Stability and Scattering Interfaces

Positive cosmological constant changes both the nonlinear stability problem and the meaning of scattering. Near global de Sitter space, sufficiently small vacuum perturbations have complete developments with smooth spacelike past and future conformal boundaries. That theorem is powerful, but it is neither a large-data cosmic-no-hair theorem nor the construction of an ordinary particle S-matrix. This page separates the geometric theorem, observer-dependent horizon physics, and model-specific scattering problems.

Required background. We use The Einstein Equation with $\Lambda\gt 0$ to state the vacuum stability theorem and its constraint hypotheses.

Helpful background. Pseudo-Riemannian Metrics and Signature fixes the causal character of conformal infinity, Affine and Variational Geodesics supplies the relevant completeness notions, Causality and Global Structure supplies maximal developments and domains of dependence, Exact Solutions of Einstein’s Equations supplies the de Sitter comparison solution, and Formulations of General Relativity supplies constrained Cauchy data and hyperbolic evolution.

Nonlinear stability near global de Sitter

The clean four-dimensional vacuum statement has several indispensable hypotheses. Let $\Sigma\cong S^3$ be a compact Cauchy hypersurface, and let $(h_{ij},K_{ij})$ be smooth initial data satisfying the Einstein vacuum constraint equations with positive cosmological constant $\Lambda\gt 0$. Suppose the data are sufficiently close, in the Sobolev norms required by the conformal hyperbolic formulation, to data induced by a global de Sitter slice. Then the maximal globally hyperbolic development solving

$$\widehat R_{ab}=\Lambda\widehat g_{ab}$$

is timelike and null geodesically complete in both time directions and admits smooth spacelike conformal boundaries $\mathscr I^-$ and $\mathscr I^+$. In particular, these nearby developments remain asymptotically simple rather than forming singularities or black holes from the small perturbation. Friedrich states this de Sitter stability result and its two-sided conformal conclusion in Friedrich 2024, §2; the original conformal-field-equation proof is Friedrich 1986, pp. 587–609.

The theorem is an open-set statement in a specified initial-data topology. “Small” is not a pointwise bound on curvature, and it is not defined merely by looking visually close to a de Sitter metric. One must compare constraint-satisfying data in the function spaces used by the well-posed evolution. Compactness of $S^3$ is also doing real work: there is no uncontrolled spatial end, and both conformal boundaries can be treated in the global conformal problem.

No conformal-flatness condition on the induced metrics at $\mathscr I^\pm$ is part of this stability theorem. Nearby solutions can carry nontrivial radiative data and have boundary metrics that are close to, but not conformally equivalent to, the round metric. Imposing $\mathcal B_{ab}=0$, with $\mathcal B_{ab}$ the magnetic part of the leading rescaled Weyl tensor, would be an additional ABK boundary restriction. It removes half of the freely specifiable gravitational data; it must not be smuggled into a stability claim. The unrestricted asymptotically de Sitter phase space and the conformally flat subspace are distinguished in Ashtekar, Bonga, and Kesavan 2015, §§IV.B and V.C.4, pp. 19–20 and 33–36.

The conclusion does not cover large perturbations, noncompact Cauchy topology, arbitrary matter, or black-hole-forming data. It also does not assert that every $\Lambda\gt 0$ cosmology is future complete. A theorem about a neighborhood of global de Sitter data is not a classification of all positive-$\Lambda$ developments.

Three different no-hair statements

The phrase “cosmic no hair” is used for conclusions of unequal strength. Keeping them separate prevents a local late-time estimate from being mistaken for a global causal theorem.

First, expansion-normalized curvature and shear may decay along a family of future-directed observers. This is an observer- and foliation-dependent asymptotic statement. Wald proved such behavior for initially expanding spatially homogeneous Bianchi models, subject to energy conditions and with specified exceptional types, in Wald 1983, pp. 2118–2120. That short theorem is not a result for arbitrary inhomogeneous data, and it does not by itself construct a smooth conformal boundary.

