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Gravitational waves are produced by an accelerated mass, similar to the production of light waves by an accelerated charge. The amount of gravitational energy released from a rotating object can be obtained by a quadrupole formula.

Consider the ordinary Schwarzschild metric of a non-rotationg and uncharged black hole and instead of the Schwarzschild radius $r_s$ plug in a slightly modified time-dependent radius of the form $$r_0(t)=r_s+\epsilon \,\sin(\omega t),$$ with $\epsilon\ll r_s$ and $\omega$ constant.

Does such a system emit gravitational waves? If yes, what could be said about their energy?

EDIT: Note that the time dependent metric corresponding to $r_0(t)$ is not thought to be a solution of Einsteins vaccum equations anymore. Classical solutions of the vacuum equations have a constant mass $M$, but this is not the case in the approach above. This is why here Birkhoff's theorem does not play any role.

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    $\begingroup$ The Birkhoff theorem forbids it $\endgroup$ – Yukterez Jul 21 '18 at 8:04
  • $\begingroup$ @СимонТыран A bit more explanation would make a good answer. $\endgroup$ – StephenG Jul 21 '18 at 8:10
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As mentioned in the comments Birkhoff’s theorem states that any spherically symmetric solution to the field equations is necessarily static and asymptotically flat. A big implication of this is the fact that the geometry exterior to a spherically symmetric matter distribution such as a star must be static and hence cannot emit gravitational radiation. A recent article by Hill & O’Leary explicitly shows that seeking radially and time dependent solutions given a spherically symmetric space time confirms Birkhoffs theorem.

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  • $\begingroup$ Actually it’s more a consequence of the theorem rather than an assumption. If you take assume a spherically symmetric line element where the metric tensor components are also spherically symmetric then you will recover the Schwarzschild solution. Or, at least a solution that is locally isometric to the Schwarzschild solution. $\endgroup$ – Rumplestillskin Jul 21 '18 at 11:33

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