Photon region boundary for stationary axisymmetric black holes

TL;DR

This paper derives explicit, parameter-independent expressions for the boundary of the photon region in stationary axisymmetric black holes, enhancing understanding of black hole shadows and photon orbits.

Contribution

It provides a general method to determine the photon region boundary using background metric functions, applicable to both separable and non-separable spacetimes.

Findings

Explicit boundary expressions for photon regions are derived.

The boundary overlaps with ergoregions and indicates SPO rotation sense.

Validation against exact solutions confirms the approach's accuracy.

Abstract

The black hole shadow is fundamentally connected to the structure of light rings and the photon region in the background geometry. We investigate the photon region boundary in a generic asymptotically flat, stationary, axisymmetric black hole spacetime that admits spherical photon orbits (SPOs). Explicit expressions possessing real solutions are provided for the photon region's boundary purely in terms of the background metric functions, independent of the photon's parameters like energy or angular momentum, which are applicable to both separable and non-separable spacetimes. We further analyze its common features, including overlap with the ergoregion and rotation sense of SPOs. Additionally, light rings are identified at the extrema of the photon region boundary curves in the $(r,\theta)$ plane. Our approach is validated against a few exact black hole solutions. Implications are discussed.