Hydrodynamic Long-Time tails From Anti de Sitter Space

TL;DR

This work demonstrates that non-linear hydrodynamic long-time tails, a universal feature of finite-temperature field theories, can be reproduced from a one-loop gravity calculation in AdS/CFT. By formulating the real-time Schwinger-Keldysh problem in the AdS bulk, separating the bulk into causally relevant regions, and focusing on the horizon-adjacent dynamics, the authors derive the boundary tails for conserved currents and the stress tensor that agree with hydrodynamic predictions in arbitrary dimensions. The calculation identifies the dominant bulk interactions and shows how horizon-encoded fluctuations propagate to produce the same power-law decay as predicted by non-linear hydrodynamics, with the correct scaling and prefactors. This result strengthens the holographic connection between quantum gravity effects and emergent hydrodynamic behavior, highlighting the universality and infrared nature of the tails and suggesting avenues for exploring nonlinear phenomena such as turbulence within holography.

Abstract

For generic field theories at finite temperature, a power-law falloff of correlation functions of conserved currents at long times is a prediction of non-linear hydrodynamics. We demonstrate, through a one-loop computation in Einstein gravity in Anti de Sitter space, that this effect is reproduced by the dynamics of black hole horizons. The result is in agreement with the gauge-gravity correspondence.

Figures (6)

• Figure 1: Schwinger-Keldysh time contour for the finite temperature path integral.
• Figure 2: The maximally extended geometry of an AdS-Schwartzshild black hole. The shaded region depicts the causal diamond associated to a causal boundary correlator.
• Figure 3: Feynman diagrams contributing to (a) current correlator (b) stress tensor correlator. Four-point vertices (c) will be found to have negligible effects. Wavy lines are bulk Yang-Mills fields and double lines are gravitons.
• Figure 4: Space-time structure of the integration region for the loop vertices ("half-almond"). The external wavefunctions force the vertices to localize in the shaded region. Hydrodynamics interactions are generated in region $\mathcal{M}_>$ when both vertices are at radii of order $T^{-1}$.
• Figure 5: (a) Keldysh index structure of the one-loop diagrams contributing to a retarded two-point function. (b) A diagram which vanishes due to a closed loop of retarded propagators. (c) A diagram with a four-point vertex (which is subleading in the long-time limit). Arrows indicate the flow of time along retarded propagators. The cut propagator is the $G_{rr}$ propagator ("symmetric propagator").