Carrollian Physics at the Black Hole Horizon

TL;DR

This work recasts black hole horizon dynamics in terms of Carrollian (ultra-relativistic) geometry, showing that near-horizon dynamics are governed by Carrollian conservation laws derived from the membrane energy–momentum tensor. It identifies Carrollian Killing and conformal Killing symmetries on the horizon, constructing associated conserved charges that generalize angular momentum for non-stationary horizons and connect to covariant phase space charges. The analysis unifies horizon constraint equations (Raychaudhuri and Damour) with Carrollian momenta, providing a geometric framework that links horizon thermodynamics, BMS-like symmetries, and ultra-relativistic holographic ideas. These insights offer a new lens on horizon microphysics, potential entropy current formulations, and the role of soft hair in black hole information.

Abstract

We show that the geometry of a black hole horizon can be described as a Carrollian geometry emerging from an ultra-relativistic limit where the near-horizon radial coordinate plays the role of a virtual velocity of light tending to zero. We prove that the laws governing the dynamics of a black hole horizon, the null Raychaudhuri and Damour equations, are Carrollian conservation laws obtained by taking the ultra-relativistic limit of the conservation of an energy-momentum tensor; we also discuss their physical interpretation. We show that the vector fields preserving the Carrollian geometry of the horizon, dubbed Carrollian Killing vectors, include BMS-like supertranslations and superrotations and that they have non-trivial associated conserved charges on the horizon. In particular, we build a generalization of the angular momentum to the case of non-stationary black holes. Finally, we discuss the relation of these conserved quantities to the infinite tower of charges of the covariant phase space formalism.

Figures (1)

• Figure 1: The horizon is a null hypersurface situated at $\rho=0$ and $\Sigma_{\rho}$ is a timelike constant $\rho$ hypersurface near the horizon. We define also four vectors that are useful for our analysis, the null vector $\vec{L}$ is the normal to the horizon while $\vec{N}$ is transverse but also null. The spacelike vector $\vec{n}$ is the normal to $\Sigma_{\rho}$ and the timelike vector $\vec{\ell}$ is the normal to a constant $v$ section of $\Sigma_{\rho}$.