Asymptotic structure with a positive cosmological constant

Abstract

Abstract This is the second of two papers (Fernández-Álvarez F and Senovilla J M M 2021 Class. Quantum Grav. 39 165011) that study the asymptotic structure of space–times with a non-negative cosmological constant Λ. This paper deals with the case Λ &gt; 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="script">J</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mi>h</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mi>D</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , where the first element of the triplet is a three-dimensional manifold, the second is a representative of a conformal class of Riemannian metrics on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">J</mml:mi> </mml:math> , and the third element is a traceless symmetric tensor field on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">J</mml:mi> </mml:math> . The full set of physically relevant properties of the space–time cannot be characterised at infinity without taking D ab into consideration, and our radiation criterion takes this fully into account. We similarly propose a no-incoming radiation criterion based also on the triplet <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="script">J</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mi>h</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mi>D</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> and on radiant supermomenta deduced from the rescaled Bel–Robinson tensor too. We search for news tensors encoding the two degrees of freedom of gravitational radiation and argue that any news-like object must be associated to, and depends on, two-dimensional cross-sections of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">J</mml:mi> </mml:math> . We identify one component of news for every such cross-section and present a general strategy to find the second component, which depends on the particular physical situation. We put in connection the radiation condition and the news-like tensors with the directional structure of the gravitational field at infinity and the criterion of no-incoming radiation. We also introduce the concept of equipped <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">J</mml:mi> </mml:math> by endowing the conformal boundary with a selected congruence of curves which may be determined by the algebraic structure of the asymptotic Weyl tensor. We also define a group of asymptotic symmetries preserving the new structures. We consider the limit Λ → 0, and apply all our results to selected exact solutions of Einstein field equations in order to illustrate their validity.

Topics & Concepts