A complete stationary boost band with a global local-isometry certificate

Fix $0<\delta <\eta$. The product circle action and the boost define $A_t$ for every $t\in \mathbb{R}$, preserve $UV$, and commute. Every $q\in \mathcal{B}$ has a parameter $t$ with $A_{-t}q\in W$. If $UV\mathrm{\ne }0$, the equation $e^{\kappa t}\mathrm{\mid }U\mathrm{\mid }=e^{-\kappa t}\mathrm{\mid }V\mathrm{\mid }$ has a real solution because $\kappa >0$, and both balanced absolute coordinates equal $\sqrt{\mathrm{\mid }UV\mathrm{\mid }}<\delta <\eta$. If exactly one coordinate is nonzero, choose the sign and size of $t$ to shrink it below $\eta$; if both vanish, every parameter works. Define $F(q)={\mathrm{\Phi }}_t(F_0(A_{-t}q))$ using any such parameter. This construction uses only the explicitly assumed local geometry, with the circle factor omitted in the nonrotating case.

The working parameters for each $q$ form an interval: each absolute normal coordinate is monotone in $t$, and the circle action keeps the $S$ factor inside $S$. For any two working parameters the entire intervening segment therefore stays in $W$. On that segment, $F_0$ carries the generator $K-\mathrm{\Omega }Z$ to $\widehat{T}$, so uniqueness of integral curves gives equality of the two definitions of $F(q)$. This also applies at $U=V=0$, where the full circle orbit stays in $W$. A fixed working parameter remains working near $q$ by openness; on this neighborhood $F={\mathrm{\Phi }}_t\circ F_0\circ A_{-t}$ is smooth and a local diffeomorphism. The parameter $t=0$ gives $F=F_0$ on $W\cap \mathcal{B}$. If $t$ works for $q$, then $t+s$ works for $A_{s}q$; the flow law yields $F(A_{s}q)={\mathrm{\Phi }}_s(F(q))$ for every real $s$.

Set $g_{\mathcal{B}}=F^{\ast }\widehat{g}$. The pullback evaluation formula and the invertibility of $dF$ give a smooth nondegenerate Lorentzian metric of the same signature as $\widehat{g}$, agreeing with $g_0$ on $W\cap \mathcal{B}$. Pull back the ambient time orientation through $dF$: the inverse image of any ambient smooth future timelike orientation field is smooth and timelike, and selects the future cone on the band. Thus $F$ is a time-orientation-preserving local isometry.

To verify the vacuum assertion locally, fix $q\in \mathcal{B}$ and restrict $F$ to a diffeomorphism from a neighborhood $O$ of $q$ onto a chart neighborhood $(Q,x)$ in $\widehat{M}$. Use $y=x\circ F$ as the coordinates on $O$. In these coordinates the coefficient functions of $g_{\mathcal{B}}$ and $\widehat{g}$, viewed on the same open subset of ${\mathbb{R}}^4$, are identical by the pullback formula. Their inverse matrices and all derivatives are therefore identical. Countable Choice from [A1] permits the Levi–Civita formula in [F6] for these smooth nondegenerate four-dimensional metrics. That formula gives identical Christoffel coefficient functions, and the curvature formula and Ricci contraction of [F6] give identical Ricci coefficient functions. The ambient ones vanish by hypothesis, hence $\mathrm{Ric}(g_{\mathcal{B}})=0$ on $O$. Since $q$ was arbitrary, the band is vacuum everywhere. This proves the needed local naturality directly in matching charts.

By [F5], ${\mathcal{L}}_{\widehat{T}}\widehat{g}=0$ implies ${\mathrm{\Phi }}_t^{\ast }\widehat{g}=\widehat{g}$ for every real $t$. Equivariance from step 2.1 and the pullback formula then give $A_t^{\ast }{g}_{\mathcal{B}}=g_{\mathcal{B}}$. In the rotating case write $R_s(U,V,p)=(U,V,{\varphi }_s^{Z}p)$. Take the same working parameter in neighborhoods of $q$ and $R_{s}q$, which is possible because $R_s$ changes neither normal coordinate. The expression $F={\mathrm{\Phi }}_t\circ F_0\circ A_{-t}$ and commutation of $R_s$ with $A_{-t}$ reduce $R_s^{\ast }{g}_{\mathcal{B}}=g_{\mathcal{B}}$ to the assumed $R_s^{\ast }{g}_0=g_0$. The stationary field is complete because $A_t$ is defined for all real $t$. The commuting isometric action $A_t{R}_{\mathrm{\Omega }t}$ is the pure boost flow, proving that $K$ too is complete and Killing by differentiation at zero in [F5]. In the nonrotating case the circle terms are omitted and $A_t$ itself is the complete pure boost flow.

Apply the finite-density supplier to $\widehat{M}$ (nonempty because it contains the product neighborhood), then the volume-time theorem and the smooth Cauchy temporal splitting theorem. Their hypotheses hold: $\widehat{M}$ is a time-oriented smooth globally hyperbolic Lorentzian four-manifold, and both choice assumptions are [A1]. This produces a smooth Cauchy temporal function $\widehat{\tau }$. For $\tau =\widehat{\tau }\circ F$, the local-isometry identity gives $g_{\mathcal{B}}^{-1}(d\tau ,d\tau )={\widehat{g}}^{-1}(d\widehat{\tau },d\widehat{\tau })\circ F<0$. The pulled-back time orientation gives $d\tau (v)>0$ for every nonzero future causal $v$. Thus $\tau$ is temporal on the whole band. Neither injectivity of $F$ nor a Cauchy property of its pulled-back levels is used.