# A Note on Geodesics in Hayward Metric

## TL;DR

This paper analyzes geodesics in Hayward's non-singular black hole metric, revealing unique features like photon spheres and stable orbits, and visualizes black hole images showing bright rings even without a photon sphere.

## Contribution

It provides a detailed study of geodesic structures in Hayward's metric, including critical parameters and visualizations, highlighting differences from classical black holes.

## Key findings

- Photon sphere exists without horizon.
- Multiple marginally stable circular orbits identified.
- Bright rings appear even without photon sphere.

## Abstract

We study timelike and null geodesics in a non-singular black hole metric proposed by Hayward. The metric contains an additional length-scale parameter $\ell$ and approaches the Schwarzschild metric at large radii while approaches a constant at small radii so that the singularity is resolved. We tabulate the various critical values of $\ell$ for timelike and null geodesics: the critical values for the existence of horizon, marginally stable circular orbit and photon sphere. We find the photon sphere exists even if the horizon is absent and two marginally stable circular orbits appear if the photon sphere is absent and a stable circular orbit for photons exists for a certain range of $\ell$. We visualize the image of a black hole and find that blight rings appear even if the photon sphere is absent.