Symmetric hyperbolic reduction of the Einstein-scalar system and on-shell conservation

(Scalar symmetric system.) Write $D_0=P$ and $D_i=Q_i$. Use ${\mathrm{\partial }}_0\phi =P$, ${\mathrm{\partial }}_0{Q}_i={\mathrm{\partial }}_{i}P$ and $-g^{00}{\mathrm{\partial }}_{0}P-2g^{0i}{\mathrm{\partial }}_{i}P-g^{ij}{\mathrm{\partial }}_i{Q}_j+g^{ab}{\mathrm{\Gamma }}^c{}_{ab}{D}_c+V^{\mathrm{\prime }}(\phi )=0.$ Multiplying the $Q_i$ equations by $g^{ji}$ gives time block $\mathrm{diag}(1,-g^{00},g^{ij})$. For each spatial derivative ${\mathrm{\partial }}_k$, its $P,P$ entry is $-2g^{0k}$, and its $P,Q_j$ and $Q_j,P$ entries are both $-g^{kj}$; all other scalar spatial entries vanish. Thus all spatial matrices are symmetric and the time matrix is positive definite on the stated open set. The lower terms are smooth functions of $z$ by [L2].

(Compatibility and matter recovery.) The first and third scalar equations imply ${\mathrm{\partial }}_0(Q_i-{\mathrm{\partial }}_i\phi )={\mathrm{\partial }}_{i}P-{\mathrm{\partial }}_i{\mathrm{\partial }}_0\phi =0$. The prescribed complete data give zero initial residual, so $Q_i={\mathrm{\partial }}_i\phi$ throughout the local solution, while $P={\mathrm{\partial }}_0\phi$ by its first equation. Substitute these relations in the displayed $P$ equation: it is exactly the scalar wave equation of [L2], multiplied by $-1$. Every physical scalar solution satisfies these equations after extraction. This establishes matter recovery on the reduced metric without using its Einstein residual.

(Coupled reduction and initialization.) Append the metric block of [L1]. In partial variables use ${\mathrm{\partial }}_{0}g=p$ and ${\mathrm{\partial }}_0{q}_i={\mathrm{\partial }}_{i}p$, which give ${\mathrm{\partial }}_0(q_i-{\mathrm{\partial }}_{i}g)=0$ and recover the metric derivatives from compatible initial data; the scalar stress is a smooth algebraic function of $(g,\phi ,P,Q)$ by [L3]. Conversely the scalar connection terms use $(g,p,q)$ without derivatives of those first-order variables. The combined principal matrices are therefore the two symmetric diagonal blocks with no cross derivatives, and the positive time matrices persist on a sufficiently small open state neighborhood. Complete scalar data determine every matter variable as stated, not merely stress tensor jets. For a curved background the transformed compatibility equations include the prescribed background-curvature commutator terms; they are obtained from this partial system rather than assuming covariant derivatives commute. Smooth background Christoffel coefficients change $p,q$ to ${\bar{\mathrm{\nabla }}}_{0}g,{\bar{\mathrm{\nabla }}}_{i}g$ by adding expressions linear in $g$; this invertible zeroth-order change preserves the reduction, with a congruence of principal matrices when changing the equations.

(Conservation on the reduced solution.) Step 2.1 has already recovered the full scalar equation. Applying [L3]'s local identity gives ${\mathrm{\nabla }}^a{T}_{ab}=0$ on this reduced solution before Einstein recovery. The identity and reduction are local tensor equations, so using local orientations if needed imposes no global orientability restriction. This proves all stated conclusions without a converse from stress conservation to the scalar equation.