A Spacetime-Diffeomorphic Approach to Magnetohydrodynamics around Black Holes: Dilation Covariance, Linearization,and the No-Hair Limit
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 ΦL: (t',x',y',z')↦ (t,x,y,z)=(Lt',Lx',Ly',Lz'), L>1, and show that it is a smooth, orientation-preserving C∞ diffeomorphism of flat spacetime. Pulling the MHD conservation laws back through ΦL, we prove that the closed ideal-MHD system is form-invariant (covariant), with the current density rescaling as j=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→∞ 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.
Related
- Chance-Constrained Nonlinear Covariance Control via Robust Linearization Remainder Bounds
- Diffeomorphism Invariance of the Favre-Filtered Compressible Navier--Stokes System under Spacetime Dilation
- New families of moment-angle manifolds diffeomorphic to connected sums of products of spheres
- Knotted tori in the four-sphere with diffeomorphic exteriors
- ConSEAL: Connectivity-Informed Streamline Endpoint Alignment for Diffeomorphic Cortical Registration
BibTeX
@misc{li2026spacetimediffeomorphic,
title = {{A Spacetime-Diffeomorphic Approach to Magnetohydrodynamics around Black Holes: Dilation Covariance, Linearization,and the No-Hair Limit}},
author = {Yuanya Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04983},
eprint = {2610.04983},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04983},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Spacetime-Diffeomorphic Approach to Magnetohydrodynamics around Black Holes: Dilation Covariance, Linearization,and the No-Hair Limit
AU - Yuanya Li
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.04983
UR - https://arxiv.org/abs/2610.04983
AB - 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')↦ (t,x,y,z)=(Lt',Lx',Ly',Lz'), L>1, and show that it is a smooth, orientation-preserving C^∞ 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 j=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→∞ 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.
ER -