Causal solutions, Killing horizons, escape and metric extensions

Let $(M,g,N)$ be a smooth Hausdorff second-countable boundaryless four-manifold with smooth Lorentz metric of signature $(-,+,+,+)$ and smooth time orientation. All definitions below are mathematical. A solution of a specified covariant field system is a tuple of fields of the regularity stated by that system satisfying its equations on its stated domain; this terminology alone supplies neither existence nor a global electromagnetic potential.

(a) Global causal and Cauchy conventions. Use exactly Causal curves, chronological and causal reachability, and inextendibility: a future causal curve is locally absolutely continuous in coordinate charts, has nonzero future causal derivative almost everywhere, and has no constant subinterval. Inextendibility and causal reachability $J^{\pm }$ have that item's endpoint-extension meanings. A Cauchy set meets every inextendible causal curve at exactly one parameter value: its inverse image under each such curve is a singleton, as in Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons. A spacelike hypersurface is a smooth embedded codimension-one submanifold with positive-definite induced metric; this local condition does not assert the global Cauchy property. Strong causality has the small-neighborhood no-leaving-and-reentering meaning of Chronology, causality, strong causality, global hyperbolicity and time functions. Global hyperbolicity means strong causality and compactness of every $J^{+}(p)\cap J^{-}(q)$, exactly as there; no equivalence theorem is added here.

(b) Null and Killing horizons. Here a hypersurface in $M$ is a smooth embedded three-dimensional submanifold, using the slice-chart convention recorded in Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons clause (a); that clause's positive-definiteness condition is not imposed unless “spacelike” is stated. A smooth embedded hypersurface $H\subset M$ is null when at every $p\in H$ the radical $\{v\in T_{p}H:g_p(v,X)=0\text{ for every }X\in T_{p}H\}$ is nonzero. Equivalently, the slice-chart definition supplies near each point a smooth local defining function $f$ with $H=\{f=0\}$ and $df\mathrm{\ne }0$ on $H$, and $g^{-1}(df,df)=0$ on $H$. A specified smooth vector field $\xi$ is Killing when ${\mathcal{L}}_{\xi }g=0$, equivalently ${\mathrm{\nabla }}_{(a}{\xi }_{b)}=0$, using Killing fields of a Lorentzian metric and Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor. A Killing horizon for $\xi$ is a null hypersurface whose nonzero-generator part $H_{\ast }:=\{p\in H:{\xi }_p\mathrm{\ne }0\}$ is nonempty and dense in $H$, and on that part the specified field is tangent, null and normal: ${\xi }_p\in T_{p}H$ and $g({\xi }_p,X)=0$ for every $X\in T_{p}H$. A generator parameter or asymptotic normalization, when required, must be specified separately.

On $H_{\ast }$ the signed surface gravity for this precise $\xi$ is the unique smooth scalar ${\kappa }_H$ such that

$d(g(\xi ,\xi ){)}_p=-2{\kappa }_H(p){\xi }_p^{\mathrm{♭}}(p\in H_{\ast })\mathrm{.}$

This is an equality of ambient covectors at points of $H$, not merely of their pullbacks to $TH$. No constancy or positivity is asserted. Under a nonzero constant rescaling ${\xi }^{\mathrm{\prime }}=a\xi$, ${\kappa }_H^{\mathrm{\prime }}=a{\kappa }_H$. A bifurcation surface for $\xi$ consists of a specified smooth spacelike embedded two-dimensional submanifold $B\subset M$ on which $\xi =0$, together with two distinct specified smooth embedded null hypersurfaces $H_{+}$ and $H_{-}$ in a common neighborhood of $B$, each a Killing horizon for $\xi$, such that near each $p\in B$ their intersection is $B$ and $T_p{H}_{+}\cap T_p{H}_{-}=T_{p}B$. Write $H_{\pm ,\ast }:=\{q\in H_{\pm }:{\xi }_q\mathrm{\ne }0\}$. At $p\in B$, a surface-gravity value is defined only if the two branch scalars on $H_{+,\ast }$ and $H_{-,\ast }$ each extend smoothly to a neighborhood of $p$ in their respective hypersurfaces and those extensions agree at $p$; the value is their common value. Such a value is unique whenever it exists, since each $H_{\pm ,\ast }$ is dense in its horizon branch. No bifurcation surface or such extension is asserted to exist, and the defining equation at $\xi =0$ alone determines no value.

