AIK perturbative global regularity and bifurcation package

The AIK perturbative GR/SBS package means the following specified data and conditions of Alexakis–Ionescu–Klainerman (AIK), arXiv:0904.0982v1 §1.1. Work with a connected oriented time-oriented smooth paracompact boundaryless four-spacetime $(M,g)$, $\mathrm{Ric}(g)=0$, a smooth Killing field $T$, and an embedded spacelike partial Cauchy surface ${\mathrm{\Sigma }}^0$ (acausal and edgeless) parametrized by ${\mathrm{\Phi }}_0:E_{1\mathrm{/}2}\to {\mathrm{\Sigma }}^0$, where $E_a=\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }>a\}$.

Here acausal has the causal-set interface's meaning: no two distinct points of the surface are joined by a causal curve in $M$. For an achronal set $A\subset M$, its causal edge consists of those $q\in {\bar{A}}^M$ for which every open neighborhood $U$ of $q$ admits $a\in I_U^{-}(q)$, $b\in I_U^{+}(q)$ and a future timelike curve from $a$ to $b$ contained in $U\setminus A$. The sets $I_U^{\pm }(q)$ use the restricted spacetime $U$. Edgeless means that this edge is empty. The partial Cauchy condition on ${\mathrm{\Sigma }}^0$ requires acausality and edgelessness in the ambient $M$; neither follows here merely from spacelikeness.

GR supplies an open end $M^{\mathrm{e}\mathrm{n}\mathrm{d}}$ with a chart ${\mathrm{\Phi }}_0:\mathbb{R}\times E_{R_0}\to M^{\mathrm{e}\mathrm{n}\mathrm{d}}$ extending the slice chart, $T={\mathrm{\partial }}_t$, and $g_{00}=-1+2M_0\mathrm{/}r+O_6(r^{-2}),g_{ij}={\delta }_{ij}+O_6(r^{-1}),g_{0i}=-2{ϵ}_{ijk}{S}^j{x}^k{r}^{-3}+O_6(r^{-3}),$ where $M_0>0$, $r=\mathrm{\mid }x\mathrm{\mid }$, ${ϵ}_{123}=1$, and $J=\mathrm{\mid }S\mathrm{\mid }\in [0,M_0^2)$. All index sums are over their displayed ranges; the differentiated convention is the source one (including time derivatives, which vanish on the stationary end). Put $E=I_M^{-}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})\cap I_M^{+}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})$. Require $E$ globally hyperbolic, ${\mathrm{\Sigma }}^0\cap I_M^{\pm }(M^{\mathrm{e}\mathrm{n}\mathrm{d}})={\mathrm{\Phi }}_0(E_1)$, $T$ nowhere zero on $E$, and every exterior $T$-orbit complete and meeting ${\mathrm{\Sigma }}^0$.

SBS requires $S_0={\mathrm{\Phi }}_0(\{r=1\})$ to be the bifurcation sphere of the two event boundaries. In a neighborhood of $S_0$, $H^{+}={\mathrm{\partial }}_M{I}_M^{-}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})$ and $H^{-}={\mathrm{\partial }}_M{I}_M^{+}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})$ are smooth embedded null nonexpanding hypersurfaces, transverse with intersection $S_0$. Require $T$ tangent to both and $T{\mathrm{\mid }}_{S_0}\equiv ̸0$. Nonexpanding means zero null second-fundamental-form trace. This perturbative package does not add maximality of ${\mathrm{\Sigma }}^0$; maximality belongs to other source contracts below.

For either local sheet $H$, null means that $g{\mathrm{\mid }}_{TH}$ has signature $(0,+,+)$ and radical ${\mathcal{R}}_p=\{v\in T_{p}H:g(v,w)=0\text{ for all }w\in T_{p}H\}$. A local generator field $L$ is a smooth nonzero future-directed section of this null normal line. The screen is ${\mathcal{S}}_p=T_{p}H\mathrm{/}{\mathcal{R}}_p$, with positive metric $\bar{q}([X],[Y])=g(X,Y)$. The null second fundamental form on the screen and its expansion are ${\bar{\chi }}_L([X],[Y])=g({\mathrm{\nabla }}_{X}L,Y)$ and ${\theta }_L={\mathrm{tr}}_{\bar{q}}{\bar{\chi }}_L$. On a transverse spacelike two-section $Q\subset H$, this is ${\mathrm{tr}}_{g{\mathrm{\mid }}_{TQ}}{\chi }_L$ with ${\chi }_L(X,Y)=g({\mathrm{\nabla }}_{X}L,Y)$. Nonexpansion in SBS requires ${\theta }_L=0$ everywhere on each local sheet, including $S_0$, for every such local generator field. The trace is on the positive screen or section metric; the degenerate metric on $TH$ is not inverted.

Stationary Lorentzian data and tensor conventions

Causal exterior, chronological boundaries and Cauchy data

Spacelike sections, maximality and lapse–shift data

Null expansions, screen quotients and focal data

Differentiated asymptotic bounds and parameter uniformity

Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons

Regular bifurcate null horizon data before rigidity

AIK, section 1.1, GR and SBS

Minguzzi, Definitions 2.130 and 3.35

Gourgoulhon–Jaramillo, local null-hypersurface geometry

Each clause is a predicate on the supplied metric, field, embeddings and charts. The end coordinate derivatives and the exterior chronological sets are computed in their explicitly specified domains. The surface parametrization fixes the particular sphere $S_0$, and the separate horizon conditions test its event boundaries; they do not assert that a bifurcate Killing generator has already been constructed. No implication from a weaker regularity definition to this package is asserted. In particular the explicit angular leading term, strict subextremal parameter bound and nonzero-horizon-field clause remain part of this source contract.

The acausality interface and the imported causal-edge definition make the partial Cauchy requirement precise, with ambient closure retained. The cited null-hypersurface construction supplies the well-defined form on the positive screen and its section trace; its rescaling rule makes zero expansion independent of positive generator normalization. Thus the section formula agrees with the supplied null-screen convention without treating a two-surface definition alone as a definition on an entire hypersurface. These are geometric definitions and imported well-definedness facts, not a physical field law or a locally reconstructed horizon theorem.

AIK, section 1.1 supplies the GR/SBS assumptions transcribed above, with footnotes 3, 5 and 6 fixing the manifold, differentiated-decay and nonexpansion conventions. This item records those assumptions only; the PK hypothesis and global rigidity theorem are separate inputs.

Minguzzi, Definitions 2.130 and 3.35 supplies the causal-edge and partial Cauchy definitions. In particular the edge test ranges over the ambient closure, and all curves in a neighborhood test remain in that neighborhood. No theorem that a supplied surface satisfies this test is asserted.

Gourgoulhon–Jaramillo, sections 2.2, 2.6–2.7, 5.6 and 5.8 supplies the local null-hypersurface construction and its well-definedness facts. The form annihilates the radical, the trace can be computed in any transverse spacelike section, and positive normal rescaling preserves its zero set. These geometric facts apply to the smooth local sheets in SBS without field equations or a global section. The full locators, hypotheses and reading limits are in the third source reference.