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 $\Lambda$, 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 $D^{\prime }{\Psi }_2$ 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 $\Lambda \ne 0$ 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 ${\Psi }_0^o=0$ on some initial null hypersurface, for $\Lambda \ne 0$, no incoming radiation requires ${\Psi }_0^o=0$ everywhere.

38 pages, 1 figure, this is the final version, accepted by Physical Review D

References in corpus (1)

• GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence

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