Topology of light rings for extremal and nonextremal Kerr-Newman-Taub-NUT black holes without <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> symmetry
Shan-Ping Wu, Shao-Wen Wei
Abstract
Understanding the light ring, one kind of fundamental orbit, shall provide us with novel insight into the astronomical phenomena, such as the ringdown of binary mergers and shadows of black holes. Recently, a topological approach has preliminarily demonstrated its potential advantages on the properties of the light rings. However, extremal spinning black holes remain to be tested. In this paper, we aim to address this issue. Because of the Newman-Unti-Tamburino (NUT) charge, the Kerr-Newman-Taub-NUT solution has no ${\mathbb{Z}}_{2}$ symmetry. By constructing the corresponding topology for the nonextremal spinning black holes, we find the topological number remains unchanged. This indicates that ${\mathbb{Z}}_{2}$ symmetry has no influence on the topological number, while it indeed affects the locations of the light rings and deviates them off the equatorial plane. For the extremal spinning black holes, we find its topology is critically dependent on the leading term of the vector's radial component at the zero point of its angular component on the black hole horizon. The findings state that there exists a topological phase transition, where the topological number changes, for the prograde light rings; while no phase transition occurs for the retrograde light rings. Our study uncovers some universal topological properties for the extremal and nonextremal spinning black holes with or without ${\mathbb{Z}}_{2}$ symmetry. It also has enlightening significance for understanding the light rings in a more general black hole background.