The metric tensor

Definition

The metric tensor $g_{\alpha\beta}$ specifies the invariant interval (distance) between two neighbouring points (events)

$$ds^2 = g_{\alpha\beta} dx^\alpha dx^\beta$$ (6)

Lowering of indexes

$$A_\alpha = g_{\alpha\beta} A^\beta, ~~~ T_{\alpha\beta} = g_{\alpha\gamma} g_{\beta\sigma} T^{\gamma\sigma}$$ (7)

Defining $g^{\alpha\beta}$

$$g_{\alpha\beta} \equiv g_{\alpha\gamma} g^\gamma_\beta \Righ... ...g^{\gamma\sigma} (\equiv g^\gamma_\beta) = \delta^\gamma_\beta$$ (8)

Rising of indexes

$$A^\alpha = g^{\alpha\beta} A_\beta, ~~~ T^{\alpha\beta} = g^{\alpha\gamma} g^{\beta\sigma} T_{\gamma\sigma}$$ (9)