2

I'm trying to do some exercises about the manipulations of the indexes. I have the tensor $X^{\mu\nu}$ represented by the following matrix $$[X^{\mu\nu}] = [X] = \begin{pmatrix} 2 & 3 & 1 & 0 \\ 1 & 1 & 2 & 0 \\ 0 & 1 & 1 & 3 \\ 1 & 2 & 3 & 0 \end{pmatrix}$$

I have to calculate $X^{\mu}_{\phantom{a}\nu}$ so I understodood that

$$X^{\mu}_{\phantom{a}\nu}= X^{\mu\sigma}\eta_{\sigma\nu} = \sum_{k=0}^3 X^{\mu k}\eta_{k \nu}$$

The above object if expanded reads

$$X^{\mu 0}\eta_{0 \nu} + X^{\mu 1} \eta_{1 \nu} + \ldots$$

But the metric tensor $\eta_{\mu \nu} = diag(1, -1, -1, -1)$ has nonzero components only on the diagonal, hence I'd get

$$X^{\mu 0} \eta_{00} + \ldots $$

hence

$$X^{\mu 0} - X^{\mu 1} - X^{\mu 2} - X^{\mu 3}$$

The notation now makes me rather confused because $X^{\mu 0}$ is what? An object from $X^{\mu \nu}$ where the components $\mu$ are the first term of each row, since the column is the $0$-th one and reains fixed, that is

$$X^{\mu 0} = \begin{pmatrix} 2 \\ 1 \\ 0 \\ 1 \end{pmatrix}$$

If this holds for the other objects too, then what I am doing is summing four $4\times 1$ matrices, and I would get a $4\times 1$ matrix, not a $4\times 4 $ matrix in the end.

I am not understanding this...

Heidegger
  • 393

1 Answers1

2

The above object if expanded reads

$$X^{\mu}_{\nu} =X^{\mu 0}\eta_{0 \nu} + X^{\mu 1} \eta_{1 \nu} + \ldots$$

Up to here we are fine because notice we started with a rank 2 tensor (2 free indices) on the left and we still have the same 2 free indices on the right. The next step,

$$X^{\mu 0} \eta_{00} + \ldots $$

is unwarranted because you lose a free index. This is no longer equal to the line above.

If you were summing over $\nu$ then you would be right to drop all the terms that had any non-diagonal elements of $\eta$ as you noted. For instance consider if we contract $\nu$ with some rank 1 tensor $Y$:

$$Y^{\nu}X^{\mu}_{\nu} =Y^{\nu}X^{\mu 0}\eta_{0 \nu} + Y^{\nu}X^{\mu 1} \eta_{1 \nu} + \ldots $$

$$Y^{\nu}X^{\mu}_{\nu} =Y^{0}X^{\mu 0}\eta_{0 0} + Y^{1}X^{\mu 1} \eta_{1 1} + \ldots $$ which would now be a rank one tensor. Note how I indeed dropped terms that had $\eta_{01}$ like you wanted. In summary, never drop a free index, always have the same free indices on the left and the right.

JohnA.
  • 1,793