they decrease by a factor of two). The physically meaningful thing, the product of standard ruler length and the number of rulers I needed to use (the product of a contravariant and a covariant quantity), is invariant.

Orodruin's explanation is better - this is just an example of the kind of thing he's talking about.

 

[olgerm]

Contravariant unitvector $\vec{x^i}$ is pointed to direction where i'th contravariant coordinate changes most per distance.
Covariant unitvector $\vec{x_i}$ is pointed to direction where only i'th contravariant coordinate changes(other contravariant coordinates remain same).

 

[Orodruin]

Orodruin's explanation is better - this is just an example of the kind of thing he's talking about.

I am not sure it is better, it is just a different illustration that is restricted to the one-dimensional case where there is only one direction so you only have to worry about scaling of the base vector (and not scalings and rotations as in the general case).

 

[Orodruin]

Contravariant unitvector $\vec{x^i}$ is pointed to direction where i'th contravariant coordinate changes most per distance.
Covariant unitvector $\vec{x_i}$ is pointed to direction where only i'th contravariant coordinate changes(other contravariant coordinates remain same).

This is a bit misleading since coordinates in general are neither co- nor contravariant, they are just functions on the base space.

The more accurate description is to say that the tangent vector basis $E_\mu = \partial_\mu$ (the basis whose transformations define what covariant means) point in the direction of the respective coordinate lines and that the dual basis $E^\mu = dx^\mu$ consists of the one-forms that map the corresponding tangent basis vectors to one and the others to zero. The interpretation in terms of "change per distance" is contingent on the existence of a metric.

 

[Ibix]

I am not sure it is better

You've also covered (or at least introduced) the role of the metric in all of this, where I've blithely assumed I have one.

Maybe I am being a bit harsh to my answer. The formal mathematical machinery is extremely powerful and necessary, but it was an example of the type I gave (in @bcrowell's book, I think) that started to give me a feel for what co- and contra-variant quantities could be, physically.