Radu small-ADM global and bifurcation package

The Radu ADM GR/SBS package consists of a connected oriented time-oriented smooth vacuum four-spacetime $(M,g)$ with a smooth stationary Killing field $T$, and an embedded spacelike acausal edgeless partial Cauchy surface ${\mathrm{\Sigma }}^0$, supplied with a diffeomorphism ${\mathrm{\Phi }}_0:E_{1\mathrm{/}2}\to {\mathrm{\Sigma }}^0$ onto the whole surface, where $E_a=\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }>a\}$ and $r=\mathrm{\mid }x\mathrm{\mid }$. Put ${\mathrm{\Sigma }}_1={\mathrm{\Phi }}_0(E_1)$ and $S_0={\mathrm{\Phi }}_0(\{r=1\})$. The end chart extends to $M^{\mathrm{e}\mathrm{n}\mathrm{d}}\simeq \mathbb{R}\times E_{R_0}$ with $T={\mathrm{\partial }}_t$ and zero-time section agreeing with ${\mathrm{\Sigma }}^0$. Its coefficients are $g_{00}=-1+2M_0\mathrm{/}r+O_6(r^{-2}),g_{ij}={\delta }_{ij}+O_6(r^{-1}),g_{0i}=-2{ϵ}_{ijk}{S}^j{x}^k{r}^{-3}+O_6(r^{-3}),$ with $M_0>0$ the source ADM mass, $S=(S^1,S^2,S^3)$ its ADM angular momentum vector, and $J=\mathrm{\mid }S\mathrm{\mid }=\sqrt{{\sum }_{i=1}^3(S^i{)}^2}$ its nonnegative Euclidean magnitude. Here $i,j,k\in \{1,2,3\}$ and ${ϵ}_{ijk}$ is the alternating symbol with ${ϵ}_{123}=1$. The exterior $E=I_M^{-}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})\cap I_M^{+}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})$ is globally hyperbolic, ${\mathrm{\Sigma }}^0\cap I_M^{\pm }(M^{\mathrm{e}\mathrm{n}\mathrm{d}})={\mathrm{\Sigma }}_1$, and $T$ is nowhere zero on $E$, with every orbit complete and intersecting ${\mathrm{\Sigma }}_1$ transversally.

Precisely, for every $p\in E$, let ${\gamma }_p:\mathbb{R}\to M$ be its complete smooth stationary orbit, with ${\gamma }_p(0)=p$ and ${\dot{\gamma }}_p(t)=T_{{\gamma }_p(t)}$. The orbit-intersection requirement is ${\gamma }_p^{-1}({\mathrm{\Sigma }}_1)\mathrm{\ne }\mathrm{\varnothing }$, and transversality means that for every $t\in {\gamma }_p^{-1}({\mathrm{\Sigma }}_1)$, writing $q={\gamma }_p(t)$, one has $d{\gamma }_p{\mathrm{\mid }}_t(T_t\mathbb{R})+T_q{\mathrm{\Sigma }}_1=T_{q}M$.

Here a set $A\subset M$ is acausal when no nonconstant future causal curve has both endpoints in $A$, including coincident endpoints; it is achronal when $I_M^{+}(A)\cap A=\mathrm{\varnothing }$, which also excludes a closed timelike curve through any point of $A$. For an achronal set $A\subset M$, a point $q\in {\bar{A}}^M$ lies in $\mathrm{edge}(A)$ exactly when every open neighborhood $U$ of $q$ admits points $p\in I_U^{-}(q)$ and $r\in I_U^{+}(q)$ joined by a future timelike curve entirely in $U\setminus A$. The notation $I_U^{\pm }(q)$ uses curves contained in $U$ with the restricted metric and time orientation. Edgeless means $\mathrm{edge}(A)=\mathrm{\varnothing }$; a partial Cauchy hypersurface means an acausal edgeless set, additionally embedded, smooth and spacelike here. These causal predicates are relative to the ambient spacetime $M$.

Radu's ADM source (§1.1 GR) uses acausality to exclude causal curves joining distinct points of the hypersurface. This local package retains that source requirement and additionally excludes nonconstant future causal curves whose two endpoints coincide at a point of ${\mathrm{\Sigma }}^0$, as required by the Minguzzi convention just stated. The coincident-endpoint exclusion is an explicit additional local condition, not a consequence attributed here to Radu's GR. Thus the local condition supplies the source's distinct-point acausality; the converse and equivalence of conventions are not asserted.

Near $S_0$ the two event boundaries $H^{\pm }={\mathrm{\partial }}_M{I}_M^{\mp }(M^{\mathrm{e}\mathrm{n}\mathrm{d}})$ are smooth transverse nonexpanding null hypersurfaces, with $H^{+}\cap H^{-}=S_0$, $T$ tangent to both, and $T{\mathrm{\mid }}_{S_0}\equiv ̸0$. The ADM source thus requires a nonzero horizon field; Radu's On rigidity of stationary black holes under a nontrapping assumption (arXiv:2609.23872v1, §1.1 SBS and §1.2) permits $T{\mathrm{\mid }}_{S_0}\equiv 0$. Maximality, $J<M_0^2$ and the quantitative smallness assumptions are additional separately named conditions, not silently included in GR/SBS here.

A local smooth null hypersurface is a smooth embedded three-manifold without boundary whose induced tangent form is positive semidefinite of rank two. The two local horizon pieces are transverse when $T_q{H}^{+}+T_q{H}^{-}=T_{q}M$ for every $q\in S_0$. These conditions apply to the horizon pieces near $S_0$ specified above.

The supplied global diffeomorphism determines the slice subsets and embeds the specified coordinate sphere; its end restriction and stationary extension determine the end arrays. The ambient causal relations determine the exterior, event boundaries and neighborhood-restricted chronological relations used in the edge predicate. The imported definitions make acausality, achronality and edgelessness definite conditions, with the edge predicate used only on achronal sets. The declared complete stationary orbits supply smooth orbit maps, and the smooth-map transversality definition gives exactly the displayed tangent-space condition at each slice intersection; the existence of an intersection remains a separate required predicate. The null-hypersurface interface applies in the supplied smooth time-oriented Lorentzian four-spacetime with the choice convention inherited from the null-expansion interface. Its rank-two induced-form condition and the embedded-submanifold transversality interface make the specified local horizon predicates definite. Smoothness and spacelikeness make the expansion trace and the surface metric meaningful. The displayed list retains Radu's GR/SBS predicates together with the explicitly additional local coincident-endpoint exclusion. The local acausality condition implies the source's distinct-point exclusion by restricting its endpoint quantifier to distinct points; no converse or equivalence is asserted or used. No implication is asserted between this local package and the original article's differently stated regularity package, and the theorem's separate maximality and quantitative assumptions are not folded into a generic GR label.