(c) Sections, area and ergoregions. A horizon section is a specified smooth spacelike embedded two-dimensional submanifold $S$ of the horizon, with positive induced metric $h$. Its area density is ${\mu }_h=\sqrt{\det (h_{ij})}\mathrm{\mid }d{y}^{1}d{y}^2\mathrm{\mid }$ of Riemannian volume density. Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for intrinsic density integration, as in Riemannian volume of a compactly supported smooth density. Its possibly infinite area is the nonnegative extended value

${\mathrm{Area}}_g(S):=\sup \{{\int }_S\chi {\mu }_h:\chi \in C_c^{\mathrm{\infty }}(S),0\le \chi \le 1\}\in [0,+\mathrm{\infty }],$

with the extended order of The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined. On compact $S$ this is ${\int }_S{\mu }_h$; orientation is unnecessary. The density glues by The riemannian volume density is coordinate independent. A stationary field here is a specified Killing field timelike on a specified nonempty open region; no normalization is inferred from this adjective. Its ergoregion in the chosen model is $\{g(\xi ,\xi )>0\}$, its stationary-limit set is $\{g(\xi ,\xi )=0\}$, and its ergosurface is the relative boundary of its ergoregion. Every ergosurface point lies in the stationary-limit set; the converse requires a separate sign-change argument. Nullness, Killing-horizon status and escape-boundary status are separate claims; a coordinate root alone proves none of them.

(d) Selected-model escape. Fix a specified open spacetime model $U\subseteq M$ and specified smooth radius observable $r:U\to \mathbb{R}$. An admissible escape path from $p$ is a continuous map $\gamma :[0,T)\to U$, $0<T\le +\mathrm{\infty }$, with $\gamma (0)=p$, such that for every compact interval $[a,b]\subset [0,T)$ there is a finite subdivision $a=t_0<\cdots <t_m=b$ and, for each $j$, a spacetime chart $(V_j,x_j)$ with $\gamma ([t_{j-1},t_j])\subset V_j$ for which $x_j\circ \gamma$ is $C^1$ on $[t_{j-1},t_j]$; its derivative is required to be nonzero and future causal in the chart interior and in the one-sided sense at subdivision endpoints. Define

${\mathcal{E}}_{U,r}:=\{p\in U:\mathrm{\exists }\gamma \text{ admissible from }p\mathrm{\forall }R>0\mathrm{\exists }t\in [0,T):r(\gamma (t))>R\}\mathrm{.}$

The selected escape boundary is ${\mathrm{\partial }}_U{\mathcal{E}}_{U,r}$ in the relative topology of $U$. This explicitly selected piecewise-$C^1$ path class is contained in the global causal-curve class of (a); it does not replace that class in Cauchy definitions. The definition includes no completion of $U$ or extension of $M$.

(e) Approached metric extensions and curvature obstructions. A $C^2$ nondegenerate metric extension of an open model $(U,g)$ consists of a smooth Hausdorff second-countable four-manifold $\tilde{M}$, a $C^2$ Lorentz metric $\tilde{g}$ of the same signature, and a smooth open embedding $j:U\to \tilde{M}$ with $j^{\ast }\tilde{g}=g$. An approached added point is specified either by $p\in \tilde{M}\setminus j(U)$ and a sequence $x_n\in U$ with $j(x_n)\to p$, or by such a $p$ and a continuous curve $\sigma :[0,1)\to U$ with $j(\sigma (t))\to p$ as $t\uparrow 1$. An extension obstruction at specified approach data means no such extension and added point realize that same sequence or curve limit. A polynomial complete curvature contraction is a scalar formed from a finite sum of tensor products of copies of the Riemann curvature tensor and the inverse metric, with every tensor index contracted. If its absolute values are unbounded along the specified sequence (as $n\to \mathrm{\infty }$), or along the specified curve (as $t\uparrow 1$), then that approach obstructs a $C^2$ nondegenerate metric extension. This criterion concerns curvature without its covariant derivatives. A smooth metric extension is the same definition with smooth $\tilde{g}$. These definitions assert no field equation and no additional global extension property.

(f) Observers and spatial fields. Observers, smooth observer congruences and their integral curves, adapted charts and tetrads retain the distinct meanings of Oriented Lorentzian fields, observers and coordinate conventions. Relative to a future unit observer $n$, an electric or magnetic spatial covector is identified with a spatial vector only by raising with the positive rest metric $h=g+n^{\mathrm{♭}}\otimes n^{\mathrm{♭}}$ on $n^{\perp }$. Such spatial vectors $e,b$ are not tetrad basis vectors. No singular-source product or action is defined by this vocabulary.