Shadows and photon motions in the axially symmetric Finslerian extension of a Schwarzschild black hole

In this paper, we have conducted a thorough investigation into the shadow characteristics exhibited by the axially symmetric Finslerian extension of a Schwarzschild black hole and supplementary numerically simulated the motion behavior of photons near the black hole with respect to varying Finsler parameter and observation inclinations ₎. In this spacetime, the apparent shape and size of the black hole shadow remain the same as the one in the Schwarzschild black hole, regardless of changes in the Finsler parameter. However, the apparent position of shadow in the observer's sky plane appears to shift along the horizontal axis, and the distance moved is positively correlated with the value of the Finsler parameter. Interestingly, with a fixed value of, the displacement of the visual position of shadow is rectified as the observation angle decreases. In addition, it is demonstrated that the motion behavior of photons around the axially symmetric Finslerian extension of a Schwarzschild black hole has a greater degree of deflection than a Schwarzschild black hole, although their shadows are in some cases consistent in shape, size, and apparent position. These results can serve as a valuable tool in distinguishing the axially symmetric Finslerian extension of a Schwarzschild black hole from its counterparts in general relativity.