\begin{definition}[Definition:Pseudo-Euclidean Metric]
Let $\struct {M, g}$ be a pseudo-Riemannian manifold.
Let $\tuple {\xi^1, \dotsc, \xi^r, \tau^1, \dotsc, \tau^s}$ be the standard coordinates of $\R^{r + s}$.
The '''pseudo-Euclidean metric (of signature $\tuple {r, s}$)''' is the pseudo-Riemannian metric, which in the standard coordinates reads:
:$g = \paren {\d \xi^1}^2 + \dotsb + \paren {\d \xi^r}^2 - \paren {\d \tau^1}^2 - \dotsb - \paren {\d \tau^s}^2$
\end{definition}