Axial Obstructions to Rotating Bumblebee Vacuum Solutions

Abstract

We establish two obstructions to globally regular rotating constant-norm vector vacua. At an axial fixed point of a regular nondegenerate bifurcate Killing horizon, horizon-boost and axial invariance force every smooth symmetry-inheriting one-form to vanish, contradicting a strictly nonzero constant norm. Independently, varying axial conicity produces an orthonormal curvature component diverging as the inverse proper distance and precludes a \(C^2\) extension. Applied to a three-parameter Kerr--disformal family in Einstein--bumblebee gravity, these results reveal a nonsmooth bumblebee one-form, while direct calculation shows that the Kretschmann scalar diverges as the inverse square of the transverse proper distance to either open exterior axis. Even the distinguished nonextremal branch selected by nonpolar outer-horizon regularity is therefore an exact nonpolar rotating exterior solution, not a globally regular isolated black hole. Together, these field- and metric-level obstructions provide a two-pronged no-go framework for globally regular rotating constant-norm vector vacua.

Minor corrections, references added, conclusions unchanged