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The metric tensor

Definition
The metric tensor $g_{\alpha\beta}$ specifies the invariant interval (distance) between two neighbouring points (events)
\begin{displaymath}
ds^2 = g_{\alpha\beta} dx^\alpha dx^\beta
\end{displaymath} (6)

Lowering of indexes

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

Defining $g^{\alpha\beta}$

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

Rising of indexes

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



Dmitri Pogosyan 2006-09-28