Light bending near non-asymptotically flat black holes

The gravitational deflection of light is a crucial test for modified gravity. A few years ago, Gibbons and Werner introduced a definition of the deflection angle based on the Gauss-Bonnet theorem. A related idea was proposed by Arakida for defining the deflection angle in non-asymptotically flat spacetimes We revisit this idea in the Kottler geometry and in a non-asymptotically flat solution to Horndeski gravity. Our analytic and numerical calculations show that a triangular array of laser beams can be designed so that the proposed definition of deflection angle is sensitive to a cosmological constant, whose contribution is amplified by the black hole mass. Moreover, we find that near the photon sphere, the deflection angle in the Horndeski solution is similar to its Schwarzschild counterpart, and we confirm that the shadows seen by a static observer would be identical. Our results offer insights that could be useful for designing future theoretical or experimental investigations aimed to detect sources of curvature in the universe.