We have thus defined three fundamental tensors \[ g_{\mu\nu},\quad g_{\mu}^{\nu},\quad g^{\mu\nu} \] of covariant, mixed, and contravariant characters respectively. \Section{26.}{Associated tensors} \index{Associated tensors}% We now define the operation of raising or lowering a suffix. Raising the \index{Suffixes, raising and lowering of}% suffix of a vector is defined by the equation \[ A^{\mu} = g^{\mu\nu} A_{\nu}, \] and lowering by the equation \[ A_{\mu} = g_{\mu\nu} A^{\nu}. \] For a more general tensor such as~$A_{\alpha\beta\mu}^{\gamma\delta}$, the operation of raising~$\mu$ is defined in the same way, viz.\ \[ A_{\alpha\beta}^{\gamma\delta\mu} = g^{\mu\nu} A_{\alpha\beta\nu}^{\gamma\delta}, \Tag{(26.1)} \] and for lowering \[ A_{\alpha\beta\mu}^{\gamma\delta} = g_{\mu\nu} A_{\alpha\beta}^{\gamma\delta\nu}. \Tag{(26.2)} \] These definitions are consistent, since if we raise a suffix and then lower it we reproduce the original tensor. Thus if in~\Eq{(26.1)} we multiply by~$g_{\mu\sigma}$ in order to lower the suffix on the left, we have \begin{align*} g_{\mu\sigma} A_{\alpha\beta}^{\gamma\delta\mu} &= g_{\mu\sigma} g^{\mu\nu} A_{\alpha\beta\nu}^{\gamma\delta} \\ &= g_{\sigma}^{\nu} A_{\alpha\beta\nu}^{\gamma\delta} \\ &= A_{\alpha\beta\sigma}^{\gamma\delta}\qquad\text{by \Eq{(25.2)},} \end{align*} which is the rule expressed by~\Eq{(26.2)}. \PageSep{57} It will be noticed that the raising of a suffix~$\nu$ by means of~$g^{\mu\nu}$ is accompanied by the substitution of~$\mu$ for~$\nu$. The whole operation is closely akin to the plain substitution of~$\mu$ for~$\nu$ by means of~$g_{\mu}^{\nu}$. Thus \begin{align*} &\text{multiplication by $g^{\mu\nu}$ gives substitution with raising,} \\ &\text{multiplication by $g_{\mu}^{\nu}$ gives plain substitution,} \\ &\text{multiplication by $g_{\mu\nu}$ gives substitution with lowering.} \end{align*} In the case of non-symmetrical tensors it may be necessary to distinguish the place from which the raised suffix has been brought, e.g.\ to distinguish between ${A_{\mu}}^{\nu}$ and~${A^{\nu}}_{\mu}$. It is easily seen that this rule of association between tensors with suffixes in different positions is fulfilled in the case of $g^{\mu\nu}$, $g_{\mu}^{\nu}$, $g_{\mu\nu}$; in fact the definition of~$g_{\mu}^{\nu}$ in~\Eq{(25.1)} is a special case of~\Eq{(26.1)}.