Existence of Halos Outside Schwarzschild-$f(R)$ Black Holes

TL;DR

This paper explores the existence of stable photon orbits, or halos, outside Schwarzschild-$f(R)$ black holes, revealing new potential observational signatures in modified gravity theories through analytical and numerical analysis.

Contribution

It demonstrates that Schwarzschild-$f(R)$ black holes can support multiple photon orbits, including stable halos, unlike standard Schwarzschild black holes, expanding understanding of photon structures in alternative gravity models.

Findings

Multiple photon orbits can exist outside the event horizon for certain $f(R)$ models.

Schwarzschild-$f(R)$ black holes can have an additional stable photon orbit (halo) without instability.

Potential observational signatures include deviations in black hole shadow size or shape.

Abstract

We investigate the possibility of photon halos (stable photon orbits) forming outside Schwarzschild-$f(R)$ black holes by analyzing null geodesics in these spacetimes. Using methods inspired by studies of spherical photon orbits around Kerr-Newman black holes, we derive conditions for the existence of such halos. We examine several f(R) gravity models, including quadratic, logarithmic, exponential, cubic, power-law, and hyperbolic forms, and find that multiple photon orbits -- both stable and unstable -- can appear outside the event horizon for certain parameter ranges. These additional orbits (halos) provide new insights into spacetime geometry and potential observational signatures of black holes in modified gravity. We present analytical expressions for the orbital radii, perform a numerical stability analysis, and discuss possible observational implications for black hole shadows. Our results indicate that while the standard Schwarzschild black hole admits only a single unstable light ring, Schwarzschild-$f(R)$ black holes can support an additional outer stable photon orbit (a halo) without triggering a black-hole bomb instability. This work deepens the understanding of photon-orbit structures in alternative theories of gravity and highlights how such effects could be detected through deviations in black hole shadow size or morphology.