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:

$$a_\mu = \sum_\nu g_{\mu\nu} a^\nu \Rightarrow a_0=a^0, a_k=-a^k .$$ (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:

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

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