Topological Charge and Black Hole Photon Spheres

Abstract

Black hole photon spheres or light rings are closely linked to astronomical phenomena, such as the gravitational waves and the shadow in spherically symmetric or axi-symmetric spacetime. In Cunha and Herdeiro [Phys. Rev. Lett. 124, 181101 (2020)], a topological argument was applied for the four dimensional stationary, axi-symmetric, asymptotically flat black hole, and the result indicates that at least, there exists one standard light ring outside the black hole horizon for each rotation sense. Inspired by it, in this paper, we would like to consider a similar issue for a nonrotating, static, spherically symmetric black hole not only with asymptotically flat behavior, but also with AdS and dS behaviors. Following Duan's topological current $\varphi$-mapping theory, the topological current and charge for the photon spheres are introduced. The topological current is nonzero only at the zero point of the vector field determining the location of the photon sphere. So each photon sphere can be assigned a topological charge. Considering the full exterior region, we find the total topological charge always equals -1. This result confirms that there exists at least one standard photon sphere outside of the black hole not only in asymptotically flat spacetime, but also in asymptotically AdS and dS spacetime. Then we apply the study to the dyonic black holes. We observe that even when more photon spheres are included, the total topological charge stays unchanged. Moreover, for a naked singularity, it has a vanishing topological charge, indicating that the black hole and naked singularity are in different topological classes. It is expected that this novel topological argument could provide an insightful idea on the study of the black hole photon spheres or light rings, and further cast new light on the black hole astronomical phenomena.

12 pages, 8 figures, and 2 tables. Dedicated to the memory of Prof. Yi-Shi Duan

References in corpus (7)

• First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole
• First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole
• First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring
• Light ring stability in ultra-compact objects
• Connection between black-hole quasinormal modes and lensing in the strong deflection limit
• Upper bound on the radii of black-hole photonspheres
• Post-Kerr black hole spectroscopy

Cited by in corpus (8)

• Black Hole Solutions as Topological Thermodynamic Defects
• Topology of black hole thermodynamics
• Topology of critical points and Hawking-Page transition
• Classifying topology of consistent thermodynamics of the four-dimensional neutral Lorentzian NUT-charged spacetimes
• Light rings and shadows of static black holes in effective quantum gravity
• Light rings of stationary spacetimes
• On Thermal Stability of Hairy Black Holes
• Topology of the charged AdS black hole in restricted phase space