A Spacetime-Diffeomorphic Approach to Magnetohydrodynamics around Black Holes: Dilation Covariance, Linearization,and the No-Hair Limit
The ideal magnetohydrodynamic (MHD) system that governs a conducting plasma---the Maxwell equations together with the conservation laws of mass, momentum and energy---is the same local balance structure that describes hot accretion flows around black holes. We introduce a uniform four-dimensional spacetime dilation \[ Φ_L:\;(t',x',y',z')\mapsto (t,x,y,z)=(Lt',Lx',Ly',Lz'),\qquad L>1, \] and show that it is a smooth, orientation-preserving C^infty diffeomorphism of flat spacetime. Pulling the MHD conservation laws back through Φ_L, we prove that the closed ideal-MHD system is form-invariant (covariant), with the current density rescaling as bm j=bm j'/L, the magnetic diffusivity as η'=η/L, and external sources rescaled by L. The transformation realises three physically distinct effects on the perturbed flow: a linearization to the tangent-space (linearised MHD) system, a movie slow-motion dilation of dissipative and dynamical timescales, and an electron-microscope magnification of small-scale structure. Taking the formal limit Ltoinfty localises the dynamics on a neighbourhood of a horizon: resistivity vanishes, the asymptotic time derivative freezes, and all transient multipolar structure of the infalling plasma redshifts away. The remaining stationary, axisymmetric, asymptotically flat electrovacuum is uniquely the Kerr--Newman geometry, so that the exterior of the compact object is characterised only by its mass M, angular momentum J and charge Q. The result provides a local, conservation-law-based perspective on the black hole no-hair theorem that is complementary to the classical Israel--Carter-- Robinson theorems and to contemporary gravitational-wave and Event Horizon Telescope tests.
