Strong-Field Regime

While the weak-field limit guarantees consistency with all classical tests, the defining features of a singularity-free theory manifest themselves in the strong-field regime. In this section we analyze particle dynamics deep inside the gravitational potential, where classical general relativity predicts pathological behavior. We show that the regular interior geometry leads to well-defined motion, modified effective potentials, and geodesically complete trajectories without introducing observationally excluded effects.

Radial Infall of Massive Particles

Consider a massive test particle falling radially from rest at infinity. In Schwarzschild spacetime, such a particle reaches the central singularity at $r=0$ in finite proper time, where the classical description breaks down. In the regularized geometry introduced here, the metric function remains finite as $r \to 0$, and the radial geodesic equation takes the schematic form

with $V_{\text{eff}}(r)$ finite everywhere.

As the particle approaches the core region $r \sim g$, the effective gravitational attraction is gradually weakened by the regularization mechanism. The infall velocity reaches a maximum and then decreases smoothly, preventing divergent acceleration. The particle reaches the center with finite proper time, finite tidal forces, and finite curvature invariants. No singular endpoint exists, and the trajectory can be smoothly extended beyond the core if the spacetime is analytically continued. This behavior demonstrates explicit geodesic completeness for radial timelike geodesics.

Non-Radial Orbits

For particles with nonzero angular momentum, the motion is governed by an effective potential incorporating both centrifugal and gravitational terms. At large radii, the potential coincides with the Schwarzschild case, yielding standard bound and unbound orbits. As the particle probes smaller radii, deviations appear due to the modified interior geometry.

Numerical integration of non-radial geodesics shows that bound orbits remain stable down to radii close to the regular core. Unlike the Schwarzschild solution, no trajectory encounters an infinite potential barrier or divergent force. Highly eccentric orbits experience a smooth modification of their periapsis behavior rather than catastrophic plunge. This implies that the notion of orbital motion remains meaningful throughout the spacetime, including regions where classical general relativity fails.

Effective Potential Structure

The qualitative behavior of geodesics can be understood by analyzing the effective potential for massive particles,

where $f(r)$ is the metric function and $L$ the conserved angular momentum. In the regular model, $f(r)$ approaches a finite value as $r \to 0$, causing the effective potential to flatten rather than diverge.

This structure eliminates the unphysical infinite well associated with the Schwarzschild singularity. Instead, the potential develops a smooth minimum or plateau near the core, depending on the angular momentum. The absence of divergences ensures that classical notions of stability and boundedness remain applicable even in the deepest strong-field region.

Innermost Stable Circular Orbits

An important observable feature of strong gravity is the existence of an innermost stable circular orbit (ISCO). In Schwarzschild spacetime, the ISCO for massive particles occurs at $r = 6GM$. In the regularized geometry, the ISCO condition is modified by terms depending on the regularization scale $g$.

Solving the conditions

we find that the ISCO radius is shifted by an amount

For realistic values of $g \ll GM$, this shift is negligible, implying that accretion disk dynamics and orbital frequencies remain essentially unchanged from their general relativistic predictions. Only when the orbit approaches the regular core do genuinely new features emerge.

In summary, the strong-field regime of the theory replaces singular behavior with smooth, finite dynamics while preserving all externally observable properties of black holes. The interior becomes physically well-defined without compromising the phenomenological success of classical gravity.