This paper proposes a new, simple way to remove the central singularities that usually appear inside black hole models. The author assumes that the quantity they call gravitational tension — a measure built from spacetime curvature — does not grow without bound at very small distances. Instead it saturates to a finite critical value. That saturation changes the short-distance geometry and produces a smooth “bounce” instead of a singular point.

How the idea works is explained with an explicit, effective model. The paper builds a screening factor Γ that looks like exp(−Fc/F), where F is the vacuum gravitational tension and Fc is the chosen critical tension. The effective tension is then Feff = F(1 − Γ). When curvature is low (large distances) Γ is tiny and Feff ≈ F, so the usual spacetime is recovered. When curvature is very high, Γ approaches one and Feff approaches the finite value Fc. This change is fed back into the short-distance scale function of the metric, for example by modifying the extended radial coordinate l^2 → l^2 + correction that depends on the screening. The result is a regular bounce geometry without inserting an ad hoc regular core.

The construction produces several kinds of regular geometries. For different choices of the transverse two-dimensional section — spherical, planar, or hyperbolic — the model yields regular black holes, extremal regular black holes, and traversable wormholes. A notable outcome is that the location of the bounce is not put in by hand. It appears dynamically from the competition between the gravitational tension and the screening. Depending on parameters, the bounce can remain at very short scales or move outward to a larger finite radius. That means saturation effects could change not only the inner core of compact objects but also some aspects of their global shape.

The paper also checks energy conditions, which are rules about reasonable matter behavior in general relativity. It finds that planar and hyperbolic regular black holes may satisfy the usual energy conditions near the bounce. The hyperbolic case shows additional interesting features, such as regular negative-mass configurations and a strong dependence of the energy conditions on model parameters. As expected, the matter required to hold open traversable wormholes violates the energy conditions near the throat.