Constant homothety preserves the original vacuum and maximal section

In the Christoffel formula, ${\widehat{g}}^{-1}=m^2{g}^{-1}$ and $\mathrm{\partial }\widehat{g}=m^{-2}\mathrm{\partial }g$ have cancelling factors because $m$ is constant. Thus $\widehat{\mathrm{\nabla }}=\mathrm{\nabla }$. The curvature operator is built from this connection alone, and its Ricci tensor is the trace of $z\mapsto R(z,w)v$, so both are unchanged. This proves the Ricci equality and the stated equivalence of Ricci-flatness.

For any smooth field $X$ with local flow ${\varphi }^X$, differentiating $({\varphi }_s^X{)}^{\ast }\widehat{g}=m^{-2}({\varphi }_s^X{)}^{\ast }g$ at zero gives ${\mathcal{L}}_X\widehat{g}=m^{-2}{\mathcal{L}}_{X}g$. Thus $X$ is Killing for one metric exactly when it is Killing for the other. The map $\widehat{\varphi }(s,p)={\varphi }^T(ms,p)$ on $\{(s,p):(ms,p)\in \mathrm{dom}{\varphi }^T\}$ is smooth, starts at $p$, has derivative $mT$ and satisfies the flow law. Replacing $m$ by $m^{-1}$ reverses this correspondence.

For every tangent vector $v$, $\widehat{g}(v,v)=m^{-2}g(v,v)$ and $\widehat{g}(v,N)=m^{-2}g(v,N)$. Both signs are unchanged. The chart regularity and almost-everywhere nonzero-velocity conditions are metric independent, so exactly the same parametrized causal and timelike curves, with the same directions, are allowed in each fixed $D$. Endpoint limits use the unchanged topology, so inextendibility is identical. The reachability sets are therefore identical, and each Cauchy-set predicate counts the same parameter values over exactly the same inextendible curves.

If $\gamma :I\to M$ is a maximal integral curve of $T$, then $s\mapsto \gamma (ms)$ on $m^{-1}I$ is an integral curve of $mT$. Any extension would, by the inverse reparameterization, extend $\gamma$, so it is maximal as well. Parameter multiplication by $m$ is a bijection of these intervals and of $\mathbb{R}$; hence the orbit images agree and every maximal interval is $\mathbb{R}$ for one field exactly when it is for the other. This proves the conditional completeness claim.

The vector $mn$ is future-directed, is orthogonal to $T\mathrm{\Sigma }$, and satisfies $\widehat{g}(mn,mn)=-1$, so it is the hatted future unit normal. Restriction gives $\widehat{h}=m^{-2}h$. With the common connection and constant $m$, $\widehat{k}(U,V)=-m^{-2}g({\mathrm{\nabla }}_U(mn),V)=m^{-1}k(U,V)$. Contracting with ${\widehat{h}}^{-1}=m^2{h}^{-1}$ gives ${\mathrm{tr}}_{\widehat{h}}\widehat{k}=m{\mathrm{tr}}_{h}k$, which preserves zero trace on this same section.

For supplied bifurcation data, the induced forms on $T{H}_{\pm }$ are multiplied by $m^{-2}$, so their radicals and nullness are unchanged. Spacelikeness of $S$ is preserved by the positive factor; the branch intersections, transversality and sphere topology use the unchanged embeddings. If the additional regular-null-horizon conditions hold, step 2.1 preserves the local causal futures and pasts of $S$ and achronality in $O$. Their boundaries are unchanged because the topology is fixed. The common connection preserves the generating null geodesics, and $\widehat{g}(L,V)=m^{-2}g(L,V)$ preserves their orthogonality to $S$. Thus the same pair and sphere retain all the specified conditions, without invoking any Killing generator.

In four dimensions the oriented volume form scales by $\sqrt{\mathrm{\mid }\det (m^{-2}g)\mathrm{\mid }}\mathrm{/}\sqrt{\mathrm{\mid }\det g\mathrm{\mid }}=m^{-4}$. The inverse-metric pairing on two-forms scales by $m^4$. Therefore for all two-forms $\alpha ,\beta$ the defining identity gives $\alpha \wedge {\star }_{\widehat{g}}\beta =⟨\alpha ,\beta {⟩}_g{\mathrm{v}\mathrm{o}\mathrm{l}}_g=\alpha \wedge {\star }_g\beta$. Uniqueness of the Hodge map in that identity proves ${\star }_{\widehat{g}}={\star }_g$, completing all claims.