Mass-loss of an isolated gravitating system due to energy carried away by gravitational waves with a cosmological constant

Abstract

We derive the asymptotic solutions for vacuum spacetimes with non-zero cosmological constant Λ, using the Newman-Penrose formalism. Our approach is based exclusively on the physical spacetime, i.e. we do not explicitly deal with conformal rescaling nor the conformal spacetime. By investigating the Schwarzschild-de Sitter spacetime in spherical coordinates, we subsequently stipulate the fall-offs of the null tetrad and spin coefficients for asymptotically de Sitter spacetimes such that the terms which would give rise to the Bondi mass-loss due to energy carried by gravitational radiation (i.e. involving σ o ) must be non-zero. After solving the vacuum Newman-Penrose equations asymptotically, we propose a generalisation to the Bondi mass involving Λ and obtain a positive-definite mass-loss formula by integrating the Bianchi identity involving D Ψ 2 over a compact 2-surface on I.Whilst our original intention was to study asymptotically de Sitter spacetimes, the use of spherical coordinates implies that this readily applies for Λ < 0, and yields exactly the known asymptotically flat spacetimes when Λ = 0. In other words, our asymptotic vacuum solutions with Λ = 0 reduce smoothly to those where Λ = 0, in spite of the distinct characters of I being spacelike, timelike and null for de Sitter, anti-de Sitter and Minkowski, respectively. Unlike for Λ = 0 where no incoming radiation corresponds to setting Ψ o 0 = 0 on some initial null hypersurface, for Λ = 0, no incoming radiation requires Ψ o 0 = 0 everywhere. * VeeLiem@maths.otago.ac.nz arXiv:1605.05151v3 [gr-qc]

Mass loss due to gravitational waves with Λ > 0

“…[5]). A positive cosmological constant Λ > 0 provides a simple explanation for the accelerated rate of expansion.…”

“…[5], the mass-loss formula with Λ > 0 (obtained from that same Bianchi identity as in the case for Λ = 0) is:…”

“…As a result, the integration is not carried out over a round 2-sphere -but instead over a topological 2-sphere of constant u on I [5,9] [10]. These compact 2-surfaces of constant u on I being topological instead of round 2-spheres give rise to the second term on the right-hand side of Eq.…”

“…the mass of an isolated gravitating system strictly decreases due to energy carried away by gravitational waves [5].…”

“…As opposed to the case where Λ = 0, here I is non-conformally flat for Λ > 0 -its Cotton-York tensor is non-zero, when those outgoing gravitational waves carry energy away from the source [5,8]. As a result, the integration is not carried out over a round 2-sphere -but instead over a topological 2-sphere of constant u on I [5,9] [10].…”