Fluid-gravity and membrane-gravity dualities. Comparison at subleading orders

Abstract

In this note we have compared two different perturbation techniques that could be used to generate solutions of Einstein's equations in presence of negative cosmological constant. One of these two methods is derivative expansion and the other is an expansion in inverse powers of dimension. Both the techniques generate space-time with a singularity shielded by a dynamical event horizon. We have shown that in the appropriate regime of parameter space and with appropriate choice of coordinates, the metrics and corresponding horizon dynamics, generated by these two different techniques, are exactly equal to the order the solutions are known both sides. This work is essentially extension of [1] where the authors have shown the equivalence of the two techniques upto the first non-trivial order.

Black rings in large D membrane paradigm at the first order

“…While the other branch, known as the "large D membrane paradigm" [53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69] states that the dynamical black holes in large D have one-to-one correspondence to the dynamics of a codimension one hypersurface, called a "membrane", which doesn't backreact on its background. In this nongravitational dual, the membrane dynamics is governed by what are called the "membrane equations of motion", and the dynamical data associated with the membrane completely defines the corresponding black hole solution.…”

Stress tensor for large-D membrane at subleading orders

“…• A detailed matching with the hydrodynamic stress tensor dual to the same gravity system in the regime of overlap for these two perturbation techniques (namely 1 D expansion and derivative expansion (see [35,36])). Now after computing the membrane stress tensor, we could extend this matching to include the effect of the gravitational radiation as well.…”

Comparison between fluid-gravity and membrane-gravity dualities for Einstein-Maxwell system

“…These questions have been answered in [1,2] for pure gravity systems and here we shall extend their work to Einstein-Maxwell systems. We shall show that the gravity solutions generated by these two perturbation techniques are equivalent also for Einstein-Maxwell systems in an appropriate regime of parameter space.…”

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Derivative expansion and large-D expansion are two perturbation techniques, which are used to generate dynamical black-brane solutions to Einstein's equations in presence of negative cosmological constant. In this note we have compared these two techniques and established the equivalence of the gravity solutions generated by these two different techniques in appropriate regime of parameter space up to first non-trivial order in both the perturbation parameters for Einstein-Maxwell systems, generalizing the earlier works of [1,2] for non-charged systems. An one-toone map between dynamical black-brane geometry and AdS space, which also exists at finite number of dimensions, has also been established.

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