The large D limit of General Relativity

TL;DR

This work introduces a controlled $1/D$ expansion for classical General Relativity, revealing that black holes act as non-interacting objects outside a near-horizon shell of thickness $r_0/D$ and that exterior gravity effectively vanishes. By employing matched asymptotic expansions, the authors derive an effective theory for scalar waves that captures absorption across a wide range of frequencies and angular momenta, including a clear high-frequency regime where the black hole is a nearly perfect absorber. They also solve the black-brane Gregory-Laflamme instability analytically to NNLO in $1/D$, achieving excellent agreement with numerical results and strengthening the case that higher-dimensional gravity can be tackled analytically. The results demonstrate the broad utility of the large-$D$ expansion for tractable, accurate analytic insights into gravitational dynamics, horizon physics, and holographic connections across diverse spacetimes.

Abstract

General Relativity simplifies dramatically in the limit that the number of spacetime dimensions D is infinite: it reduces to a theory of non-interacting particles, of finite radius but vanishingly small cross sections, which do not emit nor absorb radiation of any finite frequency. Non-trivial black hole dynamics occurs at length scales that are 1/D times smaller than the horizon radius, and at frequencies D times larger than the inverse of this radius. This separation of scales at large D, which is due to the large gradient of the gravitational potential near the horizon, allows an effective theory of black hole dynamics. We develop to leading order in 1/D this effective description for massless scalar fields and compute analytically the scalar absorption probability. We solve to next-to-next-to-leading order the black brane instability, with very accurate results that improve on previous approximations with other methods. These examples demonstrate that problems that can be formulated in an arbitrary number of dimensions may be tractable in analytic form, and very efficiently so, in the large D expansion.

Figures (2)

• Figure 1: Dispersion relation $\Omega(k)$ of unstable modes for $n=3,4,5,6,7$: the solid line is our analytic approximation eq. \ref{['Omvsk']}; the dots are the numerical solution (the same as in Camps:2010br, courtesy of P. Figueras).
• Figure 2: Dispersion relation $\Omega(k)$ for $n=50,100$ for $k$ near the threshold zero-mode. The solid line is our analytic approximation eq. \ref{['Omvsk']}; the dots are the numerical solution; the dashed line is the analytic approximation of Caldarelli:2012hy.