Currents and Radiation from the large $D$ Black Hole Membrane

TL;DR

This paper explores the dynamics of black hole membranes in large dimensions, defining stress tensors and charge currents, analyzing radiation, and establishing a local second law of thermodynamics for membrane evolution.

Contribution

It explicitly determines membrane stress tensor and charge current at low orders in 1/D, linking membrane dynamics to conservation laws and radiation behavior.

Findings

Membrane radiation is suppressed by 1/D^D

Membrane degrees of freedom are decoupled from flat space at large D

Entropy current divergence is non-negative, confirming a local second law

Abstract

It has recently been demonstrated that black hole dynamics in a large number of dimensions $D$ reduces to the dynamics of a codimension one membrane propagating in flat space. In this paper we define a stress tensor and charge current on this membrane and explicitly determine these currents at low orders in the expansion in $\frac{1}{D}$. We demonstrate that dynamical membrane equations of motion derived in earlier work are simply conservation equations for our stress tensor and charge current. Through the paper we focus on solutions of the membrane equations which vary on a time scale of order unity. Even though the charge current and stress tensor are not parametrically small in such solutions, we show that the radiation sourced by the corresponding membrane currents is generically of order $\frac{1}{D^D}$. In this regime it follows that the `near horizon' membrane degrees of freedom are decoupled from asymptotic flat space at every perturbative order in the $\frac{1}{D}$ expansion. We also define an entropy current on the membrane and use the Hawking area theorem to demonstrate that the divergence of the entropy current is point wise non negative. We view this result as a local form of the second law of thermodynamics for membrane motion.