Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 357 / 1 / e / 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 aswhere ${\Gamma }_{\beta }=g_{\beta \lambda }{g}^{\mu \nu }{\Gamma }_{\mu \nu }^{\lambda }$. Harmonic gauge sets ${\Gamma }_{\beta }=0$, soThe 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.