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

For a fixed scalar function $f$, the equation ${\square }_{g}f=0$ is coordinate invariant. A coordinate label $x^{\lambda }$, however, is not a geometrically specified scalar under an arbitrary change of chart: a nonlinear redefinition of harmonic coordinates need not remain harmonic. Thus ${\square }_g{x}^{\lambda }=0$ is a coordinate or gauge condition selecting a class of charts, rather than a tensor equation asserting the vanishing of a tensor field with index $\lambda$.

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Treating $x^{\lambda }$ as a scalar during differentiation,HenceMetric compatibility and the determinant identity give the contracted-Christoffel formulaTherefore

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Differentiate the determinant using $\partial \gamma /\partial {\gamma }_{ij}=\gamma {\gamma }^{ij}$:The advection term is $\gamma {\gamma }^{ij}{\beta }^m{\partial }_m{\gamma }_{ij}={\beta }^m{\partial }_m\gamma$. The symmetrized shift-gradient term contracts to $2\gamma {\partial }_m{\beta }^m$, and the curvature term to $-2\alpha \gamma K$. Thus

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For $\lambda =0$, $\sqrt{-g}=\alpha \sqrt{\gamma }$, $g^{00}=-{\alpha }^{-2}$, and $g^{0i}={\beta }^i{\alpha }^{-2}$. The harmonic coordinate condition isUsingfrom the determinant evolution, all shift-divergence and volume terms cancel, leavingThus harmonic slicing is the Bona--Masso slicing condition with

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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.

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