Abstract
In this note, we have compared two different perturbation techniques that are used to generate dynamical black-brane solutions to Einstein’s equations in the 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 a 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 up to the first non-trivial order in the expansion parameter on both sides.
Highlights
It is natural to ask whether it is possible to apply both the perturbation techniques simultaneously in any regime(s) of the parameter space of the solutions, and if so, how the two solutions compare in those regimes
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
As we have mentioned before, both ‘hydrodynamic metric’ and ‘large-D’ metric is determined in terms of data, defined on a codimension one hypersurfaces — in the first case it is the velocity and temperature of a relativistic fluid living on the boundary of asymptotic AdS and in the second case it is the horizon viewed as a membrane embedded in the background with fluctuating shape and velocity
Summary
We shall discuss whether we could apply both ‘derivative expansion’ and expansion simultaneously. We shall first define the perturbation parameters for each of these two techniques in a precise way and fix the range of their validity. We shall see that these two parameters are completely independent of each other and their ratio could be tuned to any value, large or small. We shall compare the forms of the two metrics, determined using these two techniques, assuming the ratio (between the two perturbation parameters) to have any arbitrary value
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