Stationary Solutions from the Large D Membrane Paradigm

TL;DR

This paper derives effective equations for stationary membranes in large D spacetimes, linking membrane dynamics to black hole thermodynamics and presenting solutions like black rings and charged rotating black holes.

Contribution

It provides the first derivation of stationary membrane equations in arbitrary backgrounds, connecting membrane dynamics with black hole thermodynamics and solutions.

Findings

Stationary membrane equations satisfy the first law of thermodynamics.

Solutions include charged rotating black holes and black rings.

Effective equations apply in both charged and uncharged backgrounds.

Abstract

It has recently been shown that the dynamics of black holes in large number of dimensions D can be recast as the dynamics of a probe membrane propagating in the background spacetime which solves Einstein equations without matter. The equations of motion of this membrane are simply the statement of conservation of the stress tensor and charge current defined on this membrane. In this paper we obtain the effective equations of motion for stationary membranes in any empty background both in presence and absence of charge. It turns out that the thermodynamic quantities associated with the stationary membranes that satisfy these effective equations also satisfy the first law of black hole thermodynamics. These stationary membrane equations have some interesting solutions such as charged rotating black holes in flat and AdS backgrounds as well as black ring solutions in large D.