Asymptotics with a positive cosmological constant: IV. The `no-incoming radiation' condition

TL;DR

The paper develops a gauge-invariant framework for gravitational radiation with a positive cosmological constant by introducing a physically relevant past boundary I^-_Rel and enforcing a no incoming radiation condition via non-expanding and weakly isolated horizon structures. It analyzes the symmetry content, revealing an infinite-dimensional BMS-like structure that reduces to a seven-dimensional group when a cross-section i^o_Loc is specified, enabling well-defined notions of mass M and angular momentum, and a Hamiltonian first law on a covariant phase space. The work clarifies how Λ>0 modifies the Bondi–Sachs–Penrose picture, demonstrates consistency with Λ→0 limits, and connects global-horizon methods with quasi-local horizon techniques, laying groundwork for defining gravitational radiation and conserved charges in asymptotically de Sitter spacetimes. These constructions provide a principled path to gauge-invariant radiation analysis and conserved quantities for isolated systems in a universe with dark energy, with explicit results for de Sitter, Schwarzschild–de Sitter, and Kerr–de Sitter spacetimes.

Abstract

Consider compact objects --such as neutron star or black hole binaries-- in \emph{full, non-linear} general relativity. In the case with zero cosmological constant $Λ$, the gravitational radiation emitted by such systems is described by the well established, 50+ year old framework due to Bondi, Sachs, Penrose and others. However, so far we do not have a satisfactory extension of this framework to include a \emph{positive} cosmological constant --or, more generally, the dark energy responsible for the accelerated expansion of the universe. In particular, we do not yet have an adequate gauge invariant characterization of gravitational waves in this context. As the next step in extending the Bondi et al framework to the $Λ>0$ case, in this paper we address the following questions: How do we impose the `no incoming radiation' condition for such isolated systems in a gauge invariant manner? What is the relevant past boundary where these conditions should be imposed, i.e., what is the \emph{physically relevant} analog of past null infinity $\mathcal{I}^{-}_{0}$ used in the $Λ=0$ case? What is the symmetry group at this boundary? How is it related to the Bondi-Metzner-Sachs (BMS) group? What are the associated conserved charges? What happens in the $Λ\to 0$ limit? Do we systematically recover the Bondi-Sachs-Penrose structure at $\mathcal{I}^{-}_{0}$ of the $Λ=0$ theory, or do some differences persist even in the limit? We will find that while there are many close similarities, there are also some subtle but important differences from the asymptotically flat case. Interestingly, to analyze these issues one has to combine conceptual structures and mathematical techniques introduced by Bondi et al with those associated with \emph{quasi-local horizons}.

Figures (3)

• Figure 1: A linearized compact binary on de Sitter background. The binary is depicted by intertwined lines on the left edge of the figure. It pierces space-like $\mathcal{I}^{\pm}$ of de Sitter space-time at points $i^{\pm}$. Solid (black) arrows denote the emitted radiation. Left Panel: The thick (blue) diagonal line represents the future event horizon $E^{+} (i^{-})$ of $i^{-}$. Observers whose world-lines are confined to the portion of space-time to the past of $E^{+}(i^{-})$ --i.e. to the past Poincaré patch-- cannot see the source, nor the radiation it emits. Therefore in the investigation of the isolated system, the relevant part $M_{\rm Rel}$ of space-time is only the future Poincaré patch. It's past boundary, denoted in the figure by $\mathcal{I}^{\,-}_{\rm Rel}$ serves as the relevant$\mathcal{I}^{-}$. Right Panel: For the future boundary, there are two choices: (i) space-like $\mathcal{I}^{+}$; or, ii) local$\mathcal{I}^{+}$, the portion of the past event horizon $E^{-}(i^{+})$ of $i^{+}$ that lies in $M_{\rm Rel}$, denoted in the figure by $\mathcal{I}^{\,+}_{\rm Loc}$. It intersects $\mathcal{I}^{\,-}_{\rm Rel}$ in a (bifurcation) 2-sphere, denoted by $i^{o}_{\rm Loc}$. The (red) dashed lines with arrows represent integral curves of a de Sitter 'time-translation' Killing field adapted to the center of mass of the linearized source. It is time-like near the source but space-like near $\mathcal{I}^{+}$.
• Figure 2: Left panel: Collapse of a spherical star in general relativity with a positive $\Lambda$. The collapse results in a space-like singularity in the future, denoted by the wiggly (magenta) line. The singularity is hidden from the exterior region by a black hole horizon, and we also have the future cosmological horizon of $i^{-}$ which serves as $\mathcal{I}^{\,-}_{\rm Rel}$, and (portion of) the past cosmological horizon of $i^{+}$ that serves as $\mathcal{I}^{\,+}_{\rm Loc}$, and intersects $\mathcal{I}^{\,-}_{\rm Rel}$ in a 2-sphere cross-section $i^{o}_{\rm Loc}$. The relevant space-time $M_{\rm Rel}$ is the portion to the causal future of $i^{-}$. There is a static Killing field $T^{a}$ outside the star, whose integral curves are denoted by dashed (red) lines with arrows. It is time-like in the region bounded by the black hole horizon, $\mathcal{I}^{\,+}_{\rm Loc}$ and $\mathcal{I}^{\,-}_{\rm Rel}$, but space-like near $\mathcal{I}^{+}$. Right panel: Eternal spherically symmetric black hole in general relativity with a positive $\Lambda$. Because $\mathcal{I}^{\pm}$ are space-like, the future (past) boundary of the maximally extended solution consists of an infinite sequence of singularities flanked by $\mathcal{I}^{+}$ (respectively, $\mathcal{I}^{-}$). Thus in contrast to the asymptotically flat, $\Lambda=0$ case, the space-time diagram continues ad-infinitum. However, following the strategy discussed in section \ref{['s2']}, for us the relevant part $M_{\rm Rel}$ of space-time is the causal future of $i^{-}$ which contains only one future singularity and one $\mathcal{I}^{+}$. Situation with $\mathcal{I}^{\,+}_{\rm Loc}$, $i^{o}_{\rm Loc}$ and the static Killing field is the same as in the figure in the left panel. The shaded portion represents $M_{\rm Loc}$, the intersection of the causal future of $i^{-}$ with the causal past of $i^{+}$.
• Figure 3: Kerr-de Sitter space-time. The future and past boundaries, $\mathcal{I}^{\pm}$, of the asymptotic region are space-like because we have a positive $\Lambda$. The future event horizon $\mathcal{I}^{\,-}_{\rm Rel}$ of $i^{-}$ intersects the past cosmological horizon of $i^{+}$ in a 2-sphere $i^{o}_{\rm Loc}$ just as in Figs 1 and 2. The vertical wiggly lines depict the singularities. We now have three horizons separating the singularity from the asymptotic regions near $\mathcal{I}^{+}$: the inner black hole horizon $r=r_{-}$, the outer black hole horizon $r=r_{+}$ and the cosmological $r=r_{c}$. As in Figures 1 and 2, black hole and cosmological horizons serve as the past and the future boundaries of the (shaded) local space-time region $M_{\rm Loc}$, the intersection of the causal future of $i^{-}$ with the causal past of $i^{+}$. The full past boundary $\mathcal{I}^{\,-}_{\rm Rel}$ is the extension of $\mathcal{I}^{\,-}_{\rm Loc}$ all the way to spatial infinity $i^{o}$, the 'right end' of $\mathcal{I}^{+}$.