Kubo formulas for relativistic fluids in strong magnetic fields

Magnetohydrodynamics of strongly magnetized relativistic fluids is derived in the ideal and dissipative cases, taking into account the breaking of spatial symmetries by a quantizing magnetic field. A complete set of transport coefficients, consistent with the Curie and Onsager principles, is derived for thermal conduction, as well as shear and bulk viscosities. It is shown that in the most general case the dissipative function contains five shear viscosities, two bulk viscosities, and three thermal conductivity coefficients. We use Zubarev’s non-equilibrium statistical operator method to relate these transport coefficients to correlation functions of equilibrium theory. The desired relations emerge at linear order in the expansion of the non-equilibrium statistical operator with respect to the gradients of relevant statistical parameters (temperature, chemical potential, and velocity.) The transport coefficients are cast in a form that can be conveniently computed using equilibrium (imaginary-time) infrared Green’s functions defined with respect to the equilibrium statistical operator.

💡 Research Summary

The paper presents a comprehensive derivation of relativistic magnetohydrodynamics (MHD) for fluids subjected to strong, quantizing magnetic fields, covering both the ideal (non‑dissipative) and first‑order dissipative regimes. The authors start by decomposing the electromagnetic field tensor (F_{\mu\nu}) into electric and magnetic components measured in the fluid rest frame, but then focus on the regime where the electric field is negligible compared to the magnetic field—a realistic situation for the interior of neutron stars and magnetars. By introducing the unit magnetic‑field four‑vector (b^\mu) (orthogonal to the fluid four‑velocity (u^\mu)) and the antisymmetric tensor (b^{\mu\nu}), they rewrite the matter part of the energy‑momentum tensor in a compact form that explicitly displays the anisotropy induced by the magnetic field: the transverse pressure (P_\perp = P - MB) and the longitudinal pressure (P_\parallel = P + MB). This anisotropy breaks the spatial rotational symmetry and necessitates a richer tensorial structure for transport coefficients.

In the dissipative sector, the entropy‑production inequality (T\partial_\mu s^\mu \ge 0) is used to constrain the form of the viscous stress tensor (\tau^{\mu\nu}) and the charge/heat diffusion currents (j^\mu) and (j_s^\mu). The authors adopt the Landau‑Lifshitz frame (fluid velocity aligned with the energy flow) which eliminates the energy flux term (h^\mu). The linear response relations are written as \