Abstract
In this note, we have compared two different perturbation techniques that could be used to generate solutions of Einstein’s equations in the 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 an 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 an extension of [1] where the authors have shown the equivalence of the two techniques up to the first non-trivial order.
Highlights
Classical evolution of the space-time is governed by Einstein’s equations, which are a set of nonlinear partial differential equations
We have compared two dynamical ‘black-hole’ type solutions of Einstein’s equations in the presence of negative cosmological constant. These two solutions were already known and were determined using two different perturbation techniques — one is the ‘derivative expansion’ and the other is an expansion in inverse powers of dimensions
We have shown that in the regime of overlap of the two perturbation parameters, the metric of these two apparently different spaces are exactly the same, to the order the solutions are known on both sides
Summary
Classical evolution of the space-time is governed by Einstein’s equations, which are a set of nonlinear partial differential equations. It has been impossible to solve these equations in full generality, when the geometry has nontrivial dynamics. In such situations, if we want to handle the problem analytically, perturbation becomes the most useful tool. We shall compare two perturbation techniques, developed to handle both the nonlinearity and the dynamics in Einstein’s equations in presence of negative cosmological constant, namely ‘derivative expansion’ [2,3,4] and ‘large-D expansion’ [5,6,7,8,9].1. We shall essentially extend it to the second subleading order. What we have done is to re express the metric dual to second order hydrodynamics [3], derived using ‘derivative expansion technique’ in the form of the metric dual to membrane dynamics [9]
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