Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 357 / 1 / b / Solution

First take the spacetime trace of the generalized field equation. Since $n_{\mu }{Z}^{\mu }=-\Theta$ and spacetime has dimension four, this givesNext contract twice with the unit normal. The result isAdding the trace equation to twice this normal projection cancels $T$. The Scalar Gauss equation then converts the curvature terms to

For the derivative terms, part (iii) givesDifferentiating $n^{\nu }{Z}_{\nu }=-\Theta$ along $n^{\mu }$ yieldsCombining these identities produces

Because $\Theta$ is a scalar field and $n^{\mu }={\alpha }^{-1}(1,-{\beta }^i)$,The normal acceleration is spatial, so $Z_{\mu }{a}^{\mu }={\Theta }_{\mu }{a}^{\mu }$. Part (iv) also gives $D^{\mu }{Z}_{\mu }=D^{\mu }{\Theta }_{\mu }-K\Theta$. Solving the preceding constraint for ${\partial }_t\Theta$ givesThus the constants in this Z4 formulation evolution equation are