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arXiv · 2610.04983

A Spacetime-Diffeomorphic Approach to Magnetohydrodynamics around Black Holes: Dilation Covariance, Linearization,and the No-Hair Limit

Yuanya Li
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Abstract

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 \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 $L\to\infty$ 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.

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Yuanya Li. 2026-10-04. A Spacetime-Diffeomorphic Approach to Magnetohydrodynamics around Black Holes: Dilation Covariance, Linearization,and the No-Hair Limit. https://arxiv.org/abs/2610.04983

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