Local Origin of Hidden Symmetry in Rotating Spacetimes

TL;DR

This paper demonstrates that Kerr-type hidden symmetries in rotating spacetimes originate from local Einstein equations constraints, revealing a universal local structure underlying Kerr geometry.

Contribution

It shows that Kerr-like separability and symmetry emerge as local consequences of Einstein equations without global assumptions or special symmetries.

Findings

Local Einstein equations enforce a projective alignment in stationary-axisymmetric geometries.

A universal classification of local solutions into M"obius, exponential, and trigonometric branches.

Global regularity selects the Kerr-type sector from local solutions.

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 M\"obius, 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.