On asymptotically flat solutions of Einstein's equations periodic in time: II. Spacetimes with scalar-field sources

We extend the work in our earlier paper (Bičák J et al 2010 Class. Quantum Grav. 27 055007) to show that time-periodic, asymptotically flat solutions of the Einstein equations analytic at {\\cal I}, whose source is one of a range of scalar-field models, are necessarily stationary. We also show that, for some of these scalar-field sources, in stationary, asymptotically flat solutions analytic at {\\cal I}, the scalar field necessarily inherits the symmetry. To prove these results we investigate miscellaneous properties of massless and conformal scalar fields coupled to gravity, in particular Bondi mass and its loss.