# 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

## 1 Answer

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.

• 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. – Rumplestillskin Jul 21 '18 at 11:33