Zero scalar with V(0)=0 reduces to electrovacuum

Since $\phi =0$ and $\pi =0$, the scalar equation is $0-V^{\mathrm{\prime }}(0)0=0$ and $j^a=0$. Also $L_{\phi }=V(0)=0$, so $T_{ab}^{\phi }=0$. The condition $V(0)=0$ is what removes the remaining term $-\frac{1}{2}{K}_{\phi }{g}_{ab}V(0)$.

Substituting $j=0$ and $T^{\phi }=0$ into the system of [F1] leaves exactly $G_{ab}+\mathrm{\Lambda }{g}_{ab}=\kappa T_{ab}^{\mathrm{E}\mathrm{M}}$, $dF=0$, and ${\mathrm{\nabla }}_a{F}^{ab}=0$. Conversely, an exact-potential source-free solution with $\phi =\pi =0$ satisfies each charged-scalar equation by step 1.1 and each remaining equation by its source-free hypothesis.

For exact-potential electrovacuum initial data, setting ${\phi }_0=0$, ${\pi }_n=0$, and hence ${\pi }_i=0$ makes $j=0$ and $T^{\phi }=0$ on the initial slice. The sourced Maxwell and total-stress Einstein constraints in [L1] then reduce to the source-free electrovacuum constraints. Applying [L1] under [A2] gives a smooth local solution. On that solution the scalar block for $(\phi ,\pi )$ is a homogeneous first-order wave/transport system with the metric, $A$ and $F$ as given smooth coefficients: the potential term is the smooth coefficient $V\mathrm{\prime }(\mathrm{\mid }\phi {\mathrm{\mid }}^2)$ times $\phi$, and every other scalar source is linear in $\phi ,\pi$. The symmetric scalar-block energy with zero initial scalar data therefore satisfies $E\mathrm{\prime }\le CE$ and is zero by the same local uniqueness argument in [L1]. Thus $\phi =\pi =0$ persists. Step 2.1 then identifies the resulting equations with the zero-scalar branch. No global existence claim follows.