Geodesically Complete Regularized Schwarzschild Black Holes

Classical general relativity predicts a singularity at the center of a black hole, where known laws of physics break down. This suggests the existence of deeper, yet unknown principles of Nature. Among various theoretical possibilities, one of the most promising proposals is a transition to a de Sitter phase at the BH core. This transition, originally proposed by Gliner and Sakharov, ensures the regularity of metric coefficients and avoids the singularity. In search for such a regular BH solution with finite curvature scalar, we propose a metric $g_{rr}$ that exhibits a dS core in the central region. An appealing feature of this metric is the existence of a $single$ event horizon resembling the Schwarzschild black hole. Furthermore, the entire spacetime geometry is determined by the black hole mass alone, in agreement with the Isarel-Carter $no-hair$ $theorem$ for a charge-less, non-rotating black hole. To determine the gravitational action consistent with such a solution, we consider a general Lagrangian density $f(R)$ in place of the Einstein-Hilbert action. By numerically solving the resulting field equation, we find that, in addition to the Einstein-Hilbert term, a Pad\'e approximant in the Ricci scalar $R$ can produce such regular black hole solutions. To assess the physical viability of these black hole solutions, we verify that the proposed metric satisfies the principal energy conditions: DEC, WEC, and NEC, throughout spacetime. Furthermore, in agreement with Zaslavskii's regularity criterion, the metric satisfies the SEC in the range $r\geq r_h/2$, where $r_h$ is the event horizon. Furthermore, with the proposed regularized metric, the expansion scalar in the Raychaudhuri equation remains finite and its derivative vanishes at $r=0$, thereby preventing formation of caustic. This confirms that the spacetime is geodesically complete and free from true physical singularities.