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

We derive the asymptotic solutions for vacuum spacetimes with non-zero cosmological constant $\Lambda$, 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 $\sigma^o$) must be non-zero. After solving the vacuum Newman-Penrose equations asymptotically, we propose a generalisation to the Bondi mass involving $\Lambda$ and obtain a positive-definite mass-loss formula by integrating the Bianchi identity involving $D'\Psi_2$ over a compact 2-surface on $\mathcal{I}$. Whilst our original intention was to study asymptotically de Sitter spacetimes, the use of spherical coordinates implies that this readily applies for $\Lambda<0$, and yields exactly the known asymptotically flat spacetimes when $\Lambda=0$. In other words, our asymptotic vacuum solutions with $\Lambda\neq0$ reduce smoothly to those where $\Lambda=0$, in spite of the distinct characters of $\mathcal{I}$ being spacelike, timelike and null for de Sitter, anti-de Sitter and Minkowski, respectively. Unlike for $\Lambda=0$ where no incoming radiation corresponds to setting $\Psi^o_0=0$ on some initial null hypersurface, for $\Lambda\neq0$, no incoming radiation requires $\Psi^o_0=0$ everywhere.