I am reading the original paper by Bondi, van der Berg and Metzner (link) regarding gravitational waves in asymptotically flat axisymmetric spacetimes. In the introduction, he makes the following comment -
The conservation of mass effectively prohibits purely spherically symmetric waves and similarly, conservation of momentum prohibits waves of dipole symmetry.
I know that Birkhoff's theorem tells us that spherically symmetric asymptotically flat solution to GR is necessarily static, and therefore contains a timelike Killing vector, which implies conservation of mass (energy). Bondi et. al. seem to be stating the converse of this theorem, whose validity I do not immediately see. How do we show this?
Also, what is the corresponding proof of the second statement made above?