Second, a spacetime may possess a smooth future conformal completion. This gives a geometric description of $\mathscr I^+$ and, with suitable hypotheses, future null and timelike completeness. It is stronger than decay of a selected list of variables but says nothing automatically about $\mathscr I^-$. A cosmological model can have a regular future boundary and a big-bang singularity in the past.

Third, two-sided nonlinear stability says that a whole open set of initial data has both smooth $\mathscr I^-$ and smooth $\mathscr I^+$, together with complete evolution between them. This is the Friedrich result above. Even here, “approaches de Sitter” does not mean that the metric becomes exactly de Sitter or that all conformal boundary data become round. The freely varying boundary geometry can retain information about the solution while physical curvature invariants approach their de Sitter values in suitable frames.

Matter changes every level of this hierarchy. Conformally regular systems can sometimes be incorporated into conformal evolution, whereas massive fields, fluids, or slow falloff can alter differentiability and constraints at infinity. The minimal ABK framework assumes that $\Omega^{-1}\widehat T_{ab}$ has a smooth limit at $\mathscr I$; it is not a theorem that arbitrary matter satisfies this decay. A no-hair statement must therefore name the matter model, energy assumptions, data class, and the norm or geometric sense in which convergence is claimed.

Why there is no ordinary global S-matrix

An asymptotically flat particle S-matrix compares free incoming and outgoing states: massless particles are associated with null infinity, whereas massive particles require timelike asymptotic data. Its standard interpretation relies on asymptotic time translations, a positive-frequency splitting, and asymptotic regions accessible to the relevant states. Global de Sitter geometry does not supply these structures in the same way.

Both $\mathscr I^-$ and $\mathscr I^+$ are spacelike because $\Lambda\gt 0$ gives a timelike conformal normal. Exact de Sitter space has no Killing field that is timelike everywhere, and a generic asymptotically de Sitter boundary permits the much larger group $\operatorname{Diff}(\mathscr I)$ rather than a preferred translation subgroup. Moreover, each timelike observer has cosmological horizons: its causal diamond intersects only part of the global spacetime and cannot receive signals from all of $\mathscr I^-$ or send signals to all of $\mathscr I^+$. These facts obstruct a canonical, observer-independent notion of global in/out particles and positive energy. The resulting asymptotic difficulties are set out in Ashtekar, Bonga, and Kesavan 2015, Introduction and §VI, pp. 1–4 and 35–38.

This obstruction must be stated narrowly. It rules out importing the ordinary asymptotically flat LSZ or particle-S-matrix construction without extra choices. It does not prove that every scattering problem on a de Sitter background is meaningless. One may study classical conformal evolution between spacelike boundaries, quantum fields in a selected state, static-patch response, or transmission across horizons. Each construction has different data spaces and observables.

Classical conformal scattering

Within the two-sided asymptotically simple class, the Einstein evolution can be viewed schematically as a map

$$\mathcal S_{\rm conf}:\mathcal D^-\longrightarrow\mathcal D^+,$$

where $\mathcal D^-$ and $\mathcal D^+$ are equivalence classes of admissible geometric conformal data at $\mathscr I^-$ and $\mathscr I^+$. In four-dimensional vacuum gravity, such data can be represented by a Riemannian conformal metric together with the transverse, trace-free electric part of the leading rescaled Weyl tensor, subject to conformal gauge equivalence. The spacelike boundary data are Cauchy-like geometric data, not collections of asymptotic particles.

Near de Sitter space, the conformal field equations and continuous dependence provide a local small-data version of this map: admissible data near the de Sitter data determine a regular development and corresponding data at the other boundary. Friedrich’s earlier existence analysis makes the free conformal data and their constraint explicit in Friedrich 1986, pp. 101–117. Whether the map is global, injective, surjective, symplectic in a chosen completion, or extendible past singular solutions is a separate theorem, not a consequence of stability alone.

