Trapping and the asymptotic black-hole region

Let $(M,g,N)$ be a smooth time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$. Let $S\subset M$ be a nonempty smooth embedded two-manifold, compact without boundary, whose induced metric $q$ is positive definite. At each point its normal plane $(T_{p}S{)}^{\perp }$ is a Lorentzian two-plane with two future null rays. On any open patch carrying a smooth ordered pair of future null normals $\mathrm{\ell },k$ with $g(\mathrm{\ell },k)=-2$, define ${\chi }_{\mathrm{\ell }}(X,Y)=g({\mathrm{\nabla }}_X\mathrm{\ell },Y),{\theta }_{\mathrm{\ell }}={\mathrm{tr}}_q{\chi }_{\mathrm{\ell }},{\chi }_k(X,Y)=g({\mathrm{\nabla }}_{X}k,Y),{\theta }_k={\mathrm{tr}}_q{\chi }_k$ for tangent vectors $X,Y$ to $S$. These are the null second fundamental forms and their expansions. A future trapped surface is such an $S$ for which, on every local patch and for both future null normal rays, the corresponding expansion is strictly negative at every point. Positive rescaling of a null normal multiplies its expansion by the same positive factor, so the sign condition is independent of its affine normalization and of a local interchange of the two rays.

When a specified conformal completion is supplied, write it as $(\bar{M},\bar{g},\bar{N},\mathrm{\Omega },j,{\mathcal{I}}^{+})$. Here $\bar{M}$ is a smooth four-dimensional manifold with boundary in the senses of Topological manifolds with boundary and Smooth charts, atlases, and structures with boundary. At each point $\bar{g}$ is a symmetric bilinear form on the tangent space, its components extend smoothly in every boundary chart, and its pointwise inertia is $(3,1,0)$ in the convention of Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space. The vector field $\bar{N}$ likewise has components that extend smoothly in every boundary chart; it satisfies $\bar{g}(\bar{N},\bar{N})<0$ everywhere and $\bar{g}(\bar{N},j_{\ast }N)<0$ on the interior. The map $j:M\to {\bar{M}}^{\circ }$ is a diffeomorphism onto the interior. The function $\mathrm{\Omega }:\bar{M}\to \mathbb{R}$ is smooth in the boundary-chart sense, positive on $j(M)$, and zero with nonzero differential on the designated smooth three-dimensional boundary subset ${\mathcal{I}}^{+}\subseteq \mathrm{\partial }\bar{M}$. On $j(M)$ the metric satisfies $\bar{g}{\mathrm{\mid }}_{j(M)}={\mathrm{\Omega }}^2(j^{-1}{)}^{\ast }g$, where pullback is as in The pullback of a covariant tensor field. The restriction of $\bar{g}$ to $T{\mathcal{I}}^{+}$ has a one-dimensional kernel at every point; this null line is the generator direction, and its future ray is the one whose vectors $v$ satisfy $\bar{g}(v,\bar{N})<0$. Thus the designated boundary is null and future-oriented; no completeness of its generators is assumed.

A curve $\gamma :[0,1]\to \bar{M}$ is locally absolutely continuous when it is continuous and, around each parameter value, its coordinate functions in a boundary or interior chart are absolutely continuous on every compact subinterval of that chart neighborhood, in the sense of Absolute continuity on a compact interval. Boundary charts take values in relatively open subsets of the closed half-space. To compare two charts around a parameter value, shrink its parameter neighborhood so the coordinate image lies in a compact convex subset of a local-extension domain of the transition map. Its derivative is bounded there, so the mean-value estimate (The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$) makes the transition Lipschitz. Applying that bound to the finite disjoint-interval condition for absolute continuity shows that the transformed coordinate functions remain absolutely continuous; the argument in both directions proves chart independence. At an interior parameter value where the coordinate derivatives exist, $\dot{\gamma }(t)$ is the tangent vector with those components; the chain rule for the chart transitions (Chain rule for smooth half-space maps) makes it independent of chart. A future causal curve in $\bar{M}$ is a locally absolutely continuous curve for which, for almost every $t\in (0,1)$, this derivative exists and satisfies $\bar{g}(\dot{\gamma },\dot{\gamma })\le 0$ and $\bar{g}(\dot{\gamma },\bar{N})<0$. These inequalities make the derivative nonzero and select the future direction. On the interior, $\bar{g}(j_{\ast }v,j_{\ast }v)={\mathrm{\Omega }}^{2}g(v,v)$ for every tangent vector $v$, so causal character is unchanged, and the specified orientation condition makes the future cone agree with that of $N$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model). Define ${\bar{J}}_{\bar{g},\bar{N}}^{-}({\mathcal{I}}^{+}):=\{x\in \bar{M}:\text{there are }y\in {\mathcal{I}}^{+}\text{ and a future causal curve in }\bar{M}\text{ from }x\text{ to }y\}$. The black region relative to this completion and its event horizon are, respectively, ${\mathcal{B}}_{{\mathcal{I}}^{+}}:=\{p\in M:j(p)\mathrm{\notin }{\bar{J}}_{\bar{g},\bar{N}}^{-}({\mathcal{I}}^{+})\},{\mathcal{H}}_{{\mathcal{I}}^{+}}:={\mathrm{\partial }}_M{\mathcal{B}}_{{\mathcal{I}}^{+}}\mathrm{.}$ If no such completion is supplied, these two sets are undefined; their existence is not a consequence of the definition of a trapped surface.

Given in addition a designated smooth future null normal ray field $[{\mathrm{\ell }}_{\mathrm{o}\mathrm{u}\mathrm{t}}]$ on $S$, called the outward direction, $S$ is marginally outer trapped relative to this choice when ${\theta }_{{\mathrm{\ell }}_{\mathrm{o}\mathrm{u}\mathrm{t}}}=0$ everywhere. If the outward label comes from a side of an embedded spacelike hypersurface or from an asymptotic end, that side or end is part of the supplied data; it is not inferred from $S$ alone. The zero condition is invariant under positive rescaling of the chosen ray.

Null normal pairs and screen geometry

Causal curves, chronological and causal reachability, and inextendibility

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

Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space

Absolute continuity on a compact interval

$C^r$ and smooth maps between smooth manifolds

The differential of a smooth map

The pullback of a covariant tensor field

Derivations at a point and the tangent space

Diffeomorphisms and local diffeomorphisms of manifolds

Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length

Chain rule for smooth half-space maps

The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$

Smooth functions on relatively open half-space sets

Topological manifolds with boundary

Smooth charts, atlases, and structures with boundary

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

The screen geometry supplier fixes the null second fundamental form and expansion and proves their boost law: under $\mathrm{\ell }\mapsto e^f\mathrm{\ell }$, one has ${\theta }_{\mathrm{\ell }}\mapsto e^f{\theta }_{\mathrm{\ell }}$. Thus the two inequalities in the trapped-surface definition are local, branch-independent conditions. The event horizon is a global boundary in a specified completion; it is not identified with a trapped surface or a marginally outer trapped surface. In the SI geometric convention $x^0=ct$ with dimensionless metric and affine parameter in metres, the expansion has units ${\mathrm{m}}^{-1}$, as in the screen geometry supplier.