Causal exterior, chronological boundaries and Cauchy data

Let $(D,g,N)$ be a time-oriented smooth Lorentzian four-manifold of signature $(-,+,+,+)$, and let $U\subset D$ be a specified open subset. Here a Killing field is a smooth vector field with ${\mathcal{L}}_{T}g=0$; future timelike means $g(T,T)<0$ and $g(T,N)<0$.

For the stationary-end terminology, supply an end chart: a spacelike embedded hypersurface $S_{\mathrm{e}\mathrm{x}\mathrm{t}}\subset D$ without manifold boundary and a specified diffeomorphism $e:\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }>R\}\to S_{\mathrm{e}\mathrm{x}\mathrm{t}}$ for some $R>0$. This is the end-chart data of the spacelike-end interface, with its finite-radius boundary removed. Also supply a local flow $\mathrm{\Phi }:\mathrm{\Omega }\to D$ of a specified Killing field $T$, with $\mathrm{\Omega }\subset \mathbb{R}\times D$ open and containing $\{0\}\times D$, in the local-flow interface's sense. Require $U=\{\mathrm{\Phi }(t,p):p\in S_{\mathrm{e}\mathrm{x}\mathrm{t}},(t,p)\in \mathrm{\Omega }\}$ and $T$ future timelike on $U$; then call $U$, with these supplied data, a stationary end. The chart supplies the spatial end topology. Metric decay, completeness of the flow and a global product identification of $U$ are additional hypotheses when needed. The causal exterior and boundary constructions below apply to any specified open $U$, independently of whether this end structure is supplied.

For $p,q\in D$, write $p{\ll }_{D}q$ when a future-directed timelike curve in $D$ joins $p$ to $q$. Curves are locally absolutely continuous, with nonzero future-directed timelike velocity almost everywhere. Define $I_D^{+}(U)=\{q\in D:\mathrm{\exists }p\in U,p{\ll }_{D}q\}$ and $I_D^{-}(U)=\{p\in D:\mathrm{\exists }q\in U,p{\ll }_{D}q\}$. The causal exterior relative to $(D,g,U)$ is $\mathcal{E}=I_D^{-}(U)\cap I_D^{+}(U)$. Its future and past event boundaries relative to $U$ are respectively ${\mathrm{\partial }}_D{I}_D^{-}(U)$ and ${\mathrm{\partial }}_D{I}_D^{+}(U)$, where ${\mathrm{\partial }}_{D}A={\bar{A}}^D\setminus {\mathrm{int}}_{D}A$. Closure and interior are taken in the manifold topology of $D$, with their topological-interface meanings. These boundaries are taken in $D$; their intersection and their relation to ${\mathrm{\partial }}_D\mathcal{E}$ are not prescribed.

A future causal curve has locally absolutely continuous coordinates and nonzero future causal velocity almost everywhere. It is inextendible in $D$ when it converges to no point of $D$ at either parameter end. A subset $S\subset D$ is a Cauchy set in $D$ when every such inextendible causal curve has exactly one parameter value in $S$. A Cauchy hypersurface is a Cauchy set that is also an embedded three-dimensional submanifold with positive-definite induced metric. These conditions are relative to $D$; for a different open domain, inextendibility and the Cauchy condition are recomputed there.

Write $J_D^{+}(p)$ for $p$ together with points reached from $p$ by future causal curves, and $J_D^{-}(q)$ for the corresponding causal past. Strong causality means that each point has arbitrarily small open neighborhoods that no future causal curve can leave and subsequently re-enter. Global hyperbolicity means strong causality and compactness in $D$ of every $J_D^{+}(p)\cap J_D^{-}(q)$. A temporal function is a smooth real function $\tau$ whose metric gradient is everywhere past-directed timelike, equivalently $g(\mathrm{\nabla }\tau ,\mathrm{\nabla }\tau )<0$ and $d\tau (N)>0$.

For this interface, $(D,g,N)$ is stably causal when there is a smooth Lorentzian metric $\widehat{g}$ on $D$, with $N$ timelike for $\widehat{g}$, such that every nonzero $g$-causal vector is $\widehat{g}$-timelike and $(D,\widehat{g},N)$ has no closed future causal curve. This is a separate condition from global hyperbolicity and from the existence of a temporal function; no implication between them is asserted here. In particular, no causal property of a different or deformed metric follows merely from a property of $g$.

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Killing fields of a Lorentzian metric

Causal curves, chronological and causal reachability, and inextendibility

Chronology, causality, strong causality, global hyperbolicity and time functions

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

Asymptotically flat spacelike end and derivative decay

Local and global flows generated by a vector field

Interior, closure, boundary, exterior, derived set and isolated point in a topological space

Given: The data $(D,g,N,U)$ in the Definition, with the end chart, $T$ and its local flow supplied when the stationary-end terminology is used.

Goal: Check that the defined sets and predicates are unambiguous for those data.

1.1 The specified end chart determines $S_{\mathrm{e}\mathrm{x}\mathrm{t}}$ and its spatial exterior coordinates. The supplied local flow defines $\mathrm{\Phi }(t,p)$ at exactly the pairs used in the displayed set; equality of that set with $U$ and the timelikeness inequalities are therefore definite predicates on the supplied data. This construction uses only the chart part of the spacelike-end interface and assumes no metric decay or flow completeness. [given, F1, F5]

1.2 The time orientation and metric determine the permitted curve classes and the relations ${\ll }_D$ and $J_D^{\pm }$; the displayed existential conditions therefore determine subsets of $D$. Intersection and the closure-minus-interior operation in the specified topology then determine $\mathcal{E}$ and both named boundaries, including when any of these sets is empty. [F1, F2, F6]

2.1 The Cauchy predicate specifies exactly which curves and parameter values are counted, while the hypersurface, strong-causality, diamond-compactness and temporal predicates use the stated interfaces. The stable-causality clause explicitly quantifies over a second metric and its causal curves, so it too is a definite predicate; it assumes neither the existence of such a metric nor any implication for another metric. [F1, F2, F3, F4, step 1.1, step 1.2] ∎

Minguzzi's Definition 2.16 supplies the endpoint convention for inextendibility; section 4.4 and Definition 4.89 supply the wider-cone definition of stable causality. The fixed orientation field $N$ selects the future convention used here for both metrics.

Gourgoulhon's section 3.2.1 supplies the conventional once-only intersection wording for a spacelike Cauchy surface. The exact parameter-counting convention used here is supplied by the declared Cauchy-set interface.