Spontaneous charge separation in accelerating relativistic plasmas

TL;DR

This work derives a relativistic electrothermal stratification framework for plasmas under constant acceleration by maximizing entropy in kinetic theory, yielding a governing PDE for the rest-frame potential $\varphi$ in curved spacetime: $K^2 \nabla_\mu( K^{-2} \nabla^\mu \varphi) = - q K (n_p - n_e)$. The proton and electron densities are given by Maxwell-J"uttner distributions with redshifted temperature $T_* = -1/\alpha^U$ and chemical potentials set by the external field and Killing flow, $n_p \propto e^{- q \varphi / T_*} K_2( m_p T_*^{-1} K )$ and $n_e \propto e^{+ q \varphi / T_*} K_2( m_e T_*^{-1} K )$. Solving in a uniformly accelerated box (Rindler) and a shell near a Schwarzschild horizon shows nonzero electric fields and charge separation persist, independent of transport coefficients, thereby extending Stewart-Tolman and Tolman effects to relativistic, curved-spacetime plasmas and highlighting MHD's limitations in extreme environments.

Abstract

The Stewart-Tolman effect posits that accelerating conductors exhibit both charge separation and rest-frame electric fields (``inertia of charge''), while the Ehrenfest-Tolman effect states that acceleration induces temperature gradients (``inertia of heat''). We study the interplay of these effects in thermodynamic equilibrium. Specifically, we derive from first principles a partial differential equation governing the electrothermal stratification of a fully ionized plasma in equilibrium under irrotational relativistic accelerations in curved spacetime. We then solve it in two settings: a plasma enclosed in a uniformly accelerated box, and a plasma shell suspended above a black hole horizon. The resulting electric fields are found not to depend on the electric conductivity of the medium.

Figures (4)

• Figure 1: Equilibrium electric field $\mathcal{E}$ (left panel) and charge separation $(n_p{-}n_e)/n_e$ (right panel) in a column of fully ionized hydrogen plasma undergoing acceleration $g$. This configuration solves the system \ref{['NewtonianStrat']}-\ref{['Coulomblaw']} with $\mathcal{Z}_p=\mathcal{Z}_e=20 q^2 T/(m_p^2 g^2)$, for a plasma confined in a region with height $10T/(m_p g)$. The boundary conditions are global charge neutrality plus the absence of externally imposed electric fields, which gives $(\partial_z\varphi)_{\text{Boundary}}{=}0$. The acceleration number for this configuration is $\mathscr{A}\sim 5$, so MHD does not apply (see footnote \ref{['footononon1']}).
• Figure 2: Qualitative Minkowski diagram (in Cartesian coordinates) illustrating the equilibrium configuration of a plasma in a container undergoing constant acceleration. The walls of the container (gray) follow hyperbolic motion, which is Born-rigid. Due to their greater mass, protons accumulate towards the rear, leading to a buildup of positive charge (red), while the lighter electrons distribute more evenly throughout the chamber, resulting in an excess of negative charge (blue) towards the front. As a consequence, the plasma is crossed by an electric field (yellow arrows) in its local rest frame. The dashed lines mark the boundary of the Rindler wedge, outside of which the curvilinear coordinates $\tau$ and $z$ lose any meaning.
• Figure 3: Equilibrium electric field (left panels) and charge separation (right panels) of a plasma column undergoing hyperbolic motion as in figure \ref{['fig:Qualitative']}. Each row corresponds to a different solution of \ref{['rescadina']}, with $\{\Tilde{z}_-,\Tilde{z}_+\}$ equal to $\{20,30\}$ (upper panels), $\{1,11\}$ (middle panels), and $\{0.01,10.01\}$ (lower panels). All solutions assume that $\Tilde{\mathcal{Z}}_p=20$ and $\Tilde{\mathcal{Z}}_e=20 \, K_2(\frac{m_e}{m_p}\Tilde{z}_-)$.
• Figure 4: Equilibrium electric field (left panels) and charge separation (right panels) of a plasma shell suspended above the event horizon of a Schwarzschild black hole. The two rows corresponds to different solutions of \ref{['eugualeaprima']}, with $\{\Tilde{r}_-,\Tilde{r}_+\}$ equal to $\{1.1,3\}$ (upper panels), and $\{1.001,3\}$ (lower panels). Both solutions assume that $m_p=10 T_\star$ and $\mathcal{Z}_{p/e}=20 \frac{r_s^2 q^2}{T_\star} K_2(\frac{m_{p/e}}{T_\star}\sqrt{1{-}\frac{r_s}{r_-}})$.