Asymptotic Structure with a positive cosmological constant

TL;DR

The paper develops a covariant, gauge-invariant framework for the asymptotic structure of space-times with a positive cosmological constant by focusing on the conformal boundary 𝒥 and the triplet (𝒥,h_{ab},D_{ab}). It introduces a radiation criterion based on the asymptotic super-Poynting vector derived from the rescaled Bel–Robinson tensor and elaborates how news-like, radiative content can be defined on cuts of 𝒥, including the concept of equipped 𝒥 to address incoming radiation. A geometric analogue of Geroch’s tensor ρ is developed to underpin the existence and properties of news-like objects, and a two-component framework for news is proposed to account for both radiative sectors and potential extrinsic data through Y_{A}. The Λ→0 limit is analyzed to connect with the standard News-based radiation notion in the asymptotically flat case, and the framework is tested on exact solutions such as de Sitter and Kerr–de Sitter to illustrate its consistency and applicability.

Abstract

This is the second of two papers that study the asymptotic structure of space-times with a non-negative cosmological constant $Λ$. This paper deals with the case $Λ>0$. Our approach is founded on the `tidal energies' built with the Weyl curvature and, specifically, we use the asymptotic super-Poynting vector computed from the rescaled Bel-Robinson tensor at infinity to provide a covariant, gauge-invariant, criterion for the existence, or absence, of gravitational radiation at infinity. The fundamental idea we put forward is that the physical asymptotic properties are encoded in $(\scri,h_{ab},D_{ab})$, where the first element of the triplet is a 3-dimensional manifold, the second is a representative of a conformal class of Riemannian metrics on $\scri$, and the third element is a traceless symmetric tensor field on $\scri$. We similarly propose a no-incoming radiation criterion based also on the triplet $(\scri,h_{ab},D_{ab})$ and on radiant supermomenta deduced from the rescaled Bel-Robinson tensor too. We search for news tensors and argue that any news-like object must be associated to, and depends on, 2-dimensional cross-sections of $\scri$. We identify one component of news for every such cross-section and present a general strategy to find the second component. We also introduce the concept of equipped $\scri$, consider the limit $Λ\rightarrow 0$ and apply all our results to selected exact solutions of Einstein Field Equations. The full-length abstract is available in the paper.

Figures (10)

• Figure 1: In the presence of a positive cosmological constant, $\mathscr{J}$ usually has $\mathbb{S}^3$-topology or $\mathbb{S}^3$ without a set of points. Also, one can consider Riemannian surfaces, or cuts, denoted by $\mathcal{S}$. The figure shows --wirh 1 dimension suppressed-- the stereographic projection of $\mathscr{J}$ to the plane, including a couple of cuts labelled by $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$. Thus, one can picture $\mathscr{J}$ as $\mathbb{R}^3$, which is how it is represented in the rest of the figures.
• Figure 2: Gravitational radiation arrives at an open region $\Delta$ on $\mathscr{J}^+$ but does not at the open region $\Delta'$. Our criterion states that the asymptotic super-Poynting is different from zero on $\Delta$ and vanishes on $\Delta'$.
• Figure 3: The lightlike decomposition \ref{['eq:n-kp-km']} on $\mathscr{J}$.
• Figure 4: Flow of the asymptotic superenergy quantities. One starts from the above middle node: strong orientation is chosen ($-\tensor{{m}}{^a}$ points along the spatial projection to $\mathscr{J}$ of a PND of the rescaled Weyl tensor with highest multiplicity). Then, either the rescaled Weyl tensor is algebraically general (left-hand side of the diagram) or it is special (right-hand side of the diagram). Moving to the left, either the radiant superenergy $\tensor[^{{\,}^{\!+\!}\!}]{\mathcal{W}}{}$ vanishes (above left-hand side) or not (below left-hand side). Thus, for an algebraically general rescaled Weyl tensor on $\mathscr{J}$, there are four configurations of asymptotic radiant superenergy: in two of them, there is gravitational radiation (one with $\tensor[^{{\,}^{\!+\!}\!}]{\mathcal{W}}{}\neq 0$, the other one with $\tensor[^{{\,}^{\!+\!}\!}]{\mathcal{W}}{}=0$); in the other two there is no gravitational radiation (the shaded nodes). Moving to the right, one finds the algebraically special cases. There are four possibilities, from which just one corresponds to no radiation (the shaded node, for Petrov type D or 0, the only case in which both radiant supermomenta vanish).
• Figure 5: In the $\Lambda=0$ limit vector fields of the class described in \ref{['thm:limitM']} become collinear with $\tensor{{N}}{^\alpha}|_{_{\Lambda=0}}$, the vector field tangent to the null generators of $\mathscr{J}_0$.

Theorems & Definitions (146)

• Definition II.1
• Definition II.2
• Lemma II.1
• Lemma II.2
• Lemma II.3
• proof
• Lemma II.4
• Corollary II.1
• proof
• Proposition II.1