Fluid dynamics of R-charged black holes (0809.2488v4)

Abstract: We construct electrically charged AdS_5 black hole solutions whose charge, mass and boost-parameters vary slowly with the space-time coordinates. From the perspective of the dual theory, these are equivalent to hydrodynamic configurations with varying chemical potential, temperature and velocity fields. We compute the boundary theory transport coefficients associated with a derivative expansion of the energy momentum tensor and R-charge current up to second order. In particular, we find a first order transport coefficient associated with the axial component of the current.

• The paper constructs a second-order derivative expansion of the metric and gauge fields, determining novel transport coefficients including vorticity effects.
• It derives numerical expressions for key transport coefficients as functions of chemical potential and temperature, confirming results like the universality of shear viscosity to entropy density.
• The study highlights how the Chern-Simons term introduces first-order vorticity contributions, linking gravitational anomalies to boundary fluid dynamics.

Overview of "Fluid Dynamics of R-Charged Black Holes"

The paper, titled "Fluid Dynamics of R-Charged Black Holes," investigates the hydrodynamic properties of gauge theories that are dual to charged black holes in asymptotically AdS${}_5$ spacetimes within the context of the AdS/CFT correspondence. The authors, Johanna Erdmenger, Michael Haack, Matthias Kaminski, and Amos Yarom, focus on the extension of black hole solutions by allowing the parameters—related to mass, charge, and boost—to vary with spacetime coordinates.

Main Contributions

The paper's central contribution is constructing an expansion for the metric and gauge fields to second order in derivatives. This expansion gives insight into the transport coefficients of the boundary theory in the strong coupling regime, defined by $\mathcal{N}=4$ supersymmetric Yang-Mills theory with a non-zero chemical potential.

To achieve this, the authors employ the method of hydrodynamic expansion, following previous breakthroughs that relate hydrodynamic modes of gauge theories to the bulk geometry of black holes. This approach builds upon the universality feature of the ratio of shear viscosity to entropy density ($\eta/s = 1/4\pi$) and extends it to analyze charged black holes.

Here are the critical findings presented in the paper:

1. Derivative Expansion: By enhancing the current understanding of the hydrodynamic gradient expansions, the paper calculates both linear and nonlinear transport coefficients up to second order in derivatives. This includes several previously unexplored coefficients, such as those associated with the vorticity of the fluid, which showcase the influence of the Chern-Simons term in the bulk action.
2. Numerical Results: The work provides expressions for the transport coefficients ($\eta$, $\kappa$, $\Omega$, and others) concerning chemical potential $\mu$ and temperature $T$. The robust derivation confirms known results in specific limits, such as $\eta/s$, and predicts new coefficient values at finite $\mu/T$, demonstrating the consistency of these results with theoretical expectations.
3. Implications of the Chern-Simons Term: The inclusion of a Chern-Simons term is significant as it contributes to novel first-order terms in the current $\Upsilon_\nu$, manifested in additional vorticity-driven contributions. This insight adds a layer of understanding to how gravitational anomalies in the bulk can translate into specific hydrodynamic phenomena on the boundary theory.
4. Boundary Analysis: The paper explores rigorous boundary analysis, exploring the conditions under which the solutions extend seamlessly, avoiding singularities in physical setups as viable, and ensuring regular behavior beyond a certain scale dictated by temperature and chemical potential.

Implications and Conclusions

This paper's results enhance the comprehension of strongly coupled fluids in theoretical physics, potentially bridging connections between condensed matter systems and relativistic hydrodynamics. The successful derivation of second-order transport coefficients may inform future holographic models aimed at mimicking real-world systems exhibiting similar behavior.

Moreover, this work has implications for the understanding of thermalization processes and entropy production in gauge theory plasmas, thereby enriching the dialogue concerning the universality of hydrodynamic behavior across diverse physical systems. The newly identified transport coefficients related to vorticity suggest further exploration into systems where angular momentum and magnetic effects play crucial roles.

Future Directions

The insights gleaned from this paper suggest several avenues for further inquiry. These include extending the approach to other gauge theories with different field contents or higher dimensions, systematically analyzing the influence of finite 't Hooft coupling corrections, and probing the stability criteria of the fluid configurations, especially as they relate to the near-extremal limits of black holes.

Additional investigations could explore the impact of different symmetry-breaking patterns or conformal anomalies on the hydrodynamic regime, shedding light on novel sectors of the AdS/CFT correspondence and potentially informing the development of theoretical models with observable analogs in heavy-ion collisions or other high-energy systems.