Local potential homotopy and Lorenz gauge reachability

Translate the coordinates so the star center is $0$. Differentiating under the integral gives ${\mathrm{\partial }}_k{A}_a-{\mathrm{\partial }}_a{A}_k={\int }_0^1(2t{F}_{ka}(tx)+t^2{x}^b({\mathrm{\partial }}_k{F}_{ba}-{\mathrm{\partial }}_a{F}_{bk})(tx))dt$. The coordinate identity $dF=0$ in [F1] gives ${\mathrm{\partial }}_k{F}_{ba}-{\mathrm{\partial }}_a{F}_{bk}={\mathrm{\partial }}_b{F}_{ka}$. The integrand is therefore $\frac{d}{dt}(t^2{F}_{ka}(tx))$, whose integral is $F_{ka}(x)$ because the lower endpoint is zero. Thus $dA=F$. On every compact coordinate rectangle around a point of $U$, the integrand and all its coordinate derivatives are continuous on the product with $[0,1]$; apply [F8] one derivative at a time to prove smoothness of $A$.

For a smooth scalar $\chi$, [F1] gives $d(A+d\chi )=dA+d^2\chi =dA$, and metric compatibility in [F7] gives ${\mathrm{\nabla }}^a(A_a+{\mathrm{\nabla }}_a\chi )={\mathrm{\nabla }}^a{A}_a+{\mathrm{\square }}_g\chi$. Hence the Lorenz condition is obtained by solving ${\mathrm{\square }}_g\chi =-{\mathrm{\nabla }}^a{A}_a$ with zero initial value and zero initial normal derivative on a smaller spacelike patch. The operator is exactly the scalar operator defined in [F11].

Fix $p\in S$. By [F4] the future unit normal $n_y$ varies smoothly in a slice neighborhood. By [F5] the geodesic flow $(t,y,v)\mapsto {\gamma }_{y,v}(t)$ is smooth near $(0,p,n_p)$, so $\mathrm{\Phi }(t,y):={\gamma }_{y,n_y}(t)$ is smooth. At $(0,p)$ its differential sends the time direction to $n_p$ and slice tangent vectors to themselves; these span $T_{p}M$, so [F5] makes $\mathrm{\Phi }$ a coordinate chart after shrinking. Put $T={\mathrm{\partial }}_0$ and $J_i={\mathrm{\partial }}_i$. The geodesic equation gives ${\mathrm{\nabla }}_{T}T=0$ and [F6] gives $g(T,T)=-1$. Since $[T,J_i]=0$, torsion-freeness in [F7] gives ${\mathrm{\nabla }}_T{J}_i={\mathrm{\nabla }}_{J_i}T$, so ${\mathrm{\partial }}_{0}g(T,J_i)=g({\mathrm{\nabla }}_{T}T,J_i)+g(T,{\mathrm{\nabla }}_T{J}_i)=\frac{1}{2}{J}_i(g(T,T))=0$. At $t=0$, $g(T,J_i)=g(n,J_i)=0$ by [F4]; hence $g_{0i}=0$ and $g_{00}=-1$ throughout the smaller chart. By continuity the spatial metric stays positive on a smaller relatively compact chart, so $(g^{ij})$ is positive definite.

Write $f=-{\mathrm{\nabla }}^a{A}_a$, $p_0={\mathrm{\partial }}_0\chi$, and $q_i={\mathrm{\partial }}_i\chi$. Put $W^c:=g^{ab}{\mathrm{\Gamma }}^c{}_{ab}$ and define the smooth lower-order term $L(p_0,q):=W^0{p}_0+{\sum }_i{W}^i{q}_i$. By [F11], $-{\mathrm{\square }}_g\chi =-g^{00}{\mathrm{\partial }}_0{p}_0-2g^{0k}{\mathrm{\partial }}_k{p}_0-g^{ij}{\mathrm{\partial }}_i{q}_j+L(p_0,q)$. The wave equation ${\mathrm{\square }}_g\chi =f$, together with ${\mathrm{\partial }}_0\chi =p_0$ and ${\mathrm{\partial }}_0{q}_i={\mathrm{\partial }}_i{p}_0$, is therefore equivalent to ${\mathrm{\partial }}_0\chi -p_0=0$, $-g^{00}{\mathrm{\partial }}_0{p}_0-2g^{0k}{\mathrm{\partial }}_k{p}_0-g^{ij}{\mathrm{\partial }}_i{q}_j+L(p_0,q)=-f$, and $g^{ij}({\mathrm{\partial }}_0{q}_j-{\mathrm{\partial }}_j{p}_0)=0$. In unknown order $(\chi ,p_0,q_1,q_2,q_3)$, the time matrix is $A^0=\mathrm{diag}(1,-g^{00},(g^{ij}))$. The $k$th spatial matrix has $p_0,p_0$ entry $-2g^{0k}$, $p_0,q_j$ entry $-g^{kj}$, $q_i,p_0$ entry $-g^{ik}$, and all other principal entries zero, so it is symmetric. By the chart conditions, $A^0$ is positive definite on a smaller compact cylinder.

