Birkhoff rigidity from a covariant optical seed

Abstract

We present a local seed-to--Kerr--Schild route to Birkhoff rigidity in four-dimensional spherical vacuum gravity. On the two-dimensional orbit space, the areal radius \(r\) determines a scalar \(F:=-(\nabla r)^2\), and the reduced vacuum equations imply \(F(r)=1-2M/r\). We show that the normalized one-forms \(dr/F\) and \((*dr)/F\) are closed, so that the null combinations \(F^{-1}(dr\pm *dr)\) are exact null seed forms. Integrating these yields local Eddington--Finkelstein coordinates in which the metric takes Kerr--Schild form over a flat background. We then prove the corresponding uniqueness statement in the stationary optical sector: spherical symmetry forces the inverse optical seed \(\mathcal R\) to equal \(\pm r\), equivalently the optical seed \(\rho\) to equal \(\mp 1/r\), and the resulting seed data reconstruct the Schwarzschild family. Thus, Birkhoff rigidity is paired with a spherical converse theorem in the stationary optical framework: Schwarzschild is the unique spherically symmetric stationary vacuum Kerr--Schild geometry generated by a nowhere-vanishing optical seed.