Definitions of Asymptotically de Sitter Spacetimes

There is no useful definition of “asymptotically de Sitter” that mentions only positive cosmological constant or curvature approaching a constant. A definition must specify a conformal completion, differentiability, matter decay, which temporal end is completed, completeness of the boundary geometry, and the topology appropriate to the problem. In four dimensions these choices separate global cosmologies, expanding patches, and isolated systems with black holes. They also prevent a restrictive conformal-flatness condition from being mistaken for generic radiative asymptotics.

Required background. We use The Einstein Equation with $\Lambda\gt 0$ and a stated stress–energy falloff near the conformal boundary.

Helpful background. Pseudo-Riemannian Metrics and Signature fixes the boundary-character convention, Affine and Variational Geodesics distinguishes boundary completeness from physical geodesic completeness, Causality and Global Structure supplies causal and conformal completions, Exact Solutions of Einstein’s Equations supplies de Sitter and Schwarzschild–de Sitter examples, and Formulations of General Relativity supplies the initial-value language.

A conformal definition in four dimensions

Let $(\widehat M,\widehat g_{ab})$ be a smooth, time-oriented, four-dimensional physical spacetime satisfying

$$\widehat G_{ab}+\Lambda\widehat g_{ab} =8\pi G\widehat T_{ab}, \qquad \Lambda\gt 0.$$

Call it weakly asymptotically de Sitter at a chosen temporal end if there are a smooth manifold-with-boundary $M$, a smooth Lorentzian metric $g_{ab}$ on $M$, and a smooth defining function $\Omega$ such that:

1. the interior $M\setminus\mathscr I$ is diffeomorphic to $\widehat M$ in a neighborhood of that end;
2. $g_{ab}=\Omega^2\widehat g_{ab}$ and $\Omega\gt 0$ in the physical region;
3. $\Omega=0$ and $d\Omega\ne0$ on the boundary component $\mathscr I$; and
4. $\Omega^{-1}\widehat T_{ab}$ has a smooth limit at $\mathscr I$.

The differentiability class must be stated if it is lower than smooth. A $C^k$ completion supports only the derivatives available at that order, and a polyhomogeneous expansion containing logarithms is not a smooth completion. The definition also fixes a conformal gauge only up to

$$\Omega'=\omega\Omega, \qquad g'_{ab}=\omega^2g_{ab},$$

where $\omega$ is smooth and positive at $\mathscr I$. Physical conclusions must be invariant under this change.

With the stated four-dimensional Einstein and matter hypotheses, the boundary normal satisfies $n^2=-\Lambda/3$ and $\mathscr I$ is spacelike. This is a consequence, not an extra clause. Ashtekar, Bonga, and Kesavan give the definition and derive its immediate boundary equations in Ashtekar, Bonga, and Kesavan 2015, §II.A–B, pp. 5–9.

Completeness and the three principal topologies

The weak definition is local near infinity and can be satisfied by an artificially truncated boundary. To exclude that defect, require the induced Riemannian metric $q_{ab}$ on $\mathscr I$ to be geodesically complete in the chosen admissible completion. This produces the stronger completeness condition that this chapter calls simply asymptotically de Sitter. On a compact boundary completeness follows from smoothness, but on $\mathbb R^3$ or $S^2\times\mathbb R$ it is a real restriction. Not every conformal rescaling preserves it.

Three topologies organize common applications:

• Global type: $\mathscr I\cong S^3$. Exact global de Sitter has both $\mathscr I^-$ and $\mathscr I^+$ of this type.
• Poincaré-patch type: $\mathscr I\cong\mathbb R^3\cong S^3\setminus\{p\}$. Spatially flat expanding cosmologies naturally approach such a future boundary; the missing point represents spatial infinity of the patch.
• Isolated-system type: $\mathscr I\cong S^2\times\mathbb R\cong S^3\setminus\{p_1,p_2\}$. Schwarzschild–de Sitter supplies the stationary model, with punctures representing the appropriate spatial and timelike ends.

These names describe a completion, not an immutable label attached to every local metric. Exact de Sitter admits both its global $S^3$ completion and a punctured $\mathbb R^3$ completion of a Poincaré patch. By contrast, a black-hole singularity cannot generally be filled by changing $\Omega$. The topology and completeness distinctions, including why the punctures matter, are stated in Ashtekar, Bonga, and Kesavan 2015, §II.A, pp. 5–7.

Future, past, and two-sided conditions

A spacetime may be future asymptotically de Sitter, past asymptotically de Sitter, or both. The conditions are independent. A spatially flat dust-plus-$\Lambda$ cosmology can have a smooth, complete $\mathbb R^3$ future boundary while ending at a big-bang singularity to the past. Its future property does not license the symbol $\mathscr I^-$ or a two-sided scattering problem; the explicit late-time completion is checked in Ashtekar, Bonga, and Kesavan 2015, §III.E, pp. 15–18.

