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 $${\mathrm{\Phi }}_L:(t^{\mathrm{\prime }},x^{\mathrm{\prime }},y^{\mathrm{\prime }},z^{\mathrm{\prime }})\mapsto (t,x,y,z)=(L{t}^{\mathrm{\prime }},L{x}^{\mathrm{\prime }},L{y}^{\mathrm{\prime }},L{z}^{\mathrm{\prime }}),L>1,$$ and show that it is a smooth, orientation-preserving $C^{\mathrm{\infty }}$ diffeomorphism of flat spacetime. Pulling the MHD conservation laws back through ${\mathrm{\Phi }}_L$, we prove that the closed ideal-MHD system is form-invariant (covariant), with the current density rescaling as $\mathit{j}={\mathit{j}}^{\mathrm{\prime }}\mathrm{/}L$, the magnetic diffusivity as ${\eta }^{\mathrm{\prime }}=\eta \mathrm{/}L$, and external sources rescaled by $L$. The transformation realises three physically distinct effects on the perturbed flow: a \emph{linearization} to the tangent-space (linearised MHD) system, a \emph{movie slow-motion} dilation of dissipative and dynamical timescales, and an \emph{electron-microscope} magnification of small-scale structure. Taking the formal limit $L\to \mathrm{\infty }$ 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.