What exactly do we mean when we say that something is covariant? How is it different from being invariant?
Am i right if I say that the word invariant is used when we're talking about a physical quantity (example-length and momentum 4-vectors are Lorentz invariant). Whereas the word covariant is used when we're talking about equations.
I'm a bit lost with the terminologies.
Strictly speaking, covariant means that an object transforms covariantly. This means that it transforms along with the basis vectors (co- means with, or along).
If the basis vectors transform like $$\tilde {\mathbf e}_i={\mathbf e}_j{F^j}_i$$ Then a covariant vector is something whose components transform using the same matrix. $$\tilde \omega_i=\omega_j{F^j}_i$$ A contravariant vector, which we often think of as a regular vector, transforms with the inverse of this matrix. It transform against (contra) the basis vectors. $$\tilde V^i={B^i}_jV^j$$ where $B$ is the inverse of $F$, or in other words $B\cdot F=1\!\!1$ or ${B^i}_j{F^j}_k=\delta^i_k$. I used $F$ for the forward transformation and $B$ for the backward transformation. Check out this playlist on youtube by Eigenchris for a more in-depth explanation using the same notation.
But... the word covariant is also used often to denote something that is invariant under such transformations, as can be seen enter link description here on wikipedia. So covariant could mean either what I meant above or it could mean that something is invariant.
Note that vectors are invariant as well under a change of basis. The vector components change, but the vector as a whole stays invariant. You can write $$\mathbf V=V^i\mathbf {e}_i$$ where $V^i$ transforms covariantly, but $V$ doesn't transform. You can check that $\tilde{ \mathbf V}=\mathbf V$
To make things even more complicated there are active and passive transformations. A passive transformation is like a change of basis, like I mentioned before. Vectors do not transform under passive transformations. An active transformation means that vectors change as well. As an example you could lay down a magnet on your lab table. The magnetic produces a magnetic field $\mathbf B$ around the magnet. If we now rotate our lab table we perform an active transformation on $B$: we rotate each vector with the same angle as we rotated the table.
Usually the author specifies at the beginning of a book if the transformations are active or passive and if they don't they deserve to stub their toe on a heavy table.
