Extended uncertainty principle inspired black hole in a G\"odel Universe

We explore analytically the implications of a curvature-modified extended uncertainty principle (EUP) derived in a rotating G\"odel spacetime and apply it to the construction of a semiclassical black hole model. Adapting techniques from corpuscular black hole frameworks, we reinterpret the G\"odel-type uncertainty relation as an effective energy bound, leading to a modified lapse function with explicit dependence on the global rotation parameter a and the radial coordinate r₀. Analytic expressions are derived for key gravitational features, including the event horizon, photon spherehere, shadow radius, and deflection angle, with curvature corrections scaling as a⁻² and r₀² / a⁴. Series expansion in the limit a → ∞ shows that global rotation consistently increases all observables relative to the Schwarzschild case. Applying these results to astrophysical data, we use Event Horizon Telescope (EHT) measurements of Sgr A* and M87* to infer lower bounds of a/M ~ 10⁵, while solar system light-bending observations in the parametrized post-Newtonian (PPN) framework yield a / M_ ~ 5 × 10⁴. These large but finite values validate the asymptotic expansion and confirm that G\"odel-type rotation remains observationally suppressed, yet theoretically coherent. Our results demonstrate that global rotation, when treated semiclassically via curvature-modified uncertainty, introduces detectable signatures in principle, though well below current observational sensitivity. The framework offers a consistent path toward exploring the quantum-gravitational interplay between global geometry and local black hole structure.