Notes on emergent conformal symmetry for black holes

TL;DR

This paper demonstrates the emergence of conformal symmetry in the near-ring region of black holes for scalar fields and null geodesics, extending the understanding of hidden symmetries beyond symmetric black hole configurations.

Contribution

It reveals that conformal symmetry persists in the near-ring region of black holes with acceleration or gravitomagnetic mass, even without north-south symmetry, and extends to timelike geodesics and scalar fields.

Findings

Conformal symmetry appears in scalar fields near black holes.

Null geodesics forming the photon ring exhibit conformal symmetry.

Symmetry persists even for black holes lacking north-south reflection symmetry.

Abstract

We examine the motion of the massless scalar field and nearly bound null geodesics in the near-ring region of a black hole, which may possess either acceleration or a gravitomagnetic mass. Around such black holes, the photon ring deviates from the equatorial plane. In the large angular momentum limit, we demonstrate that the massless scalar field exhibits an emergent conformal symmetry in this near-ring region. Additionally, in the nearly bound limit, we observe the emergence of a conformal symmetry for the null geodesics that constitute the photon ring in the black hole image. These findings suggest that the hidden conformal symmetry, associated with the Lie algebra $\mathfrak{s l}(2, \mathbb{R})$, persists even for black holes lacking north-south reflection symmetry, thereby broadening the foundation of photon ring holography. Finally, we show that the conformal symmetry also emerges for nearly bound timelike geodesics and scalar fields in proximity to the particle ring, and with specific mass around a Schwarzschild black hole.