Local Origin of Hidden Symmetry in Rotating Spacetimes

Abstract

We show that, within a broad stationary-axisymmetric class, Kerr-type separability and hidden symmetry arise as a local consequence of the Einstein equations. Without assuming separability, algebraic speciality, Killing--Yano symmetry, or global boundary conditions, we analyze stationary and axisymmetric geometries in a locally non-rotating orthonormal frame and impose a minimal local equilibrium condition, namely the absence of mixed energy-momentum fluxes. We find that the mixed Einstein equations enforce a rigid projective alignment between radial and angular sectors, uniquely characterized by a constant-Schwarzian constraint. This constraint yields a universal classification of local solutions into Möbius, exponential, and trigonometric branches, of which global regularity selects precisely the Kerr-type sector. In this sense, the kinematical core of Kerr geometry is already fixed locally, and the Schwarzian structure provides the local origin of Kerr rigidity.

22 pages, No figure, Add re-derivation of the key equation in appendix C