Peeling property and asymptotic symmetries with a cosmological constant

Abstract

This paper establishes two things in an asymptotically (anti-)de Sitter spacetime, by direct computations in the physical spacetime (i.e. with no involvement of spacetime compactification): (1) The peeling property of the Weyl spinor is guaranteed. In the case where there are Maxwell fields present, the peeling properties of both Weyl and Maxwell spinors similarly hold, if the leading order term of the spin coefficient ρ when expanded as inverse powers of r (where r is the usual spherical radial coordinate, and r → ∞ is null infinity, I) has coefficient −1.(2) In the absence of gravitational radiation (a conformally flat I), the group of asymptotic symmetries is trivial, with no room for supertranslations.Recently, the behaviour of empty asymptotically (anti-)de Sitter spacetimes has been studied [6], and further extended to contain Maxwell fields [7] (see also [8][9][10] [11]). These took the approach of exclusively studying the physical spacetime, i.e. no conformal rescaling was explicitly made in deriving those asymptotic solutions. In accordance to the case for asymptotically flat spacetimes, the imposed condition Ψ 0 = O(r −5 ) leads to the expected peeling property for the Weyl spinor and a similar condition on the dyad component of the Maxwell spinor φ 0 = O(r −3 ) yields the analogous peeling behaviour for the Maxwell fields.These peeling properties of Ψ n and φ m (where m = 0, 1, 2) with a non-zero cosmological constant should not be surprising, as the peeling property for the Weyl spinor has been proven for (weakly) asymptotically simple spacetimes that do include asymptotically (anti-)de Sitter spacetimes [12][13][14]. This follows from admitting a smooth conformal compactifiability and working with a conformally compactified version of the spacetime, involving spinors.The driving motivation in Ref. [6] was to generalise the notion of the Bondi mass of an isolated gravitating system, and the mass loss due to energy carried away by gravitational waves with a positive cosmological constant, Λ > 0 -since we now know that our universe expands at an accelerated rate (which is well described by Einstein's theory with Λ > 0).Although Bondi et al. led the initial breakthrough to this problem for the asymptotically flat case [2,15], the mass-loss formula for Λ = 0 could also be obtained using the Newman-

Gravitational radiation with Λ>0

“…Subsequently, various authors used the Newman-Penrose formalism to define and study asymptotically de Sitter spacetimes [52][53][54]. The two earlier papers by Saw used a special choice of null foliation, thereby excluding the Robinson-Trautman spacetime with a positive cosmological constant as part of their allowed class of spacetimes.…”

A review of total energy–momenta in GR with a positive cosmological constant

“…However, to derive this, one should solve the Newman-Penrose spin coefficient equations to an appropriately high order. This solution, both for the vacuum and electro-vacuum in the physical spacetime, is given by Saw in [68] and [69], respectively (see also [70]). The analogue of the mass-loss formula is derived from the Bianchi identity in [68,69], using an integral identity in [73].…”

The peeling theorem with arbitrary cosmological constant

“…There are much more recent works that aim at a generalisation of the peeling property in the presence of a non-vanishing cosmological constant without using explicitly a conformal completion of the space-time. The results presented in [8,9] rely on assumptions that otherwise can be deduced if a conformal completion is used, while the work in [10] focuses on a particular kind of metrics of the Bondi-Sachs kind and imposes boundary conditions at infinity; if a specific metric does not admit a conformal compactification with boundary J, our results cannot be applied to that case.…”