# Gravitational waves of an oscillating Schwarzschild black hole

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.

• The Birkhoff theorem forbids it – Yukterez Jul 21 '18 at 8:04
• @СимонТыран A bit more explanation would make a good answer. – StephenG Jul 21 '18 at 8:10