Physical expectation of late-time stationarity

PDF p. 1: §1.1, physical motivation

As physical motivation, expect an isolated dynamical black hole to approach a stationary state at late times through persistent emission of gravitational radiation. If such a limiting state is a regular, four-dimensional, asymptotically flat stationary vacuum black hole and the black-hole uniqueness conjecture holds for that state, its domain of outer communications is isometric to a Kerr exterior. In that conditional sense the conjecture would characterize possible asymptotic states of general vacuum evolution.

Here a stationary state has a smooth Lorentzian metric $g$ and a Killing field $T$, meaning ${\mathcal{L}}_{T}g=0$, which is future timelike near the designated distant end. Vacuum means $\mathrm{Ric}(g)=0$. For a designated end $U$, the domain of outer communications is $I^{+}(U)\cap I^{-}(U)$: the points that can both be reached from and send a future-directed timelike signal to $U$. An isometry is a diffeomorphism that pulls one metric back to the other. Kerr exteriors here are those of the supplied Kerr metric with $M>0$, $\mathrm{\mid }a\mathrm{\mid }\le M$ and $r>r_{+}=M+\sqrt{M^2-a^2}$, including the extremal case $\mathrm{\mid }a\mathrm{\mid }=M$; for example, the supplied subextremal family has $M>0$, $\mathrm{\mid }a\mathrm{\mid }<M$ and exterior coordinate range $r>M+\sqrt{M^2-a^2}$. The expectation itself is not restricted to that subfamily.

For a precise instance of the regularity qualification in the conditional claim, take a specified asymptotically flat spacelike end $S_{\mathrm{e}\mathrm{x}\mathrm{t}}$: in its end chart, the induced metric $\gamma$ and second fundamental form $k$ satisfy ${\gamma }_{ij}-{\delta }_{ij}=O_m(r^{-\alpha })$ and $k_{ij}=O_{m-1}(r^{-1-\alpha })$ for some $\alpha >0$ and integer $m\ge 2$, with the derivative-decay convention of the supplied end interface. Require $T$ to be complete and future timelike on a sufficiently distant tail, and let $U={\bigcup }_{t\in \mathbb{R}}{\mathrm{\Phi }}_t(S_{\mathrm{e}\mathrm{x}\mathrm{t}})$ be the flow saturation of that tail. All chronological sets are computed in the ambient spacetime $\mathcal{M}$. The black-hole region relative to this end is $\mathcal{B}=\mathcal{M}\setminus I_{\mathcal{M}}^{-}(U)$, required to be nonempty; its event horizon is ${\mathrm{\partial }}_{\mathcal{M}}\mathcal{B}$. In this precise instance, regularity means the supplied Chruściel–Costa $I^{+}$-regularity predicate: completeness of $T$, global hyperbolicity of the domain of outer communications, and the specified acausal spacelike section with finite ends and a boundary meeting each future exterior-boundary generator once. This is a particular mathematical formulation of regularity, not an identification with every use of the word regular in the article's qualitative motivation.

The conditional uniqueness assumption for this formulation is explicit: every smooth four-dimensional stationary vacuum spacetime with the preceding end, nonempty black-hole region and $I^{+}$-regularity has its domain of outer communications isometric to a Kerr exterior with $M>0$ and $\mathrm{\mid }a\mathrm{\mid }\le M$. This implication is assumed here, not established. Its application to a limiting state requires that state to satisfy all these hypotheses. The general late-time expectation itself is not restricted to this particular regularity formulation.

For an end admitting the separately hypothesized smooth vacuum Bondi expansion, $u$ is retarded time, $C_{AB}$ is the Bondi shear and $N_{AB}={\mathrm{\partial }}_u{C}_{AB}$ is the Bondi news, a standard asymptotic radiation datum. The expectation does not assert that every evolution has such an expansion or specify a condition on $N_{AB}$.

Stationary Lorentzian data and tensor conventions

Causal exterior, chronological boundaries and Cauchy data

A smooth vacuum Bondi expansion

Subextremal Kerr exterior in length coordinates

Asymptotically flat spacelike end and derivative decay

Complete stationary flow and its exterior region

Staticity, axisymmetry and black-hole regularity hypotheses

Kerr metric, parameters, exterior and extremality

The source gives a qualitative expectation and a conditional motivation, not a precise class of initial data, a convergence topology, a decay rate, or a proof that radiation produces stationarity. The phrase persistent emission is retained as the source's physical motivation; no quantitative flux law or assertion of nonzero radiation at every finite time is imposed here. The uniqueness conjecture and the applicability of its regularity hypotheses to a proposed limit are separate conditions. No measurement or empirical confirmation is claimed.

The black-hole-region and event-horizon convention above is the definition immediately after equation (2.9) in Chruściel–Costa section 2.2, PDF page 7. It supplies terminology only; the uniqueness implication remains an explicit assumption and the late-time expectation remains a postulate.