The Unruh Effect in Relativistic Fluids

TL;DR

The paper establishes a classical analogue of the Unruh effect within relativistic dissipative fluids by reinterpreting frame changes between Eckart and Landau–Lifshitz as a local thermodynamic boost in a heat plane, governed by a thermodynamic acceleration $a_{\rm th}$. It derives a Thermodynamic Unruh temperature $T_{\rm th}=\frac{\hbar}{2\pi k_B}|a_{\rm th}|$ that a comoving probe would perceive during transient out-of-equilibrium dynamics, with explicit expressions linking $a_{\rm th}$ to standard hydrodynamic transport through Israel–Stewart relations and extending to modern BD-NK formalisms. The results show universality at leading order: $a_{\rm th}=u\cdot\nabla\eta$ remains invariant under admissible first-order frame redefinitions, making $T_{\rm th}$ robust across a range of theories, while $T_{\rm th}$ remains distinct from the material temperature and vanishes in Tolman–Ehrenfest equilibrium. This work provides a conceptual bridge between observer-dependent quantum vacua and classical out-of-equilibrium hydrodynamics, with potential validation through simulations of strong transients in relativistic fluids.

Abstract

We identify the relativistic-fluid counterpart of the Unruh effect, in which a comoving probe measures a Thermodynamic Unruh temperature. Frame changes in first-order hydrodynamics are recast as a local, time-dependent hyperbolic rotation in a Rindler-style state space where the instantaneous map between frames is the Thermodynamic Boost and its proper-time variation defines the Thermodynamic Acceleration, which results in an Unruh-like thermal spectrum. To leading order, the Thermodynamic Unruh temperature is frame-independent and universal across out-of-equilibrium relativistic fluid descriptions, from Israel-Stewart to modern theories.

Figures (4)

• Figure 1: The space of the $1+1+2$ decomposition. The heat plane $\mathrm{span}\{u,\hat{q}\}$ carrying the hyperbolic block and the transverse 2-surface carrying anisotropic shear.
• Figure 2: Thermodynamic Rindler diagram. Visualizing the state-space geometry in the heat plane $(\mathcal{X},\mathcal{T})=(\varepsilon+p_{\parallel},\,2q)$. Dashed lines are the thermodynamic horizons $\mathcal{X}=\pm \mathcal{T}$. Hyperbolae $\mathcal{X}^{2}-\mathcal{T}^{2}=\rho^{2}$ are orbits under thermodynamic boosts, parameterized by the geometric rapidity $\eta_g$. Any energy frame with $q=0$ sits at the apex ($\eta=0$). As the fluid evolves dynamically, its state moves in this space. The rate of change of the physical rapidity $\eta$ is the thermodynamic acceleration $a_{\rm th}=u\!\cdot\!\nabla \eta$.
• Figure A1: Causal vs. acausal heat propagation. The red curve illustrates the diffusion response, characterized by instantaneous spatial tails that extend beyond the light cone. The Green function extending to infinity corresponds to the acausal character. The relativistic correction is illustrated by the Green curve, which results in a more complex Green function.
• Figure B1: Rindler diagram. Rindler coordinates as in \ref{['eq:rindler-transform']}: $x=\xi\cosh(a\eta_{\mathrm R}/c),\; ct=\xi\sinh(a\eta_{\mathrm R}/c)$. Solid curves are $\xi=\text{const}$ (uniform acceleration, $a=c^{2}/|\xi|$); dashed lines are $\eta_{\mathrm R}=\text{const}$. The null lines $x=\pm ct$ bound the right/left wedges (Rindler horizons). Along $\xi=\text{const}$, $d\tau=(a\xi/c^{2})\,d\eta_{\mathrm R}$.