General relativistic Lagrangian continuum theories -- Part II: electromagnetic fluids and solids with junction conditions

TL;DR

This work develops a covariant Lagrangian framework for relativistic electromagnetic continua, unifying fluids and solids under a Hamilton-principle approach in the material description and extending Part I to include polarization, magnetization, and elastic–electromagnetic coupling. By enforcing both spacetime and material covariance, it derives spacetime and convective reduced Lagrangians and covariant Euler-type balance equations, enabling a transparent computation of the total stress–energy–momentum tensor and natural coupling to gravity. The authors show how electromagnetic and gravitational junction conditions follow from the inclusion of Gibbons–Hawking–York boundary terms, yielding Israel–Darmois-like matching conditions at interfaces. They provide multiple equivalent forms of the stress–energy tensor (in terms of $E,B$ and in terms of the Faraday 2-form $F$) and illustrate the theory with relativistic Euler–Maxwell, linear/nonlinear constitutive models, and extensions to anisotropic and nonlinear electrodynamics. The formalism lays a robust foundation for modeling strongly coupled relativistic electromagnetic continua in astrophysical contexts such as neutron-star crusts, magnetars, jets, and accretion flows, with clear pathways to include anisotropy and nonlinear electrodynamics.

Abstract

We develop a covariant variational framework for relativistic electromagnetic continua (fluids and solid) based on Hamilton's principle formulated directly in the material description. The approach extends the geometric theory of relativistic continua introduced in Part I to include polarization, magnetization, and general elastic-electromagnetic coupling through a unified energy functional. By exploiting spacetime and material covariance, the framework yields the corresponding spacetime and convective variational principles, together with transparent expressions for the stress-energy-momentum tensor and the covariant Euler-type balance equations governing nonlinear electromagnetic continua. Coupling to general relativity is naturally incorporated, and when the action is augmented with Gibbons-Hawking-York boundary terms, the gravitational and electromagnetic junction conditions follow directly from the variational formulation. The results provide a coherent foundation for modeling nonlinear electromagnetic continua in relativistic regimes, with relevance to astrophysical systems where relativistic continuum dynamics and electromagnetic fields are known to be strongly coupled, such as neutron-star crusts, magnetar flares, relativistic jets, and accretion disks. We also offer systematic connections with several formulations appearing in the existing literature.

Figures (2)

• Figure 2.1: Illustration of the word-tube $\Phi:[\lambda_0,\lambda_1]\times\mathcal{B}\rightarrow\mathcal{M}$ and the tensor fields involved in the description of electromagnetic continua. The situation is summarized as follows: -- The fixed reference material tensor fields ($K$ in the general theory) include the vector field $\partial_\lambda$, the $(n+1)$-forms $R$ and $S$, and the $2$-covariant symmetric positive tensor $G$. Their push-forward by $\Phi$ are $w$, $\varrho$, $\varsigma$, and $\mathsf{c}$, respectively. -- The spacetime tensor field ($\gamma$ in the general theory) is $\mathsf{g}$. This tensor becomes dynamic only when the theory is coupled with Einstein's equations (§\ref{['coupling']}). From $\mathsf{g}$ and $w$, the $2$-covariant symmetric positive tensor $\mathsf{p}$ is constructed. Its pull-back by $\Phi$ yields the right Cauchy-Green tensor $C$. -- Besides $\Phi$, the other dynamic field can be chosen as either the material electromagnetic potential $\mathcal{A}$ or its spacetime version $A$, obtained by pushing $\mathcal{A}$ forward with $\Phi$. The corresponding electromagnetic fields are $\mathcal{F}={\rm d}\mathcal{A}$ and $F={\rm d}A$.
• Figure 2.2: Illustration of the variational principles in the three representations of a relativistic electromagnetic continuum. The primary dynamic fields are the world-tube $\Phi$ and the electromagnetic potential $\mathcal{A}$ (or $A$ see §\ref{['material_spacetime_A']}). Given the material and spacetime tensor fields $K$, $W$, and $\gamma$, one defines the associated dynamic fields $\kappa= \Phi _*K$, $w= \Phi _*W$, and $\Gamma = \Phi ^* \gamma$.

Theorems & Definitions (34)

• Lemma 2.1
• Theorem 2.2: Covariant Eulerian reduction for electromagnetic continua
• proof
• Lemma 2.3
• proof
• Corollary 2.4
• Remark 2.5: Isotropy subgroup
• Remark 2.6: Variation conversion
• Corollary 2.7
• Theorem 2.8: Covariant convective reduction for electromagnetic continua