Someone argues: "Birkhoff's theorem says that the exterior of any spherically…
PHYSICS · Subjective · Fermi University Physics · Relativity & Gravitation · Schwarzschild spacetime and the tests of general relativity · Schwarzschild spacetime and the tests of general relativity
The Geometry of Gravity, Ch The Schwarzschild geometry and Birkhoff's theorem
Problem
Someone argues: "Birkhoff's theorem says that the exterior of any spherically symmetric mass distribution in vacuum is the static Schwarzschild metric, with a fixed mass parameter M. The exterior is static, so nothing about the star can change with time. In particular, a spherically symmetric star cannot collapse, because collapse would make the metric depend on time, and that contradicts the theorem." Say whether this argument is correct. If it is not, identify the exact step where it fails, and state precisely what Birkhoff's theorem does and does not constrain for a spherical star whose radius R(t) is shrinking with time while its total mass M stays constant.