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

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.