Asymptotic Simplicity and Black-Hole-Compatible Variants

Asymptotic simplicity is a global conformal condition: every inextendible physical null geodesic must begin and end on the conformal boundary. That elegant requirement gives a complete view of radiation but excludes black holes, because captured null geodesics do not reach future null infinity. Black-hole theory therefore uses weaker asymptotic conditions near $\mathscr I$, supplemented by completeness and predictability assumptions that control the exterior without demanding a nonsingular interior. The distinctions are logical, not merely terminological.

Required background. The Riemann Curvature Tensor supplies the Ricci and Weyl conditions used in conformal completions.

Helpful background. Pseudo-Riemannian Metrics and Signature fixes null directions; Affine and Variational Geodesics supplies inextendibility and completeness; Causality and Global Structure provides causal pasts and domains of dependence; Formulations of General Relativity supplies Cauchy evolution; and Exact Solutions of Einstein’s Equations provides Minkowski, Schwarzschild, and Kerr examples.

Penrose asymptotic simplicity

Let $(\widehat M,\widehat g_{ab})$ be a smooth, time- and space-orientable physical spacetime. It is asymptotically simple if there are a smooth Lorentzian manifold-with-boundary $(M,g_{ab})$, a smooth function $\Omega$, and an embedding identifying $\widehat M$ with the interior of $M$ such that

$$g_{ab}=\Omega^2\widehat g_{ab} \quad\text{on }\widehat M, \qquad \Omega=0, \quad d\Omega\ne0 \quad\text{on }\mathscr I:=\partial M,$$

and every maximally extended null geodesic of $(\widehat M,\widehat g_{ab})$ has both a past and a future endpoint on $\mathscr I$. If, in addition, $\widehat R_{ab}=0$ in a neighborhood of $\mathscr I$, the spacetime is called asymptotically simple and empty; Frauendiener uses “asymptotically flat” for this strengthened class Frauendiener 2004, §2.3, pp. 15–16. Matter versions replace exact vacuum by a sufficiently strong, explicitly differentiable stress-energy decay.

The clauses do different jobs. Smooth extension controls regularity. The simple zero of $\Omega$ makes the boundary a genuine hypersurface. The endpoint condition is global: it says that the attached boundary captures every null direction of the entire physical spacetime, not only rays already known to escape. In the asymptotically empty case the field equations make $\mathscr I$ null and, under the standard global hypotheses, give components with topology $S^2\times\mathbb R$.

Because physical and unphysical null curves agree as unparametrized curves, the endpoint clause is conformally natural. Their affine parameters do not agree. Near a simple conformal zero, a null geodesic reaching $\mathscr I$ at finite unphysical parameter has infinite physical affine length. Thus asymptotic simplicity implies physical null completeness under the stated smooth hypotheses, but it says neither timelike nor spacelike geodesic completeness.

Asymptotic simplicity by itself also does not mean asymptotic flatness. De Sitter and anti-de Sitter spacetimes possess regular conformal completions of the appropriate broad type but have nonzero cosmological Ricci curvature near their boundaries; their boundaries are not the null infinity of the vacuum, zero-cosmological-constant class.

Why full simplicity excludes a black hole

Assume a specified future null infinity $\mathscr I^+$. Its black-hole region is

$$\mathcal B:=\widehat M\setminus J^-(\mathscr I^+),$$

and the future event horizon is the boundary

$$\mathcal H^+:=\partial_{\widehat M}J^-(\mathscr I^+) =\partial_{\widehat M}\mathcal B.$$

If $p\in\mathcal B$, no future-directed causal curve from $p$ reaches $\mathscr I^+$. In particular, a future-inextendible null geodesic from $p$ cannot acquire its future endpoint there. A nonempty black-hole region is therefore incompatible with the endpoint clause of asymptotic simplicity.

Schwarzschild exhibits two failures directly. Null rays inside the event horizon terminate at the curvature singularity rather than at $\mathscr I^+$. The maximally extended spacetime also contains trapped or limiting null behavior associated with the photon sphere, so not every inextendible null geodesic runs from $\mathscr I^-$ to $\mathscr I^+$. Yet a neighborhood of each asymptotic end has a smooth vacuum null completion. Frauendiener uses Schwarzschild to motivate the weak definition precisely at this point Frauendiener 2004, §2.3, pp. 16–17.

This exclusion is a virtue when proving global scattering results on geodesically complete radiative spacetimes. It is a defect when the intended subject is collapse, an event horizon, or a stationary black hole. One must weaken the global clause without weakening every asymptotic conclusion indiscriminately.

Weak asymptotic simplicity

A spacetime is weakly asymptotically simple when a neighborhood of its conformal boundary is isometric to a neighborhood of the boundary in some asymptotically simple spacetime. “Weakly asymptotically simple and empty” adds vacuum near that boundary. The reference spacetime supplies the clean asymptotic collar; the physical spacetime may differ arbitrarily in a compact or causally interior region. The standard WASE formulation is summarized in Chruściel 2002, §1.4.1, pp. 15–16.

