Photon motion and redshift effects in geometrically charged Einstein–Gauss–Bonnet black holes

Abstract In this work we study static black hole solutions with geometric charge in Einstein–Gauss–Bonnet gravity, considering both positive and negative branches. We establish conditions for asymptotic flatness and horizon formation, and delineate parameter regions corresponding to black holes and naked singularities. Photon motion is analyzed using the Hamilton–Jacobi formalism, which demonstrate that the effective potential, photon-sphere radius, and horizon structure depend sensitively on the Gauss–Bonnet coupling, geometric charge, and cosmological constant, with different trends between the two branches. We further examine the gravitational redshift of photons emitted near the black hole and show its distinct parametric behavior for each branch, approaching flat-space limits at large distances. In the presence of a positive cosmological constant, we derive a generalized frequency-shift expression that separates strong-gravity and cosmological contributions, recovering the Hubble law asymptotically.