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

## TL;DR

This paper compares two perturbation methods for solving Einstein's equations with negative cosmological constant, demonstrating their equivalence at subleading orders in certain regimes and coordinate choices.

## Contribution

It extends previous work by showing the exact equivalence of derivative expansion and inverse dimension expansion techniques at subleading orders.

## Key findings

- Metrics and horizon dynamics match exactly at known solution orders.
- The techniques are equivalent in specific parameter regimes and coordinate choices.
- This work generalizes the known first-order equivalence to 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 \cite{prevwork} where the authors have shown the equivalence of the two techniques up to the first non-trivial order.