This localization admits Schwarzschild and Kerr asymptotic regions, collapse singularities, and event horizons. It also admits less desirable interiors. For example, remove a timelike worldline from Minkowski spacetime. The result remains exactly Minkowskian near $\mathscr I$, hence weakly asymptotically simple there, but null geodesics can terminate at the missing line. Weak simplicity alone does not say that the field equations have a globally well-posed exterior evolution or that singularities are hidden.

Nor should weak simplicity be confused with the one-sided definition of asymptotic flatness at $\mathscr I^+$. A one-sided definition may ask only for a future boundary component with specified topology, differentiability, matter decay, and generator completeness. WASE normally compares a whole asymptotic neighborhood with an asymptotically simple and empty model. Authors use variants, so the actual conditions must be read rather than inferred from the adjective “weak.”

Predictability and a trustworthy event horizon

An event horizon depends on the entire future of the chosen $\mathscr I^+$. If that boundary is artificially truncated, the definition can create a fake black hole even in flat spacetime. Restrict Minkowski $\mathscr I^+$ to retarded times $u\lt 0$; points that can reach only the omitted portion are then excluded from $J^-(\mathscr I^+)$. This counterexample and the need for complete generators are described in Chruściel 2002, §1.4.1, pp. 15–16.

Future asymptotic predictability adds a partial Cauchy surface $S$ and requires

$$\mathscr I^+\subset\overline{D^+(S)},$$

where the closure is taken in the conformal completion. Informally, data on $S$ determine the asymptotic future; no naked failure of determinism reaches infinity. A commonly used strong version additionally requires

$$J^+(S)\cap\overline{J^-(\mathscr I^+)}\subset D^+(S).$$

This controls the portion of the exterior to the future of $S$ while permitting an uncontrolled singular region behind $\mathcal H^+$. The Hawking–Ellis versions and their exact domain-of-dependence clauses are reviewed in Chruściel 2002, §1.4.1, pp. 16–17.

These hypotheses are not consequences of WASE, and formulations differ in their causal regularity and completeness requirements. They must be stated theorem by theorem. Predictability also does not prove cosmic censorship; rather, it is one precise condition expressing the exterior regularity that censorship would be expected to supply.

Comparing the variants

Condition	Controls near infinity	Controls all physical null geodesics	Allows a black-hole region	Typical use
Asymptotic simplicity	Smooth full conformal boundary	Yes, both endpoints	No	Global radiation and complete scattering models
Asymptotic simplicity and emptiness	Same, plus vacuum near the boundary	Yes	No	Smooth vacuum peeling and conformal evolution
Weak asymptotic simplicity and emptiness	An asymptotic collar modeled on an ASE spacetime	No	Yes	Black-hole exteriors and collapse
One-sided null asymptotic flatness	A specified $\mathscr I^+$ or $\mathscr I^-$	No	Yes	Bondi data, flux, and BMS structure
WASE plus future predictability	Asymptotic collar and an exterior domain of dependence	No; only the visible exterior is controlled	Yes	Event-horizon and area-theorem frameworks

None of these conditions follows from spatial ADM falloff alone. Conversely, full asymptotic simplicity does not by itself choose spatial parity conditions or establish matching through $i^0$.

Common pitfalls

“Complete infinity” is not the same as “all rays escape.” Complete generators of $\mathscr I^+$ prevent an artificial truncation of the boundary. Asymptotic simplicity additionally requires every physical null geodesic to end on it.

A regular exterior does not regularize the interior. WASE deliberately leaves the interior uncontrolled. Any theorem needing predictability, an energy condition, or a regular horizon must add it explicitly.

An event horizon is global. It cannot be located from one time slice without assumptions about the future development. Apparent or trapping horizons answer different, quasi-local questions.

Exercises

Classify each example as asymptotically simple, weakly asymptotically simple near null infinity, both, or neither: (a) Minkowski spacetime; (b) Minkowski spacetime with one complete timelike line removed; (c) maximally extended Schwarzschild spacetime, considering either asymptotic end; and (d) de Sitter spacetime. Use “asymptotically flat” only when vacuum with zero cosmological constant holds near the relevant boundary.

Solution

(a) Minkowski is asymptotically simple and empty, hence also weakly asymptotically simple and asymptotically flat. (b) The removed-line spacetime agrees with Minkowski in a neighborhood of null infinity, so it is weakly asymptotically simple and empty. It is not fully asymptotically simple because null geodesics can end at the missing line. (c) Each Schwarzschild asymptotic region has a smooth vacuum collar and is therefore weakly asymptotically simple and asymptotically flat near its infinity. The full spacetime is not asymptotically simple because captured null geodesics end at the singularity or fail to reach $\mathscr I$. (d) De Sitter is asymptotically simple, but it is not asymptotically flat in the zero-$\Lambda$ sense: its Ricci tensor does not vanish near its spacelike conformal boundary.

References

• Chruściel, Piotr T. “Black Holes.” arXiv:gr-qc/0201053 [gr-qc] (2002). arXiv.
• Frauendiener, Jörg. “Conformal Infinity.” Living Reviews in Relativity 7 (2004): 1. DOI.