Asymptotic structure with a positive cosmological constant

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→0 and apply all our results to selected exact solutions of Einstein Field Equations. The full-length abstract is available in the paper.

Gravitational radiation condition at infinity with a positive cosmological constant

“…More importantly, we have argued that the criterion is equivalent to the standard one -based on the news tensor-for the Λ = 0 case. Adding appropriate structure on (J , h ab , D ab ) which allows to deal with 'incoming and outgoing' waves, we are developing a framework [45] in order to recover further properties of the asymptotically flat case. In particular, ways to define asymptotic symmetries at infinity, as well as enlightening evidences for the existence of news-like tensor fields on J in the presence of a positive cosmological constant, emerge.…”

“…There exists one proposal [41] that requires knowledge of the physical spacetime, but we aim at handling this problem using information solely from, or on top of, J + . A worth-exploring possibility arising naturally in our scheme is partly related to the algebraic structure of the rescaled Weyl tensor in relation with N α [45]. In any case, when Λ > 0 it seems necessary to endow (J + , h ab , D ab ) with further structure, essentially a distinguished or chosen congruence of curves which depends on the physical properties of the spacetime under consideration (see also [46]).…”

The partial Bondi gauge: Further enlarging the asymptotic structure of gravity

“…The analysis of the asymptotic structure of general relativity has also been extended to spacetimes which are asymptotically (A)dS 4 [63][64][65][66][67][68][69], where part of the motivations and intriguing questions pertain to the very nature, when Λ = 0, of gravitational radiation, mass, angular momentum, and news [70][71][72][73][74][75][76][77][78][79][80][81][82][83][84][85]. Even if there is no notion of S-matrix in this context, there are meaningful asymptotic symmetries and (yet poorly understood) memory effects.…”

“…The notion of radiation for spacetimes with Λ = 0 is poorly understood [70][71][72][73][74][75][76][77][78][79][80][81][82][83][84][85]. Although here we have not directly discussed notions pertaining to gravitational waves and quadrupole formulas in (A)dS, we have seen how certain well-defined quantities in asymptotically flat spacetimes generalize to the case Λ = 0 and ∂ u q AB = 0 (the question of whether these remain the relevant quantities is however open).…”

Gravitational Radiation at Infinity with Non-Negative Cosmological Constant

“…The problem of the existence of news-like objects in this case, and the question of in-and out-going radiation are discussed in section 5 and the existence of asymptotic symmetries is studied in section 6. I end the paper with a list of examples presented in [32,31] and some closing comments.…”