Make a time-independent affine change of the spatial chart coordinates, if needed, so a relatively compact spatial box about $p$ identifies with a box $V_T⋐(-\pi ,\pi {)}^3$ in ${\mathbb{T}}_{2\pi }^3$. Transform the spatial principal matrices by the constant coordinate Jacobian; they remain symmetric. Choose an open time interval $I_0\ni 0$ and, by step 2.1, a relatively compact cylinder $C⋐I_0\times V_T$ on which the time matrix is positive definite. Take a smooth cutoff $\rho \in C_c^{\mathrm{\infty }}(C)$ with $0\le \rho \le 1$ and $\rho =1$ near $(0,p)$. Extend the cutoff products by zero to $I_0\times {\mathbb{T}}_{2\pi }^3$, set $A_{\mathrm{e}\mathrm{x}\mathrm{t}}^0=\rho A^0+(1-\rho )I$ on $C$ and $A_{\mathrm{e}\mathrm{x}\mathrm{t}}^0=I$ outside $C$, and set $A_{\mathrm{e}\mathrm{x}\mathrm{t}}^k=\rho A^k$. For the unknown order $(\chi ,p_0,q_1,q_2,q_3)$, the lower-order vector before cutoff is $B=(-p_0,L(p_0,q)+f,0,0,0)$; set $B_{\mathrm{e}\mathrm{x}\mathrm{t}}=\rho B$. Both $A^0$ and $I$ are positive definite wherever the convex combination is used, so $A_{\mathrm{e}\mathrm{x}\mathrm{t}}^0$ is positive definite everywhere; each $A_{\mathrm{e}\mathrm{x}\mathrm{t}}^k$ remains symmetric. Smoothness of the cutoff products and compactness of their support give a uniform positive lower bound for $A_{\mathrm{e}\mathrm{x}\mathrm{t}}^0$ on $I_0\times {\mathbb{T}}_{2\pi }^3$. Regard the unknown as ${\mathbb{R}}^5$-valued, use $O={\mathbb{R}}^5$, $s=5$, $K=\{0\}$, $K^{\mathrm{\prime }}=\{u:\mathrm{\mid }u\mathrm{\mid }\le 1\}$, $R=0$, and zero initial state. The extended coefficients form a smooth periodic family; by [F9] the periodic system has a solution on a common interval. By [F2] this solution is $C^1$ and solves the system pointwise near $(0,p)$. Its initial data are smooth, so [F10] makes that same solution smooth on the original common interval. On the intersection of this interval with the smaller cylinder where $\rho =1$, the coefficients and source agree with the unextended system, so restriction gives the required smooth local solution.

The invertible matrix $(g^{ij})$ in the last equations of step 2.1 gives ${\mathrm{\partial }}_0{q}_i={\mathrm{\partial }}_i{p}_0$. Since initially $q_i={\mathrm{\partial }}_i\chi =0$ and ${\mathrm{\partial }}_0\chi =p_0$, the difference $q_i-{\mathrm{\partial }}_i\chi$ remains zero. Thus the middle equation in step 2.1 is exactly ${\mathrm{\square }}_g\chi =f$, and step 1.2 gives ${\mathrm{\nabla }}^a{A}_a^{\mathrm{\prime }}=0$ near $p$.

If $F=d{A}^{\mathrm{\prime }}$ also obeys the Maxwell divergence equation, [F3] and the Lorenz condition give its Ricci-coupled potential wave equation; no Ricci-flat cancellation is asserted. The construction in 1.1 is confined to its star-shaped chart and the gauge argument in 4.1 is local near each point of $S$, so neither supplies a global potential or gauge. This completes the local construction and scope assertion.