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- Definition
- The metric tensor
specifies the invariant interval
(distance) between two neighbouring points (events)
![\begin{displaymath}
ds^2 = g_{\alpha\beta} dx^\alpha dx^\beta
\end{displaymath}](img11.png) |
(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}](img12.png) |
(7) |
- Defining
![$g^{\alpha\beta}$](img13.png)
-
![\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}](img14.png) |
(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}](img15.png) |
(9) |
Dmitri Pogosyan
2006-09-28