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2026-04-09 20:08:59 -04:00
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\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:
\begin{equation}
\pdvt[2]{\phi} - \tensor{\delta}{^i^j} \Partial{i} \Partial{j} \phi = 0,
\Partial{t}^2 {\phi} - \tensor{\delta}{^i^j} \Partial{i} \Partial{j} \phi = 0,
\end{equation}
we can introduce $\pi := \Partial{t} \phi$ and $\tensor{\psi}{_i} := \Partial{_i} \phi$, and rewrite the wave equation as
\begin{equation}