Questions about metric tensor in terms of matrix

Question

1)Metric tensor is used for lowering and raising indices. Does that means that it will always be the same matrix as long as it is in the same space(e.g. flat space)? $$g_{\mu\nu}=g^{\mu\nu}=g_{\alpha\beta}=g_{\alpha\gamma}=g_{\beta\delta}=...= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0& 0& -1 \end{pmatrix} $$

E.g. $A_{\alpha\beta}=g_{\alpha\gamma}g_{\beta\delta}A^{\gamma\delta}$<--------Does this means that $g_{\alpha\gamma}$ and $g_{\beta\gamma}$ are the same matrix(metric tensor)? But when I matrix product $g_{\alpha\gamma}$ and $g_{\beta\gamma}$, I get an identity matrix instead which doesn't seem to help in raising/lowering indices.

My confusion here is because there are lack of examples involving them in matrix.

2) To raise/lower indices, is metric tensor the only way to do it?

Answer 1

In matrix notation you must write $$ (A_{\alpha\beta})= (g_{\alpha\mu})(A^{\mu\nu})( g_{\nu\beta})$$ \begin{eqnarray} =\begin{bmatrix} 1 &0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{bmatrix} \begin{bmatrix} A_{00} &A_{01}&A_{02}&A_{03}\\ A_{10}&A_{11}&A_{12}&A_{13}\\ A_{20}&A_{21}&A_{22}&A_{23}\\ A_{30}&A_{31}&A_{32}&A_{33} \end{bmatrix} \begin{bmatrix} 1 &0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{bmatrix} \end{eqnarray}

Answer 2

When the indices are at the same position the tensors are the same, thus:

$$g_{\mu\nu}=g_{\alpha\beta}=g_{\alpha\gamma}=g_{\beta\delta}\neq g^{\mu\nu}$$

One can get from one to the other by realizing that consecutive lowering and raising of indices should give the original result:

$$g_{\rho\nu} \cdot g^{\mu\nu} \cdot V_{\mu} = V_{\rho} = V_{\mu} $$

therefore

$$g_{\rho\nu} \cdot g^{\mu\nu} = \delta^{\mu}_{\rho} =\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$

This allows one to get $g^{\mu\nu}$ when $g_{\mu\nu}$ is known.

For Minkowski spacetime we find that $g^{\mu\nu} = g_{\mu\nu}$, but this is not generally true.

Note that your strange result arose because the summation convention does not work when multiplying $g_{\alpha\gamma}$ by $g_{\beta\gamma}$ since the $\gamma$ indices are both low. Multiplying two second order tensors with lowered indices will give a fourth order tensor with lowered indices.