Birkhoff rigidity from a covariant optical seed

2026/04/10 by D. A. Easson
#gr-qc #hep-th

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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:=-(∇ 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± *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 ± r, equivalently the optical seed ρ to equal ∓ 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.

Citations

• On the general spherically symmetric solutions of Einstein's gravitational equations in vacuo
• Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Collapse

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