On the Bondi mass of Maxwell-Klein-Gordon spacetimes

In this paper we calculate the Bondi mass of asymptotically flat spacetimes with interacting electromagnetic and scalar fields. The system of coupled Einstein-Maxwel-Klein-Gordon equations is investigated and corresponding field equations are written in the spinor form and in the Newman-Penrose formalism. Asymptotically flat solution of the resulting system is found near null infinity. Finally we use the asymptotic twistor equation to find the Bondi mass of the spacetime and derive the Bondi mass-loss formula. We compare the results with our previous work \cite{BST2} and show that, unlike the conformal scalar field, the (Maxwell-)Klein-Gordon field has negatively semi-definite mass-loss formula.