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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 357 / 1 / e / Solution

In harmonic gauge, the vacuum Einstein equations have principal part $-g^{\mu\nu}\partial_\mu\partial_\nu g_{\alpha\beta}/2$, forming a quasilinear wave system supplemented by constraints.


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 Incoming links: Hyperbolic reduction of Einstein's equations