The peeling property of Bondi-Sachs metrics for nonzero cosmological constants

Abstract

In this paper, we show that the peeling property still holds for Bondi-Sachs metrics with nonzero cosmological constant under the boundary condition given by Sommerfeld's radiation condition together with three nontrivial Λ-independent functions B, a, b. This should indicate the new boundary condition is natural. Moreover, we construct some nonstationary vacuum Bondi-Sachs metrics without Bondi news, which Newmann-Penrose quantities fall faster than usual. This provides a new feature of gravitational waves for nonzero cosmological constant.

Energy-Momentum in General Relativity

“…The peeling property for Λ = 0 was proved previously by Penrose in framework of conformal compactification [63], and by Saw without conformal compactification [61,62]. It was pointed out in [52] that the induced metric on conformal boundary J + are e 2B Λ 3 du 2 + 2Y dudθ + 2Xdudφ + dθ 2 + sin 2 θdφ 2 . (4.9)…”

“…So the boundary condition given in [51,52] does not seem to consist with Penrose's framework. This new boundary condition is natural in three aspects that it consists with nontrivial Bondi news, gives rise to the peeling property and features B-gravitational waves without Bondi news.…”

“…, 4, be the Newmann-Penrose quantities. It was proved in [52] that, under Sommerfeld's radiation condition together with the nontrivial B, X, Y , the following peeling property holds…”

“…where coefficients Ψ 5−k This indicates that there exist gravitational waves without Bondi news, which may be referred as B-gravitational waves. In [52], some nonstationary vacuum Bondi-Sachs metrics are constructed with γ = δ = 0. Hence c = d = 0.…”

Asymptotics with a cosmological constant: The solution space

“…One can see the cosmological constant appears for the first time in (72). But one can just apply the same method as the first two groups to solve out µ, λ, and Ψ 2 , which are (20), (21), and (13). Then, γ is derived from (70) as (22), U is derived from (74) as (26), Ψ 3 is derived from (80) as (14), ν is derived from (73) as (23), and finally Ψ 4 is derived from (81) as (15).…”