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Thermoelectric Conduction in General Relativity: A Causal, Stable, and Well Posed Theory

Lorenzo Gavassino

Abstract

We present a covariantly stable first-order framework for describing charge and heat transport in isotropic rigid media embedded in curved spacetime. Working in the Lorenz gauge, we show that the associated initial value problem is both causal and locally well-posed in the fully nonlinear regime. We then apply such framework to explore a range of gravitothermoelectric effects in metals undergoing relativistic acceleration. These include (1) the separation of charge through acceleration, (2) the non-uniformity of Joule heating across accelerating circuits due to time dilation, and (3) the effect of redshift on magnetic diffusion. As an astrophysical application, we derive a relativistic Thomas-Fermi equation governing the charge distribution inside a compact object, also accounting for Seebeck charge displacements driven by cooling.

Thermoelectric Conduction in General Relativity: A Causal, Stable, and Well Posed Theory

Abstract

We present a covariantly stable first-order framework for describing charge and heat transport in isotropic rigid media embedded in curved spacetime. Working in the Lorenz gauge, we show that the associated initial value problem is both causal and locally well-posed in the fully nonlinear regime. We then apply such framework to explore a range of gravitothermoelectric effects in metals undergoing relativistic acceleration. These include (1) the separation of charge through acceleration, (2) the non-uniformity of Joule heating across accelerating circuits due to time dilation, and (3) the effect of redshift on magnetic diffusion. As an astrophysical application, we derive a relativistic Thomas-Fermi equation governing the charge distribution inside a compact object, also accounting for Seebeck charge displacements driven by cooling.

Paper Structure

This paper contains 4 sections, 2 theorems, 44 equations, 6 figures.

Table of Contents

  1. Global mode structure
  2. The gapless mode describes heat transport
  3. Origin of the gap of the charge-diffusion mode
  4. Which modes are physical?

Key Result

Theorem 1

Suppose that themain holds, where $\sigma_i(T,\mu,x^\alpha)$ and $\kappa_i(T,\mu,x^\alpha)$ are such that $\sigma_1\kappa_3{-}\sigma_3 \kappa_1{\neq} 0$ everywhere. Then, in the Lorenz gauge ($\nabla_\mu A^\mu \,{=}\, 0$), Maxwell's system defines a well-posed Cauchy problem for the variables $\Psi{=}\{A^\nu,\mu,T\}$, given initial data on spacelike surfaces. Furthermore, the resulting dynamics p

Figures (6)

  • Figure 1: Continuity of the redshifted current. Under stationary conditions, the Lie derivative $\mathfrak{L}_{\partial_t} J$ vanishes, where $J^\nu$ is the current density, and $\partial_t$ is a Killing vector. Combining this with the conservation law $\nabla_\mu J^\mu{=}0$ and the Killing property $\nabla_\mu (\partial_t)^\mu{=}0$, we find that $\nabla_\mu [(\partial_t)^\mu J^\nu{-}J^\mu (\partial_t)^\nu]=0$. Integrating over the portion of wire in the figure, and applying Stokes' theorem, we obtain $\int_{\text{Area}_1} J^\mu(\partial_t)^\nu dS_{\mu \nu}=\int_{\text{Area}_2} J^\mu(\partial_t)^\nu dS_{\mu \nu}$, or, equivalently, $I_1 ||\partial_t||_1=I_2 ||\partial_t||_2$.
  • Figure 2: A rocket containing a circuit undergoes rigid motion in Minkowski space ($ds^2={-}d\mathcal{T}^2{+}dx^2{+}dy^2{+}d\mathcal{Z}^2$), with velocity $u^\mu$ proportional to the generator of boosts, $\mathcal{Z}\partial_\mathcal{T}{+}\mathcal{T}\partial_\mathcal{Z}$. In Rindler coordinates ($\mathcal{T}{=}z\sinh(gt)$, $\mathcal{Z}{=}z\cosh(gt)$), we have $u^\mu\partial_\mu =(gz)^{-1}\partial_t$, and the proper acceleration (blue arrows) of a piece of equipment is $a\,{=}\,1/z$, which increases towards the rear of the rocket, and diverges at Rindler's horizon ($z=0$).
  • Figure 3: Electric field (upper panel) and charge density (lower panel) of a solution of \ref{['ezparab']} with $\mathcal{E}^z(t{=}0)=\mathcal{E}^z(\text{boundary})=0$, for a piece of metal extending from $z=b$ to $z=2b$. The curves represent snapshots at $g\sigma_1 b t=0$ (gray), $0.01$ (red), $0.05$ (magenta), $0.1$ (blue), $\infty$ (dashed).
  • Figure 4: Solutions of \ref{['dzdzT']} with boundary conditions $z_1T(z_1){=}z_2 T(z_2)$. In the upper panel, we set $z_1/z_2{=}0.99$ and $\mathcal{J}^2/[\sigma_1\kappa_3T(z_2)]{=}10^5$. In the lower panel, we set $z_1/z_2{=}10^{-7}$, and $\mathcal{J}^2/[\sigma_1\kappa_3T(z_2)]=1$.
  • Figure 5: Density profile of a solution of \ref{['maxsphere']} with $\rho(t{=}0)=\text{const}$ and $\mathcal{E}_r(r{=}R_\star)=\text{const}$ (no charge escapes the star), both adjusted so that $Q/(R_\star\mu_0)=-10$. The curves represent snapshots at $\sigma_1 t=0$ (gray), $1$ (red), $3$ (blue), and $\infty$ (dashed). The dashed line also solves \ref{['debyeHuck']}.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof