Birkhoff's theorem requires the use of a spherical shell as the test mass

UPDATE January 20, 2022: Is the spatial metric inside a spherical shell not stretched? If we simply add linearly the metric perturbations caused by each particle in the shell, the spatial metric must be stretched inside the shell. The speed of light is the same to all directions, but distances might have uniform stretching? We have to check the literature, and analyze the metric inside a spherical shell.

At the first sight, linearly adding perturbations is not the right way, because energy shipping happens in the combined field of the mass shell. The spatial metric is not stretched. So, no problem.

We also removed the incorrect claim that pressure inside a spherically symmetric mass can attract a point test mass outside the mass.

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In our November 6, 2021 blog post we were worried that Birkhoff's theorem may be broken, because varying pressure inside a spherically symmetric mass may vary tidal forces on a point test mass m outside the spherical mass. Thus, the metric, as seen by a point test mass, can change outside a spherically symmetric mass. Is Birkhoff's theorem broken?

Now we realize that one must use a spherical shell of mass as the test mass in Birkhoff's theorem. A point test mass m would break the spherical symmetry of Birkhoff's theorem.

A spherical shell test mass does not cause tidal forces.

Is a spherically symmetric collapse to a black hole extremely slow?

In the syrup model of gravity on January 17, 2022 we remarked that an elephant can run fast through the syrup. That is, the merger of two equal-sized large masses happens very fast.

A large mass drags the frame with it and can move much faster than a small particle.

But in a spherically symmetric collapse there probably is no frame-dragging. If a spherically symmetric system of electric charges collapses, there is no magnetic field. On the other hand, a speeding single charge has a significant magnetic field.

We argued that even a 1 MeV photon passes the event horizon quickly, because it cannot acquire more inertia than exists in the whole black hole. The argument does not work well with a contracting shell of mass, because the total momentum of the shell is zero.

It might be that a spherically symmetric collapse into a black hole happens extremely slowly in the global Minkowski time. We need to study this.