A fluid dual to charged large D membrane paradigm

Abstract: According to the formulation of the charged large $D$ membrane paradigm, an arbitrary dynamic black hole solution to a theory of gravity with a $U(1)$ gauge field is dual to the dynamics of a membrane in a non-gravitational background. This membrane is endowed with a stress-energy tensor and a charge current, whose conservation equations govern its dynamics. In this work, we demonstrate that the dynamics of these membrane configurations (at the leading nontrivial order in $1/D$) can be mapped to a relativistic charged fluid. Establishing a correspondence for asymptotically flat black holes with a particular class of fluid systems. Unlike the standard AdS/Hydrodynamics correspondence, this dual fluid does not reside on an asymptotic boundary, but is localized strictly on the non-gravitational membrane worldvolume. By evaluating the system in both the Eckart and Landau frames, we systematically extract the out-of-equilibrium transport coefficients. We find that the fluid is governed by a negative effective thermal conductivity and a negative heat capacity, a mechanism that enforces thermodynamic stability in agreement with the quasinormal mode damping in the large $D$ Reissner-Nordström black hole geometry.

• The paper establishes a rigorous hydrodynamic dual for charged membranes by mapping geometric variables to fluid dynamics in large D gravity.
• The methodology reveals unconventional thermodynamic features such as negative effective thermal conductivity and heat capacity in the dual fluid.
• Detailed analysis in both Eckart and Landau frames identifies frame-dependent transport coefficients, with higher-order terms emerging at leading order in 1/D.

Fluid Duality in the Charged Large $D$ Membrane Paradigm

Introduction

The paper "A fluid dual to charged large D membrane paradigm" (2605.15797) establishes a rigorous hydrodynamic correspondence for the charged membrane paradigm in the large $D$ limit of asymptotically flat black hole spacetimes. By systematically mapping the geometric degrees of freedom of a codimension-one membrane (arising from black hole event horizons in large $D$ gravity) to the macroscopic fields of a charged relativistic fluid, the authors extend the well-known fluid-gravity duality to a domain where the dual fluid does not reside on the asymptotic boundary but is strictly supported on the dynamical worldvolume of the membrane. This construction achieves a comprehensive identification of transport coefficients, dynamical equations, and frame structures, while clarifying the unique thermodynamic attributes of the system.

Large $D$ Membrane Paradigm and Quasi-Normal Mode Decoupling

The membrane paradigm in large $D$ capitalizes on the fact that the gravitational and electromagnetic fields of black holes localize to a thin region near the horizon, with thickness $\mathcal{O}(r_0/D)$, leading to effective separation of scales. The dynamics outside this "membrane region" become trivial, and the geometry inside is causally disconnected. At leading order in $1/D$, the entire nontrivial evolution is relegated to a $(D-1)$-dimensional hypersurface defined solely in terms of local geometric data: the shape function, charge density, and velocity field. The associated constraint equations derived from the higher-dimensional Einstein-Maxwell system are equivalent to the conservation equations for a stress tensor and charge current defined intrinsically on the membrane.

Mapping to Hydrodynamics: Dual Fluid Construction

The duality is constructed by correlating the membrane variables with relativistic fluid variables as follows: membrane velocity $\leftrightarrow$ fluid velocity, membrane charge $\leftrightarrow$ chemical potential, and membrane curvature/shape $D$0 temperature. The stress tensor and charge current on the membrane, computed in terms of the extrinsic and intrinsic geometry, are rewritten in decomposed forms analogous to first-order viscous relativistic hydrodynamics.

Two standard frame choices in out-of-equilibrium hydrodynamics—Eckart and Landau frames—are explicitly realized. In the Eckart frame, the fluid velocity is aligned with the charge flow, whereas in the Landau frame, it aligns with the energy flow. Detailed analysis in both frames enables the precise identification of dissipative fluxes, allowing extraction of all leading-order transport coefficients.

Nonstandard Transport and Thermodynamic Structure

Crucially, the system demonstrates several nonstandard features relative to conventional relativistic hydrodynamics:

• Negative Effective Thermal Conductivity and Heat Capacity: The heat flux shows an unconventional sign; the relevant transport coefficient is always positive, but due to the constitutive structure, this results in heat flowing from cooler to hotter regions. Similarly, the total heat capacity is negative: energy input lowers the system temperature, and heat outflow raises it. These features are generic in self-gravitating systems and black holes but are manifestly encoded in the dual hydrodynamics here.
• Suppression of Pressure and Gibbs-Duhem Violation: At leading order in $D$1, the pressure term is dynamically suppressed, in contrast to standard fluids where Euler and Gibbs-Duhem relations constrain the gradient expansions. The momentum conservation is sourced entirely by extrinsic geometric forces, and the algebraic relations connecting temperature, chemical potential, and pressure gradients are absent.
• Higher-Order Terms at Leading $D$2: Unlike the first-derivative truncation of the standard gradient expansion, the leading-order dynamics in $D$3 contain terms which (from the perspective of standard hydrodynamics) would be formally higher order. This includes Laplacians of the velocity field in the stress tensor and charge current.
• Explicit Calculation of Frame-Dependent Transport: The Eckart and Landau frames admit three independent first-order transport coefficients each for the dissipative charge and energy fluxes—unlike the single thermal conductivity of usual hydrodynamics—directly reflecting the absence of the usual algebraic constraints due to the suppressed pressure sector.

Thermodynamic Stability and Dynamical Implications

Thermodynamic stability is assured in this model through the strongly negative heat capacity and thermal conductivity, which guarantee the suppression of any thermal fluctuations: heat flows in reverse to dampen temperature gradients, and the system is dynamically driven back to global equilibrium. This is entirely consistent with the quasi-normal mode analysis of large $D$4 Reissner-Nordström black holes, where perturbations always decay due to strong damping dominated by the lightest modes retained in the membrane description.

Outlook and Future Directions

The framework provided captures all leading-order dissipative effects but incorporates only specific subleading terms needed to close the equations at $D$5. A full systematic expansion to higher orders would allow for the explicit extraction of additional (second-order) transport coefficients and a more robust entropy current formalism, which is nontrivial in this setting due to the absence of a standard local hydrodynamic entropy current.

Moreover, the paper suggests that a construction analogous to the AdS/hydrodynamics correspondence on null infinity (requiring a Carrollian fluid structure) could yield further insight into flat space holography in the large $D$6 regime—a direction ripe for further investigation.

Conclusion

This work (2605.15797) presents a detailed and systematic realization of the hydrodynamic dual to the charged large $D$7 membrane paradigm, revealing a rich structure of relativistic fluid dynamics with unique transport and thermodynamic properties. The identification of negative thermal conductivity and negative heat capacity provides a first-principles microscopic justification for black hole thermodynamic peculiarities at the level of effective hydrodynamics, while the detailed frame analysis of transport structure offers a precise technical correspondence for future developments in holography and gravitational effective theory in high dimensions.