A membrane paradigm at large D

TL;DR

This paper develops a membrane paradigm for large D black holes, deriving equations of motion for the membrane shape and velocity that determine black hole dynamics in the large D limit.

Contribution

It introduces a novel membrane-based framework for analyzing black hole solutions in the large D limit, connecting horizon geometry with dynamical equations.

Findings

Derived equations governing membrane shape and velocity.

Constructed nonsingular solutions at first sub-leading order in 1/d.

Established a well-posed initial value problem for black hole dynamics.

Abstract

We study $SO(d+1)$ invariant solutions of the classical vacuum Einstein equations in $p+d+3$ dimensions. In the limit $d \to \infty$ with $p$ held fixed we construct a class of solutions labelled by the shape of a membrane (the event horizon), together with a `velocity' field that lives on this membrane. We demonstrate that our metrics can be corrected to nonsingular solutions at first sub-leading order in $\frac{1}{d}$ if and only if the membrane shape and `velocity' field obey equations of motion which we determine. These equations define a well posed initial value problem for the membrane shape and this `velocity' and so completely determinethe dynamics of the black hole. They may be viewed as governing the non-linear dynamics of the light quasi normal modes of Emparan, Suzuki and Tanabe.