Boundary completeness also differs from asymptotic simplicity. A two-sided asymptotically simple spacetime has a regular $\mathscr I^-\cup\mathscr I^+$ and every inextendible physical null geodesic has endpoints there. Exact de Sitter satisfies that condition. A Schwarzschild–de Sitter black-hole spacetime can possess regular spacelike conformal infinity while some null geodesics instead meet singularities; it is not asymptotically simple. This stronger all-geodesics requirement is excellent for stability theory but excludes the black holes that an isolated-system definition is meant to accommodate.

Similarly, a regular boundary does not by itself prove timelike or null geodesic completeness everywhere. It controls geodesics that reach the regular end transversely. Interior singularities, Cauchy horizons, or excluded regions require separate analysis.

Curvature and initial-data formulations

The conformal definition has useful curvature consequences. In a divergence-free conformal gauge, the unphysical Weyl tensor $C_{abcd}$ vanishes at $\mathscr I$, while its leading rescaling

$$K_{abcd}=\Omega^{-1}C_{abcd}$$

can have a finite, nonzero limit. Its electric and magnetic parts carry information not present in the constant-curvature de Sitter background. Therefore “the Weyl tensor tends to zero” is too weak as a definition: it can discard the first nontrivial asymptotic data, and without a conformal gauge it does not specify a coordinate-invariant rate. The boundary equations and the distinction between $C_{abcd}\rvert_{\mathscr I}=0$ and nonzero $K_{abcd}\rvert_{\mathscr I}$ are derived in Ashtekar, Bonga, and Kesavan 2015, §II.B, pp. 7–9.

There are two complementary initial-value viewpoints. One may prescribe ordinary Einstein Cauchy data $(h_{ij},K_{ij})$ on a spacelike physical slice and prove that their development admits a regular future or two-sided completion. That implication is a nonlinear stability theorem only for stated function spaces, topologies, matter models, and smallness regimes. Alternatively, because $\mathscr I$ is spacelike, one may prescribe conformal data there and solve inward. In four-dimensional vacuum gravity the free boundary data consist, schematically, of a conformal Riemannian three-metric and a conformal class of symmetric trace-free tensors satisfying a divergence constraint. Friedrich’s asymptotic initial-value construction establishes a semiglobal solution for such data in Friedrich 1986, pp. 101–117. Neither formulation makes arbitrary physical data automatically asymptotically de Sitter.

Matter decay and the conformal-flatness trap

The falloff $\Omega^{-1}\widehat T_{ab}$ smooth at $\mathscr I$ is part of the definition used here. Maxwell fields and null fluids can decay faster, while dust cosmologies motivate retaining the weaker rate because a normal-normal component can have a nonzero limit. A different matter equation, a massive field, or lower differentiability may require another conformal formulation. One must not quote a vacuum existence theorem after merely appending matter to the Einstein equation.

Ashtekar, Bonga, and Kesavan also define a strongly asymptotically de Sitter subclass by requiring $q_{ab}$ to be conformally flat. In three boundary dimensions this is equivalent to vanishing Cotton tensor and, through the boundary equations, to vanishing magnetic part $B_{ab}$ of the leading rescaled Weyl curvature. The adjective is their technical nomenclature; it does not mean “the physically correct definition.” The condition removes half of the freely radiative boundary data. It does not force every electric transverse-tracefree mode to vanish, but within the associated charge framework the remaining gravitational modes carry no de Sitter momentum flux across $\mathscr I$. The restriction and its consequences are analyzed in Ashtekar, Bonga, and Kesavan 2015, §V.C, pp. 29–36. This chapter never assumes conformal flatness unless it says so explicitly.

Exercises

Classify the following four-dimensional vacuum or cosmological examples as far as the information permits: (a) global de Sitter with its standard completion; (b) the expanding Poincaré patch of exact de Sitter; (c) a spatially flat dust-plus-$\Lambda$ cosmology with a big bang and smooth complete future boundary; and (d) Schwarzschild–de Sitter with regular $S^2\times\mathbb R$ conformal infinity and a black-hole singularity. Which are two-sided asymptotically simple?

Solution

(a) The standard completion is global type, with complete $S^3$ components at both temporal ends. Every inextendible null geodesic reaches one component in each direction, so it is two-sided asymptotically simple.

(b) The expanding patch has a complete $\mathbb R^3$ future boundary in its planar completion, but it is only part of the globally complete spacetime. As a patch it is future asymptotically de Sitter; calling it two-sided requires extending beyond its past null boundary.

(c) The cosmology is future asymptotically de Sitter of Poincaré-patch type. The big bang prevents a regular past completion and two-sided asymptotic simplicity.

(d) The regular $S^2\times\mathbb R$ component gives isolated-system asymptotically de Sitter behavior at the stated end. The black-hole singularity captures some null geodesics, so the spacetime is not asymptotically simple. None of these conclusions by itself imposes conformal flatness; that must be checked as an additional condition.

References

• Ashtekar, Abhay, Béatrice Bonga, and Aruna Kesavan. “Asymptotics with a Positive Cosmological Constant: I. Basic Framework.” Classical and Quantum Gravity 32, 025004 (2015). arXiv.
• Friedrich, Helmut. “Existence and Structure of Past Asymptotically Simple Solutions of Einstein’s Field Equations with Positive Cosmological Constant.” Journal of Geometry and Physics 3, 101–117 (1986). DOI.