Geometric Saturation of the Scale Function in Black-Bounce Spacetimes: Spherical, Planar, and Hyperbolic Transverse Sections

Abstract

We present a new construction for black-bounce spacetimes based on a deformation of the scale function that acts as a geometric saturation rather than introducing a prescribed constant minimal length, as in conventional black-bounce models. This function encodes the gravitational information of the underlying vacuum counterpart, keeping the geometric quantities of the deformed metric finite in the short-distance region where tidal forces and curvature invariants of the vacuum geometry diverge. The deformation saturates at short scales, suppressing curvature divergences without requiring a potentially unstable de Sitter core. The resulting regular geometries with spherical, planar, and hyperbolic transverse sections describe regular black holes (RBHs), extremal RBHs, and traversable wormholes. A key result is that the deformed geometry remains finite at $l=0$ in all regimes, while, depending on the parameter space, the bounce, identified with the minimum of $R(l)$, may remain in the short-distance region or shift to a larger finite value of $l$. Thus, the deformation regularizes the central region and, in some regimes, also modifies the global spacetime profile. Spherical and planar RBHs satisfy the standard energy conditions near the bounce, showing that a geometric bounce does not necessarily require exotic matter sources in this region. In the hyperbolic case, the energy conditions depend more strongly on the mass parameter, being satisfied near the bounce for positive-mass RBHs but violated for negative-mass and extremal configurations. The hyperbolic sector also admits regular negative-mass black holes. For wormhole geometries, the WEC and NEC are violated near the throat, as expected.