Light rings in static and extremal black holes

TL;DR

This paper investigates the conditions under which external light rings exist in extremal black hole spacetimes, establishing a curvature-based criterion using the Newman-Penrose formalism under specific assumptions.

Contribution

It provides a new sufficient condition involving curvature scalars for the existence of external light rings in extremal black holes, applicable across various metric gravity theories.

Findings

External light rings exist if R < 2 K_G at the horizon.

Results hold for any metric gravity theory with null geodesic photon paths.

Conditions are derived assuming conformal flatness, staticity, and the null energy condition.

Abstract

In this work we establish some results concerning the existence of external light rings in extremal black hole spacetimes through the Newman-Penrose formalism. Specifically, assuming conformal flatness, staticity and the null energy condition, we show that a sufficient condition for the existence of external light rings is $R < 2 K_{G}$, where $R$, the curvature scalar of the spacetime and $K_{G}$, the Gaussian curvature of a spacelike two-surface, are both evaluated at the outermost event horizon, which can be endowed with spherical, hyperbolic or planar geometry. Our results are valid for any metric gravity theory where photons follow null geodesics.