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