= Solution
The Ricci tensor contains second derivatives from $\partial\Gamma$ and quadratic first-derivative terms from $\Gamma\Gamma$. Its second-derivative terms can be rearranged as
$$
R_{\alpha\beta}
=-\frac12g^{\mu\nu}\partial_\mu\partial_\nu g_{\alpha\beta}
+\partial_{(\alpha}\Gamma_{\beta)}
+Q_{\alpha\beta}(g,\partial g),
$$
where $\Gamma_\beta=g_{\beta\lambda}g^{\mu\nu}\Gamma^\lambda_{\mu\nu}$. Harmonic gauge sets $\Gamma_\beta=0$, so
$$
\boxed{R_{\alpha\beta}
=-\frac12g^{\mu\nu}\partial_\mu\partial_\nu g_{\alpha\beta}
+Q_{\alpha\beta}(g,\partial g)=0}.
$$
The principal part is the spacetime wave operator acting on every metric component. This <hyperbolic reduction of Einstein's equations> turns the reduced vacuum equations into a quasilinear hyperbolic system with finite-speed propagation and a well-posed local Cauchy problem, provided the constraints and harmonic gauge constraints hold initially.
Solved by gpt-5.6-sol high.
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