A leading-order comparison between fluid-gravity and membrane-gravity dualities

TL;DR

This paper compares two perturbation methods for generating dynamical black-brane solutions in AdS space, establishing a correspondence and mapping between fluid and membrane dualities, valid even at finite dimensions.

Contribution

It demonstrates an overlap between derivative expansion and inverse dimension expansion techniques, linking fluid and membrane dualities in AdS black-brane solutions.

Findings

Matched gravity solutions from both perturbation methods.

Established a one-to-one map between black-brane geometry and AdS space.

Extended the duality correspondence to finite dimensions.

Abstract

In this note, we have compared two different perturbation techniques that are used to generate dynamical black-brane solutions to Einstein equation in presence of negative cosmological constant. One is the `derivative expansion', where the gravity solutions are in one-to-one correspondence with the solutions of relativistic Navier-Stokes equation. The second is the expansion in terms of inverse power of space-time dimensions and here the gravity solutions are dual to a co-dimension one dynamical membrane, embedded in AdS space and coupled to a velocity field. We have shown that in large number of space-time dimensions, there exists an overlap regime between these two perturbation techniques and we matched the two gravity solutions along with their dual systems upto the first non-trivial order in the expansion parameter on both sides. In the process, we established a one-to-one map between dynamical black-brane geometry and the AdS space, which exists even when the number of dimensions is finite.