Asymptotics with a positive cosmological constant: I. Basic framework

TL;DR

The paper analyzes the pitfalls of extending the Λ=0 Bondi–Sachs asymptotic framework to a universe with a positive cosmological constant, showing that the conformal boundary is spacelike and that the natural BMS structure no longer arises. It introduces weak/strong notions of asymptotically de Sitter spacetimes, explores their boundary topologies, and derives asymptotic fields, Weyl curvature behavior, and conserved gravitational charges under restricted conditions. A key result is that enforcing conformal flatness of the boundary dramatically reduces the asymptotic symmetry group and suppresses gravitational radiation flux across I, revealing a fundamental tension in defining energies, momenta, and S-matrix concepts for Λ>0. The analysis, including detailed examples (de Sitter, Schwarzschild-de Sitter, Kerr-de Sitter, Vaidya-de Sitter, FL cosmologies) and the B^ab=0 discussion, underscores the need for a new, physically robust framework to address gravitational radiation and related questions in Λ>0 spacetimes, to be developed in subsequent work.

Abstract

The asymptotic structure of the gravitational field of isolated systems has been analyzed in great detail in the case when the cosmological constant $Λ$ is zero. The resulting framework lies at the foundation of research in diverse areas in gravitational science. Examples include: i) positive energy theorems in geometric analysis; ii) the coordinate invariant characterization of gravitational waves in full, non-linear general relativity; iii) computations of the energy-momentum emission in gravitational collapse and binary mergers in numerical relativity and relativistic astrophysics; and iv) constructions of asymptotic Hilbert spaces to calculate $S$-matrices and analyze the issue of information loss in the quantum evaporation of black holes. However, by now observations have established that $Λ$ is positive in our universe. In this paper we show that, unfortunately, the standard framework does not extend from the $Λ=0$ case to the $Λ>0$ case in a physically useful manner. In particular, we do not have positive energy theorems, nor an invariant notion of gravitational waves in the non-linear regime, nor asymptotic Hilbert spaces in dynamical situations of semi-classical gravity. A suitable framework to address these conceptual issues of direct physical importance is developed in subsequent papers.

Figures (5)

• Figure 1: Isometries near $\mathcal{I}$ of Minkowski and de Sitter space-times. Left Panel: Minkowski space-time. The time translation Killing fields are time-like in a neighborhood of $\mathcal{I}$ and null on $\mathcal{I}$. Right Panel: de Sitter space-time. Since $\mathcal{I}$ is now space-like, all Killing fields of de Sitter are space-like near and on $\mathcal{I}$. The arrows represent a 'time translation' which changes its time-like versus space-like character across cosmological horizons.
• Figure 2: Conformal diagram of the Schwarzschild-de Sitter space-time. In contrast with the asymptotically flat case, this solution for an eternal black hole admits analytical continuations to the right and left of the diagram, exhibiting an infinite number of black hole and white hole singularities. Therefore, one generally makes an identification. Then the space-time has only one (white hole) singularity in the past and one (black hole) singularity in the future. But now the Cauchy surfaces have a topology $\mathbb{S}^{2}\times \mathbb{S}^{1}$ rather than $\mathbb{S}^{2}\times \mathbb{R}$ as in the asymptotically flat case. Also, we now have additional (cosmological) horizons at $r=r_{c}$ and the 'time translation' Killing field (whose orbits are shown in red dashed lines) is space-like near $\mathcal{I}$.
• Figure 3: Conformal diagram of the Vaidya-de Sitter space-time describing the gravitational collapse of an in-falling null fluid from $\mathcal{I}^{-}$. The in-falling null fluid is indicated by the shaded (yellow) region $v_1 \le v \le v_2$. In contrast to the asymptotically flat case, now the dynamical nature of the space-time geometry modifies the structure even at $\mathcal{I}^{+}$. While in the Schwarzschild-de Sitter space-time the Killing and the cosmological horizon $E^{+}(i^{-})$ coincide near $\mathcal{I}^{+}$ they are now distinct; one intersects $\mathcal{I}^{+}$ and the other meets the singularity. Furthermore, the natural identification that allowed us to consider a single black hole (and the associated white hole) in the Schwarzschild-de Sitter case is no longer available.
• Figure 4: Conformal diagram of the Friedmann-Lemaître space-time with positive $\Lambda$. This space-time corresponds only to the Poincaré patch of de Sitter space-time because of the big-bang singularity along the event horizon $E^{+}(i^{-})$ (i.e., $\eta = -\infty$, where $\eta$ is the conformal time).
• Figure 5: Conformal diagram depicting a spherical collapse. Shaded (yellow) region corresponds to the collapsing spherical star. The dashed (red) lines with arrows represent integral curves of the 'static' Killing field. Note that the space-time is incomplete to the right unless we add another collapsing star.