Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 357 (Gravitational Waves and Numerical Relativity)

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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From $\gamma =c/r+O(r^{-2})$,The first hypersurface equation givesand henceAsymptotic flatness in the standard Bondi frame sets the integration function ${\beta }_0=0$.

With this choice, the leading bracket in the second hypersurface equation isThereforeWriting $D={\partial }_{\theta }c+2c\cot \theta$ and integrating twice,Standard asymptotic flatness sets $U_0=0$; $U_3$ is free integration data at the next order.

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The azimuthal metric component isSince $\gamma =c/r+O(r^{-2})$,Thus the Bondi news function satisfies

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Only the transverse metric components are needed. At fixed $(u,r,\theta )$,Contracting this vector with the $\tilde{x},\tilde{y}$ block of the metric makes all angular factors combine to ${\tilde{\rho }}^2$:Substitution into the preceding news formula givesTherefore the requested constants are

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Decompose each index with ${\delta }^{\rho }{}_{\mu }={\perp }^{\rho }{}_{\mu }-n^{\rho }{n}_{\mu }$. Since $n^{\nu }{n}_{\nu }=-1$, one has $n^{\nu }{\nabla }_{\mu }{n}_{\nu }=0$, so the second index of ${\nabla }_{\mu }{n}_{\nu }$ is spatial. Its spatial projection in the first index is $-K_{\mu \nu }$ by definition, while its normal projection isHence

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Use $n_{\mu }=-\alpha {\nabla }_{\mu }t$. Differentiating, contracting with $n^{\rho }$, and using $n^{\rho }{\nabla }_{\rho }t=1/\alpha$ gives a gradient of $\log \alpha$ plus a component parallel to $n_{\mu }$. The acceleration is orthogonal to $n^{\mu }$, so spatial projection removes the parallel part:

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Because $n^{\mu }{\partial }_{\mu }\varphi =-2\Pi$,Writing $\varphi ={\varphi }_R+i{\varphi }_I$ and $\Pi ={\Pi }_R+i{\Pi }_I$ givesEvery quantity on the right is real, so the Noether charge density is explicitly real valued.

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By definition,Thereforewith $n_{\alpha }{Q}^{\alpha }=0$.

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Taking the divergence of the Noether current,because the two gradient-product terms cancel. The field equation and its conjugate are $\square \varphi =V^{\prime }\varphi$ and $\square \bar{\varphi }=V^{\prime }\bar{\varphi }$, with real $V^{\prime }(|\varphi {|}^2)$. Hence

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Insert $J^{\mu }=Q{n}^{\mu }+Q^{\mu }$:The extrinsic-curvature convention gives ${\nabla }_{\mu }{n}^{\mu }=-K$. For a spatial vector,ThereforeThe constants are

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In adapted coordinates,Multiplying the conservation law by $\alpha$ givesEquivalently, combining the last two terms,This is the coordinate form of the 3+1 Noether-current conservation law.

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