Curvatures, Photon Spheres and Black Hole Shadows

TL;DR

This paper extends a geometric method for determining photon spheres and black hole shadows to more general spherically symmetric black holes, showing equivalence with traditional effective potential methods.

Contribution

It generalizes a geometric approach to a broader class of spherically symmetric black holes and proves its equivalence to conventional methods.

Findings

Extended geometric approach to general spherically symmetric black holes.

Proved equivalence between geometric and effective potential methods.

Applicable to a wider class of black hole spacetimes.

Abstract

In a recent work PRD 106, L021501 (2022), a new geometric approach is proposed to obtain the photon sphere (circular photon orbit) and the black hole shadow radius. In this approach, photon spheres and the black hole shadow radius are determined using geodesic curvature and Gaussian curvature in the optical geometry of black hole spacetimes. However, the calculations in PRD 106, L021501 (2022) only restricted to a subclass of static and spherically symmetric black holes with spacetime metric $g_{tt} \cdot g_{rr}=-1$, $g_{\theta\theta}=r^{2}$ and $g_{\phi\phi}=r^{2}\sin^{2}\theta$. In this work, we extend this approach to more general spherically symmetric black holes (with spacetime metric $ds^{2}=g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}$). Furthermore, it can be proved that our results from the geometric approach are completely equivalent to those from conventional approach based on effective potentials of test particles.