Other uses of “de Sitter scattering” are model-specific. Linear scalar fields depend on mass and curvature coupling; quantum constructions depend on a state and an algebra of observables; a static-patch experiment uses horizon boundary conditions and the observer’s Killing time. Black-hole exterior scattering introduces event and cosmological horizons and is not the boundary-to-boundary problem of global de Sitter space. None of these theories should inherit conclusions from the vacuum conformal map merely because the word “scattering” appears in both.

Interfaces and limitations

The geometric framework supplies definitions, causal boundaries, and a domain in which sharper analytic questions can be posed. Detailed nonlinear stability of Kerr–de Sitter or Schwarzschild–de Sitter exteriors, decay rates, resonances, quasinormal modes, and horizon scattering belong to Mathematical Relativity and the black-hole dynamics volumes. Quantum state selection and the status of particle observables belong to Quantum Field Theory in Curved Spacetime. These subjects use the global distinctions developed here, but require equations and boundary conditions beyond them.

Topology remains consequential. The theorem stated above uses compact $S^3$ Cauchy surfaces and two compact conformal boundaries. A planar $\mathbb R^3$ presentation of half of exact de Sitter space is a different global problem, even though it is locally isometric to de Sitter. Similarly, a future-asymptotically de Sitter cosmology with $\mathscr I^+\cong\mathbb R^3$ need not possess a regular past boundary. Coordinate resemblance at late time cannot replace a theorem about maximal development, completeness, and boundary topology.

Exercise

Consider smooth vacuum initial data with $\Lambda\gt 0$ on $S^3$, sufficiently close in the required Sobolev norm to global de Sitter data. Decide which of the following conclusions follow from the nonlinear stability theorem: (a) smooth spacelike $\mathscr I^\pm$; (b) null and timelike completeness in both directions; (c) conformally flat metrics on $\mathscr I^\pm$; (d) a canonical global particle S-matrix; (e) absence of black holes for this small-data development; (f) nonlinear stability of every Schwarzschild–de Sitter exterior.

Solution

Statements (a), (b), and (e) follow. Smooth two-sided conformal completion and causal-geodesic completeness are the theorem’s conclusions; within this small neighborhood the development therefore does not form a black-hole region or singularity. Statement (c) does not follow: conformal flatness is an extra boundary restriction that removes magnetic radiative data. Statement (d) does not follow because spacelike infinity, observer horizons, and the lack of a preferred global time translation obstruct the ordinary particle interpretation; a classical conformal data map is a different object. Statement (f) concerns a different background, topology, causal region, and perturbation problem. It requires an independent black-hole stability theorem.

References

• Abhay Ashtekar, Béatrice Bonga, and Aruna Kesavan, “Asymptotics with a Positive Cosmological Constant: I. Basic Framework,” Classical and Quantum Gravity 32 (2015), 025004. doi:10.1088/0264-9381/32/2/025004. Open PDF
• Helmut Friedrich, “Cosmologies with Positive $\lambda$: Hierarchies of Future Behaviour,” Philosophical Transactions of the Royal Society A 382 (2024), 20230044. doi:10.1098/rsta.2023.0044
• Helmut Friedrich, “Existence and Structure of Past Asymptotically Simple Solutions of Einstein’s Field Equations with Positive Cosmological Constant,” Journal of Geometry and Physics 3 (1986), 101–117. doi:10.1016/0393-0440(86)90004-5
• Helmut Friedrich, “On the Existence of $n$-Geodesically Complete or Future Complete Solutions of Einstein’s Field Equations with Smooth Asymptotic Structure,” Communications in Mathematical Physics 107 (1986), 587–609. doi:10.1007/BF01205488
• Robert M. Wald, “Asymptotic Behavior of Homogeneous Cosmological Models in the Presence of a Positive Cosmological Constant,” Physical Review D 28 (1983), 2118–2120. doi:10.1103/PhysRevD.28.2118