The probe limit in MHD and its implications for magnetic transport

TL;DR

This work interrogates the validity of the relativistic MHD probe limit, where electromagnetic fluctuations decouple from energy-momentum dynamics, through both a high-form hydrodynamic EFT and a holographic AdS/CFT model. It demonstrates that the probe limit is approximate and consistent decoupling generally conflicts with exact conservation of $T^{ ho u}$ in a background magnetic field; energy-momentum transport can feed back into magnetic diffusion, altering dispersion relations. The holographic implementation provides explicit transport coefficients, revealing that Hall transport can arise from background charge density and that backreaction can trigger phase-structure changes via a magnetic BF bound. The findings have implications for dense nuclear matter in neutron stars, where full MHD dynamics may be essential to correctly capture magnetic diffusion, Alfvénic propagation, and Hall effects. Overall, the paper clarifies the limits of the probe approach and offers quantitative holographic insights into the full MHD dynamics in strong-field environments.

Abstract

Many phenomenological and effective field-theoretical (EFT) applications of magnetohydrodynamics (MHD) in the presence of a background magnetic field employ a simplifying assumption whereby the electromagnetic and the energy-momentum fluctuations decouple. In studies of magnetic transport, for example in magnetic diffusion, the conservation of energy and momentum is then neglected. In this paper, we investigate the details and the consistency of this so-called $\textit{probe limit}$ in different parametric regimes of MHD plasmas. In the first part of the paper, our discussion explores the hydrodynamic (higher-form) theory of MHD. In the second part, we then explicitly test the probe limit by using a microscopic holographic (AdS/CFT) model of a strongly coupled plasma. In the process, we develop the holographic Schwinger-Keldysh EFT prescription for describing the bulk 2-form fields and their dual 1-form symmetries. Moreover, we find evidence of a phase transition at low temperatures and show that magnetic Hall transport can emerge as a consequence of background charge density that breaks the charge conjugation symmetry of the state. Finally, we discuss the implications for magnetic transport, with a particular view towards the dynamics of dense nuclear matter in neutron stars.

Figures (10)

• Figure 1: The complex radial contour connecting the two AdS boundaries of an eternal black hole, upon which the EOMs \ref{['eq:chEOM']} are defined. $r_c$ is the stretched horizon upon which the extra boundary condition \ref{['eq:dss_condition']} is imposed.
• Figure 2: Energy density $\varepsilon$ and pressure $p$ for neutral magnetic branes ($n_\text{phys}=0$) as functions of dimensionless temperature $T/\sqrt{\rho_\text{phys}}$ for different values of the renormalization constant $\bar{\alpha}$: $\infty$ (solid), $1/2\pi$ (dashed) and $1/137$ (dotted). $\rho_\text{phys}$ is abbreviated to $\rho$ in the plots.
• Figure 3: Energy density $\varepsilon$ and pressure $p$ for charged magnetic branes ($n_\text{phys}/\rho_\text{phys}^{3/2}=1$) as functions of dimensionless temperature $T/\sqrt{\rho_\text{phys}}$ for different values of the renormalization constant $\bar{\alpha}$: $\infty$ (solid), $1/2\pi$ (dashed) and $1/137$ (dotted). $\rho_\text{phys}$ is abbreviated to $\rho$ in the plots.
• Figure 4: Enthalpy density $w=\varepsilon+p$ for neutral (left) and charged (right) magnetic branes ($n_\text{phys}/\rho_\text{phys}^{3/2}=1$) as functions of dimensionless temperature $T/\sqrt{\rho_\text{phys}}$ for different values of the renormalization constant $\bar{\alpha}$: $\infty$ (solid), $1/2\pi$ (dashed) and $1/137$ (dotted). $\rho_\text{phys}$ is abbreviated to $\rho$ in the plots.
• Figure 5: Parallel resistivity for neutral (left) and charged (right, $n_\text{phys}/\rho^{3/2}_\text{phys}=1$) magnetic branes, as a function of dimensionless temperature. The transport coefficients are normalized using the resistivity for a Schwarzschild black brane $r_0$. The gray lines correspond to the probe limit expressions \ref{['eq:pblrs']}, whereas the dashed black lines to the values calculated via \ref{['eq:hk_parz']}.