Is it legal to write $A_\mu = (a, b, c, d)$?

Question

In special relativity (theoretical physics), one uses a lot of four-vectors. With regular vectors, I would say the following is okay: $$ \vec A = \begin{pmatrix} a \\ b \\ c \end{pmatrix} \qquad A_1 = a \qquad A_2 = a \qquad A_3 = c $$

It would be incorrect, however, to write something like: $$ A_i = \begin{pmatrix} a \\ b \\ c \end{pmatrix} $$

Since $A$ (or $\vec A$) is the vector, and $A_i$ is the $i$-th component of that vector.

In special relativity, I see the following all the time: $$ A^\mu = \begin{pmatrix} a \\ b \\ c \\ d \end{pmatrix} $$

Or the four-$\nabla$: $$ \partial_\mu = \begin{pmatrix} \frac 1c \frac{\partial}{\partial t} &\frac{\partial}{\partial x} &\frac{\partial}{\partial y} &\frac{\partial}{\partial z} \end{pmatrix} \qquad \partial_0 = \frac 1c \frac{\partial}{\partial t} $$

So far, I have read in “Mathematical methods for Physicists” (Arfken & Weber) that everything is okay, as long as one does not write $\vec A = A^\mu$ or $A = A^\mu$. Some Physicists don't even understand the problem, others say that this $\partial_\mu = (\ldots)$ is wrong, but they are Physicists and ignore that.

Is writing $\partial_\mu = (\ldots)$ okay or just sloppy notation that virtually everybody uses? If it is okay, what about the ambiguity of $\partial_\mu$ and $\partial_0$?

Answer 1

If a, b, and c are real numbers the first equality is wrong, because LHS is a scalar (the i-th component of the vector, i.e., a, b, or c), while the RHS is a vector.

Answer 2

I'm not an expert in special relativity, but I guess the meaning of the notation $$ A^\mu = \begin{pmatrix} a \\ b \\ c \\ d \end{pmatrix} $$ is that $\mu = (0,1,2,3)$ is a multi-index which picks the entries $0,1,2$ and $3$ of $A$.