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

TL;DR

This paper proves that time-periodic, asymptotically-flat solutions of Einstein's equations with scalar-field sources are necessarily stationary and that the scalar field inherits the symmetry in stationary solutions, extending previous work.

Contribution

It extends prior results to show scalar-field sourced solutions are stationary and scalar fields inherit symmetry, providing new insights into Einstein-scalar field systems.

Findings

Time-periodic solutions are necessarily stationary.

Scalar fields inherit symmetry in stationary solutions.

Analysis of Bondi mass and its loss for scalar fields.

Abstract

We extend the work in our earlier article [4] to show that time-periodic, asymptotically-flat solutions of the Einstein equations analytic at scri, 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 scri, 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.