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\end{frame}
\begin{frame}{Towards first order Z4c}{First order reduction}
\subsection{First order reduction: wave equation}
\subsection{First order reduction: wave equation as an example}
We can introduce auxiliary variables to reduce a second order PDE system to a first order one. For example, for the wave equation on flat spacetime:
\begin{equation}
\Partial{t}^2 {\phi} - \tensor{\delta}{^i^j} \Partial{i} \Partial{j} \phi = 0,