The fluid/gravity correspondence (1107.5780v1)

Abstract: We review the fluid/gravity correspondence which relates the dynamics of Einstein's equations (with negative cosmological constant) to the dynamics of relativistic Navier-Stokes equations.

• The paper demonstrates how the long-wavelength limit of Einstein’s equations reduces to relativistic Navier-Stokes equations via the AdS/CFT correspondence.
• It constructs time-dependent, inhomogeneous black hole solutions that illuminate stress tensor dynamics in strongly coupled field theories.
• The study extends its analysis to non-conformal and charged fluids, paving the way for broader applications in gravitational physics and hydrodynamics.

Insights into the Fluid/Gravity Correspondence

The chapter "The fluid/gravity correspondence" by Hubeny, Minwalla, and Rangamani provides an in-depth exploration of the intricate link between the equations governing fluids and those in gravitational systems, particularly within the context of the AdS/CFT correspondence. Central to this discussion is the long wavelength limit of Einstein's equations with a negative cosmological constant in $d+1$ dimensions, which under specific conditions reduce to the equations characterizing fluid dynamics in $d$ dimensions. This chapter meticulously elucidates the theoretical groundwork and computational strategies necessary to unravel this correspondence, which finds its relevance both within and outside the field of string theory.

The AdS/CFT correspondence serves as a pivotal foundation for this work. It posits a duality wherein the dynamics of a quantum field theory at strong coupling maps onto a gravitational theory in higher-dimensional spacetime. The authors highlight how this correspondence can inform our understanding of stress tensor dynamics in strongly coupled field theories and motivate a deeper examination into the fluid/gravity duality.

A particularly insightful aspect of the paper is its explication on how Einstein's equations, within the long wavelength regime, can be methodically reduced to the well-known Navier-Stokes equations of hydrodynamics, albeit in a relativistic form appropriate for the context. The authors leverage this insight to construct solutions to Einstein's equations that manifest as time-dependent, inhomogeneous black hole solutions, thus inferring a systematic derivative expansion that aligns with solutions of fluid dynamical equations.

In the gravity description, the fluid dynamics arise from fluctuations around black hole solutions in AdS spacetime. This setup leads to the exploration of novel phase structures and the dynamics of quantum field theories under thermalization. For researchers engaged in holography or associated fields, this chapter delivers a comprehensive framework to explore the dynamics of black holes using hydrodynamic principles, enriching both theoretical versatility and potential applications.

The authors also delve into the implications of anomalies and boundary conditions for fluid dynamics. They extend their analysis to consider systems beyond conformal fluids, including non-conformal and charged fluids, each bringing unique challenges and insights into hydrodynamic theoretical frameworks. By tackling these different cases, the research presented builds a robust bridge between high-energy theoretical physics and practical fluid dynamics, paving the way for future exploration in both analytical and numerical simulations of gravitational systems.

The chapter concludes by considering the broader applications and implications of the fluid/gravity map within different physical phenomena, such as turbulence and the dynamics of non-equilibrated systems. This fluid/gravity interface thus strongly suggests a rich landscape of exploration for both gravitational physicists and fluid dynamicists, revealing deeper connections and potentially novel insights into fundamental physics.

In future developments, researchers might aim to refine this correspondence in more complex settings, perhaps extending it to non-relativistic regimes or exploring its implications in higher-dimensional gravitational theories. Such work could unlock new pathways in understanding how microscale quantum dynamics give rise to macroscale classical phenomena and vice versa, thus continuing to enrich the theoretical dialogue between fields seemingly as disparate as fluid dynamics and string theory.