Geodesically complete regularized Schwarzschild black holes

Classical general relativity predicts a singularity at the center of a black hole, where known physical laws break down, suggesting the need for deeper principles of Nature. Among various theoretical possibilities, one promising proposal is a transition to a de Sitter phase at high curvatures near the black hole core. Originally proposed by Gliner and Sakharov, this transition ensures regular metric behavior and avoids singularities. In search of a regular black hole solution with finite curvature scalar, we propose a static, spherically symmetric metric featuring a de Sitter-like core. Notably, this solution admits a single event horizon , resembling the Schwarzschild geometry, and depends solely on the black hole mass, providing a connection with the Israel-Carter no-hair theorem for uncharged non-rotating black holes. To determine the underlying theory, we adopt a generalized f ( R ) model in place of the Einstein-Hilbert action. By numerically solving the modified field equations, we find that a Padé approximant in the Ricci scalar R , added to the Einstein-Hilbert term, yielding such regular solutions. We verify the physical viability of this model by confirming that the metric satisfies the null, weak, and dominant energy conditions throughout the spacetime. Additionally, the strong energy condition holds for r ≥ r h /2, consistent with Zaslavskii’s regularity criterion. The expansion scalar in the Raychaudhuri equation remains finite, and its derivative vanishes at the core ( r = 0 ), preventing caustic formation. This ensures geodesic completeness and the absence of physical singularities.