General Effective Theories of Black Holes in the Large D Limit

TL;DR

This paper derives a comprehensive set of effective equations describing black hole dynamics in the large D limit, including stationary and fluctuating configurations, and establishes their connection to the membrane formalism.

Contribution

It provides the first general derivation of effective equations for black holes at large D, including arbitrary sources and a well-posed initial value problem.

Findings

Effective equations split into stationary and dynamical parts.

The equations form a well-posed parabolic system.

Connection established with the covariant membrane formalism.

Abstract

We derive the general form of the effective equations governing black hole dynamics in the limit of a large number of dimensions $D$. These split into a universal \emph{soap-bubble} embedding condition for stationary configurations and a set of nonlinear dynamical evolution equations describing near-horizon fluctuations of $O(1/D)$ amplitude over horizon scales of $O(1/\sqrt{D})$. We obtain these equations in full generality, including arbitrary asymptotic sources in the near-horizon region, and we show that they form a parabolic system with a well-posed initial value problem. To connect the various approaches to large-$D$ black hole dynamics, we also show that both the embedding and dynamical equations can be derived from the covariant membrane formalism. We clarify the intrinsic scope of the large-$D$ approach, emphasizing that it yields a well-posed dynamical evolution only on horizon scales of $O(1/\sqrt{D})$, which is the range where the most relevant horizon dynamics occur. Our results highlight the versatility of these effective theories for studying a wide class of black hole phenomena.