Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 52 / 3

For each fixed coordinate function $x^{\alpha }$, the expression $\square x^{\alpha }$ is a scalar: $\alpha$ labels four chosen functions and is not a free tensor index. The equation is therefore coordinate invariant when those functions are held fixed. However, the demand that the currently chosen coordinates themselves be harmonic coordinates is a chart condition, not a four-vector equation invariant under arbitrary changes of coordinate functions.

Indeed, for new functions ${\bar{x}}^{\alpha }=f^{\alpha }(x)$ the d'Alembert operator givesThe second term prevents a vector transformation law. An affine redefinition preserves harmonic coordinates; an arbitrary nonlinear one does not. For example, a Cartesian spatial coordinate $x$ in Minkowski spacetime has $\square x=0$, whereas $\square (x^2)=2$, and $x^2$ is a valid new coordinate on $x>0$. The equations are scalar equations for specified functions, while harmonic gauge is chart dependent.

Treat each $x^{\alpha }$ as a scalar field. Its second covariant derivative is ${\nabla }_{\beta }{\nabla }_{\gamma }{x}^{\alpha }=-{\Gamma }^{\alpha }{}_{\gamma \beta }$, so $\square x^{\alpha }=-H^{\alpha }$, where $H^{\alpha }=g^{\beta \gamma }{\Gamma }^{\alpha }{}_{\beta \gamma }$.

On the other hand, metric compatibility givesUsing the Christoffel trace identity in the last term yieldsSince the volume density never vanishes in a valid chart, the harmonic coordinate conditions are equivalent:

Contract the printed Riemann curvature tensor using $R_{ab}=R^{\mu }{}_{a\mu b}$. Only the derivative terms can contain second derivatives of the metric tensor:Substituting the Christoffel symbol formula, its second derivative part isSet $H_b=g_{b\lambda }{H}^{\lambda }=g^{\mu \nu }{\partial }_{\mu }{g}_{\nu b}-\frac{1}{2}{g}^{\mu \nu }{\partial }_b{g}_{\mu \nu }$. The first, second and fourth terms are the second derivative terms of $\frac{1}{2}({\partial }_a{H}_b+{\partial }_b{H}_a)$. Derivatives of the inverse metric tensor produce only products of first derivatives. Thuswhere $Q$ contains at most first metric derivatives. In harmonic coordinates, $H^{\lambda }$ vanishes throughout the chart, so $H_b$ and its derivatives vanish. The vacuum equations have the wave principal partThese are quasilinear partial differential equations with the principal part of a wave equation, because $g^{\mu \nu }$ has Lorentzian signature. This is the harmonic reduction of the vacuum Einstein equations.

First, the inverse metric tensor printed in the PDF has a missing factor: its lower-left entry must be $g^{i0}={\alpha }^{-2}{\beta }^i$, symmetric with $g^{0i}$. This correction follows either from symmetry or direct block multiplication and does not change the evolution equation used here.

Let $D_t={\partial }_t-{\beta }^k{\partial }_k$. Contract the spatial metric tensor evolution equation with ${\gamma }^{ij}$. Both shift-gradient terms contribute ${\partial }_k{\beta }^k$, givingThe Jacobi determinant derivative formula applies to every derivative of $\gamma =\det ({\gamma }_{ij})$ and gives $D_t\gamma =\gamma {\gamma }^{ij}{D}_t{\gamma }_{ij}$. Therefore the spatial determinant evolves asEquivalently, the spatial volume evolution identity is $D_t\sqrt{\gamma }=\sqrt{\gamma }({\partial }_k{\beta }^k-\alpha K)$, with $K$ the trace of the extrinsic curvature of a spatial hypersurface.

Use the equivalent density form of harmonic slicing from part (ii). With $\sqrt{-g}=\alpha \sqrt{\gamma }$, positive lapse function $\alpha$, and the corrected inverse metric tensor, its time component becomesThus $D_t(\sqrt{\gamma }/\alpha )=(\sqrt{\gamma }/\alpha ){\partial }_i{\beta }^i$, orThe spatial volume evolution identity gives $D_t\log \sqrt{\gamma }={\partial }_i{\beta }^i-\alpha K$. Subtraction yields $D_t\log \alpha =-\alpha K$, hence the harmonic lapse evolution isThe shift vector need not vanish; its transport term is essential in a general 3+1 decomposition of spacetime.