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Covariant and Contravariant Indices

We will need to distinguish between a covariant vector and a contravariant vector. In terms of indices, we define

covariant indices $\rightarrow$ $a_\mu$ (subscript),
contravariant indices $\rightarrow$ $a^\mu$ (superscript).

The metric tensor is used to convert from one type of vector to the other:

a_\mu = \sum_\nu g_{\mu\nu} a^\nu \Rightarrow a_0=a^0, a_k=-a^k .
\end{displaymath} (2.10)

Normally the sum over identical indices is implied (Einstein summation convention) and we simply write $a_\mu=g_{\mu\nu}a^\nu$. We can also rise indices: $a^\mu=g^{\mu\nu}a_\nu$, where $g^{\mu\nu}=g_{\mu\nu}$ for a Lorentz metric. Also $g_\mu^{\ \nu} = g_{\mu\rho}g^{\rho\nu} = g^\mu_{\ \nu} =
\delta_\mu^{\ \nu}$, where $\delta_\mu^{\ \nu}$ is the Kronecker symbol:

\delta_\mu^{\ \nu} = \left\{
1 \quad \textr...
...}, \\
0 \quad \textrm{if $\mu\neq\nu$}.
\end{displaymath} (2.11)

Also, notice that $g_{\mu\nu}g^{\mu\nu} = (g_{\mu\mu})^2 = 4$.

Douglas M. Gingrich (gingrich@