The universal topological charge of black hole photon spheres in higher dimensions

TL;DR

Extends a topological framework for photon spheres to higher-dimensional static, spherically symmetric, and asymptotically flat black holes, showing the total topological charge is the dimensionally invariant value $Q=-1$. This result, derived via Duan's $\phi$-mapping topological current theory and a boundary analysis of the deflection angle $\Omega$, implies at least one standard (unstable) photon sphere exists outside the horizon for any $D\ge5$. The authors validate universality by analyzing two purely gravitational regular black holes (Hayward and Dymnikova-like), demonstrating the same $Q=-1$ across dimensions. The work provides a dimension-independent, topology-based criterion for the existence and structure of photon spheres, with implications for understanding black hole shadows in higher-dimensional gravity.

Abstract

A recently developed topological approach offers novel insights into photon spheres, which are fundamental to the formation of black hole shadows. In this study, we extend this topological analysis to higher-dimensional, static, spherically symmetric, and asymptotically flat black holes. By examining the asymptotic properties of the vector field associated with the photon spheres, we demonstrate that their topological charge is consistently -1. This result is a dimensionally independent invariant, guaranteeing the existence of at least one standard (unstable) photon sphere outside the event horizon. We further explore this conclusion by analyzing two distinct regular black hole solutions derived from pure gravity theory, confirming that the topological charge remains -1 irrespective of the spacetime dimension. These results provide a robust and universal characterization of photon spheres in higher-dimensional spacetimes.

Figures (5)

• Figure 1: Two Types of limit boundaries in the ($r$, $\theta$) plane. The black arrow indicates that counterclockwise is the positive direction. $C$ is the union of line segments: $\{l_{1}:r\to +\infty, 0 \le \theta \le \pi \} \cup \{l_{2}:\theta\to\pi^{-}, r_{\text{h}} \le r < +\infty \} \cup \{l_{3}:r\to r^{+}_{\text{h}}, 0 \le \theta \le \pi \} \cup \{l_{4}:\theta\to 0^{+}, r_{\text{h}} \le r < +\infty \}$. $C_{1}$ is the union of dashed line segments: $\{l^{1}_{1}:r\to +\infty, \theta_{0} \le \theta \le \pi-\theta_{0} \} \cup \{l^{1}_{2}:\theta = \pi-\theta_{0}, r_{\text{h}} \le r < +\infty \} \cup \{l^{1}_{3}:r\to r^{+}_{\text{h}}, \theta_{0} \le \theta \le \pi-\theta_{0} \} \cup \{l^{1}_{4}:\theta=\theta_{0}, r_{\text{h}} \le r < +\infty \}$, with $\theta_{0} \in (0, \pi/2)$.
• Figure 2: The behaviour of the unit vector field $n$ in the $(r,\theta)$ plane in different dimensions, and the varying behaviour in $\phi$ space corresponds to the respective curves $C^{a}_{1}$, $C^{a}_{2}$ and $C^{a}_{3}$. The red arrow represents the direction of $n$, and the black dot represents the zero point of $n$. The blue dashed contour lines $C^{a}_{1}$, $C^{a}_{2}$ and $C^{a}_{3}$ are all closed ellipse centred on the zero point, and the solid contour lines $C^{a}_{4}$, $C^{a}_{5}$ and $C^{a}_{6}$ are the changes in the components $(\phi^{r} , \phi^{\theta})$ of $\phi$ along $C^{a}_{1}$, $C^{a}_{2}$ and $C^{a}_{3}$. (a) The unit vector field for the Hayward black hole with $D=5$, $\alpha = 1$ and $m=9$. (b) The unit vector field for the Hayward black hole with $D=7$, $\alpha = 1$ and $m=9$. (c) The unit vector field for the Hayward black hole with $D=10$, $\alpha = 1$ and $m=9$. (d) The behaviour of $\phi$ in $\phi$ space for $C^{a}_{1}$, $C^{a}_{2}$ and $C^{a}_{3}$.
• Figure 3: $\Delta\Omega$ as a function of $\vartheta$ in different dimensions.
• Figure 4: The behaviour of the unit vector field $n$ in the $(r,\theta)$ plane in different dimensions, and the varying behaviour in $\phi$ space corresponds to the respective curves $C^{b}_{1}$, $C^{b}_{2}$ and $C^{b}_{3}$. The red arrow represents the direction of $n$, and the black dot represents the zero point of $n$. The blue dashed contour lines $C^{b}_{1}$, $C^{b}_{2}$ and $C^{b}_{3}$ are all closed ellipse centred on the zero point, and the solid contour lines $C^{b}_{4}$, $C^{b}_{5}$ and $C^{b}_{6}$ are the changes in the components $(\phi^{r} , \phi^{\theta})$ of $\phi$ along $C^{b}_{1}$, $C^{b}_{2}$ and $C^{b}_{3}$. (a) The unit vector field for the Dymnikova-like black hole with $D=5$, $\alpha = 1$ and $m=9$. (b) The unit vector field for the Dymnikova-like black hole with $D=7$, $\alpha = 1$ and $m=9$. (c) The unit vector field for the Dymnikova-like black hole with $D=10$, $\alpha = 1$ and $m=9$. (d) The behaviour of $\phi$ in $\phi$ space for $C^{b}_{1}$, $C^{b}_{2}$ and $C^{b}_{3}$.
• Figure 5: $\Delta\Omega$ as a function of $\vartheta$ in different dimensions.