Bondi mass with a cosmological constant

The mass loss of an isolated gravitating system due to energy carried away by gravitational waves with a cosmological constant $\mathrm{\Lambda }\in \mathbb{R}$ was recently worked out, using the Newman-Penrose-Unti approach. In that same article, an expression for the Bondi mass of the isolated system, $M_{\mathrm{\Lambda }}$, for the $\mathrm{\Lambda }>0$ case was proposed. The stipulated mass $M_{\mathrm{\Lambda }}$ would ensure that in the absence of any incoming gravitational radiation from elsewhere the emitted gravitational waves must carry away a positive-definite energy. That suggested quantity, however, introduced a $\mathrm{\Lambda }$-correction term to the Bondi mass $M_B$ (where $M_B$ is the usual Bondi mass for asymptotically flat spacetimes), which would involve information not just on the state of the system at that moment but ostensibly also its past history. In this paper, we derive the identical mass-loss equation using an integral formula on a hypersurface formulated by Frauendiener based on the Nester-Witten identity and argue that one may adopt a generalization of the Bondi mass with $\mathrm{\Lambda }\in \mathbb{R}$ without any correction, viz., $M_{\mathrm{\Lambda }}=M_B$ for any $\mathrm{\Lambda }\in \mathbb{R}$. Furthermore, with $M_{\mathrm{\Lambda }}=M_B$, we show that for purely quadrupole gravitational waves given off by the isolated system (i.e., when the “Bondi news” ${\sigma }^o$ comprises only the $l=2$ components of the spherical harmonics with spin-weight 2) the energy carried away is manifestly positive definite for the $\mathrm{\Lambda }>0$ case. For a general ${\sigma }^o$ having higher multipole moments, this perspicuous property in the $\mathrm{\Lambda }>0$ case still holds if those $l>2$ contributions are weak—more precisely, if they satisfy any of the inequalities given in this paper.