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?


1 Answer 1


All you need to do is set up a multipole expansion of the gravitational waveform. You'll find that the monopole moment is proportional to the time derivative of the mass of the stress-energy tensor, and the dipole moment is proportional to the second time derivative of the momentum from the stress-energy tensor, both of which are conserved. Thus, the first nonzero moment comes from the quadrupole moment. This is worked out in great detail in MTW.


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