On the analytic generalization of particle deflection in the weak field regime and shadow size in light of EHT constraints for Schwarzschild-like black hole solutions

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In this paper, an analytic generalization of the weak field deflection angle (WDA) is derived by utilizing the current non-asymptotically flat generalization of the Gauss–Bonnet theorem. The derived formula is valid for any Schwarzschild-like spacetime, which deviates from the classical Schwarzschild case through some constant parameters. This work provided four examples, including Schwarzschild-like solutions in the context of Bumblebee gravity theory and the Kalb–Ramond framework, as well as one example from a black hole surrounded by soliton dark matter. These examples explore distinct mechanisms of Lorentz symmetry breaking, with results that are either new or in agreement with existing literature. The WDA formula provided a simple calculation, where approximations based on some conditions can be done directly on it, skipping the preliminary steps. For the shadow size analysis, it is shown how it depends solely on the parameter associated with the metric coefficient in the time coordinate. A general formula for the constrained parameter is also derived based on the Event Horizon Collaboration (EHT) observational results. Finally, the work realized further possible generalizations on other black hole models, such as RN-like, dS/AdS-like black hole solutions, and even black hole solutions in higher dimensions.