We can break Birkhoff's theorem with a sphere which "flexes its muscles" - or break the geodesic hypothesis

In November 2021 we wrote several blog entries about how "tidal" effects can break the metric of general relativity, and Birkhoff's theorem. Here we present yet another example.

Suppose that we have a small test mass m. It possesses the Schwarzschild solution around it. The spatial metric of the solution is not absolutely flat around the mass m.

Let us then imagine a very light sphere, which through some spherically symmetric mechanism can make itself more rigid when it wants. For example, it may contain liquid. When the sphere tightens its surface, it becomes more rigid overall.

              O                     •    test mass m

         sphere

When the sphere is more rigid it pushes m outward with a greater force.

Here we have an example of how gravity together with other force fields can cause unexpected forces. Since the force changes, the "metric" around the sphere changes, which breaks the Jebsen-Birkhoff theorem.

https://en.wikipedia.org/wiki/Birkhoff%27s_theorem_(relativity)

One could claim that we are cheating because the sphere is not totally symmetric; rather it is a little bit distorted by m. There is a "tidal" effect. However, the same objection applies to any application of Birkhoff's theorem with a pointlike test mass. Our November 2021 blog entries point out that using a spherical shell as a test mass around the sphere would save Birkhoff's theorem. There would be no tidal effects.

Let us check various proofs of the Jebsen-Birkhoff theorem. They probably assume that the "coupling" of gravity to other force fields is local and very simple. In our example, the coupling is through the structure of the sphere. The coupling is not local.

The theorem states that a spherically symmetric solution to the Einstein equatioms in a vacuum must be static and asymptotically flat.

The proofs do not assume anything about the couplings. They simply use the Einstein equations. The theorem is true.

https://en.wikipedia.org/wiki/Geodesics_in_general_relativity

We could say that our counterexample does not break Birkhoff's theorem. Rather, it breaks the geodesic hypothesis that a point mass obeys the metric derived from the Einstein equations.