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\begin{frame}{Towards first order Z4c}{First order reduction} \begin{frame}{Towards first order Z4c}{First order reduction}
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: 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} \begin{equation}
\pdvt[2]{\phi} - \tensor{\delta}{^i^j}\pdv{\phi}{x^i}{x^j} = 0, \pdvt[2]{\phi} - \nabla^2 \phi = 0,
\end{equation} \end{equation}
we can introduce $\pi := \pdvt{\phi}$ and $\psi := \pdv{\phi}{x}$, and rewrite the wave equation as we can introduce $\pi := \pdvt{\phi}$ and $\psi := \pdv{\phi}{x}$, and rewrite the wave equation as
\begin{equation} \begin{equation}