Regular Interior Geometry

Metric Ansatz and Asymptotic Behavior

To implement the structural stability and no-singularity requirements, we adopt a static, spherically symmetric metric ansatz of the form

where $d\Omega^2$ denotes the line element on the unit two-sphere. The function $f(r)$ is chosen such that the metric is asymptotically Schwarzschild at large radii,

ensuring agreement with classical general relativity in the weak-field regime. Near the origin, however, $f(r)$ departs from the classical form in order to prevent divergences in curvature invariants. This deviation is controlled by a characteristic length scale that sets the onset of non-classical behavior and encodes the regularization of the interior geometry.

Effective Interior Dynamics

The modification of the metric function $f(r)$ in the deep interior can be interpreted as arising from an effective gravitational dynamics that replaces the classical Einstein equations at high curvature. Rather than specifying a fundamental microscopic theory, we describe this regime phenomenologically through an effective stress-energy contribution that becomes relevant only near the core. In this picture, the interior geometry responds to an effective repulsive component that counteracts unlimited gravitational collapse. As a result, the interior dynamics smoothly interpolate between the classical exterior solution and a regular central region, without introducing discontinuities or singular behavior.

Regular Core Formation

As $r \to 0$, the metric approaches a regular core geometry characterized by finite curvature and well-defined causal structure. A typical behavior is

with $\alpha>0$, corresponding to a de Sitter–like or otherwise non-singular interior. In this regime, all scalar curvature invariants remain bounded, and the spacetime avoids the formation of a curvature singularity. The precise form of the core depends on the chosen regularization scheme, but the qualitative feature of a smooth, non-singular center is generic under the imposed stability constraints.

Geodesic Completeness

An essential consequence of the regular interior geometry is geodesic completeness. Timelike and null geodesics can be extended through the core region to arbitrary values of their affine parameters without encountering divergences or boundaries of the spacetime manifold. In contrast to the classical Schwarzschild solution, where geodesics terminate at a singularity in finite proper time, the regularized geometry allows infalling observers and light rays to pass smoothly through the high-curvature region. This restores predictability and ensures that the spacetime is globally well-defined, satisfying a key requirement of structural stability.