Peeling property for the Einstein scalar field equations with the nonzero cosmological constant

Abstract

Inspired by interaction of gravitational waves, dark matters and dark energy, we study the Einstein scalar field equations with the nonzero cosmological constant for the Bondi-Sachs metrics. We provide the asymptotic expansions under the outgoing radiation condition. Unlike the case of the zero cosmological constant, the asymptotic expansions depend on four additional $Λ$-independent functions $B(u, θ, ϕ)$, $\tilde{X}(u, θ, ϕ)$, $\tilde{Y}(u, θ, ϕ)$ and $\tilde{I}(u)$. We show that, under suitable coordinate transformation, we can make $B(u, θ, ϕ)=0$. We prove the peeling property for the Einstein scalar field equations if $\tilde{I}=0$, and derive a loss formula of the Bondi energy-momentum for the nonzero cosmological constant. We remark that, for certain real data from gravitational waves, the loss formula is likely to indicate the loss property of the Bondi energy-momentum.