White holes in the noncommutative Painlevé–Gullstrand coordinates

In this study, we explore white holes using noncommutative Painlevé–Gullstrand (PG) coordinates, integrating noncommutative geometry within the framework of general relativity to describe white holes and eliminate metric singularities at the event horizon. We derive modified Wheeler–DeWitt (WdW) equations and perturbative solutions to address noncommutative effects, finding that these reduce the event horizon radius, decrease entropy, and increase temperature compared to classical Schwarzschild black holes. Additionally, we analyze the probability of quantum tunneling transitions between black holes and white holes, demonstrating that noncommutativity increases this probability, especially for higher mass black holes. The dynamics of massive and massless particles in the vicinity of these noncommutative objects are also explored, revealing significant alterations in effective potential and stability of particle trajectories due to noncommutative effects. This theoretical framework offers novel insights into high-energy physics by bridging general relativity with quantum mechanics through the lens of noncommutative geometry.