Mathematical Details

Regular Metric Construction

We consider a static, spherically symmetric spacetime described by the line element

where $d\Omega^2$ is the metric on the unit two-sphere. Structural stability and the no-singularity condition require that the metric function $f(r)$ remain finite and smooth for all $r \ge 0$.

A representative regular construction is obtained by introducing an effective mass function $M(r)$ such that

where $M$ is the asymptotic mass and $g$ is a characteristic length scale associated with the regular core. This choice ensures that $M(r) \to M$ as $r \to \infty$ and $M(r) \sim r^3$ as $r \to 0$, preventing divergences in curvature.

Curvature Invariants

To verify regularity, we compute scalar curvature invariants such as the Ricci scalar $R$ and the Kretschmann invariant

For the metric above, one finds that $K$ remains finite everywhere and approaches a constant value at $r=0$. In particular,

demonstrating explicitly the absence of curvature singularities. All higher-order invariants constructed from contractions of the Riemann tensor are similarly bounded.

Weak-Field Expansions

In the weak-field regime $r \gg g$, the metric function admits an expansion

This guarantees that post-Newtonian corrections induced by the regular core are suppressed by high powers of $g/r$. As a result, standard weak-field predictions of general relativity are recovered to leading order, with deviations far below current experimental sensitivity.

Numerical Reproducibility

Integration Parameters

All numerical integrations were performed using dimensionless units with $G=c=1$. Typical simulations employed values $M=1$ and $g \in [10^{-3},10^{-1}]$ to explore both near-Schwarzschild and moderately regularized regimes. Radial grids extended from $r_{\min}=10^{-6}$ to $r_{\max}=10^{4}$ with adaptive step sizes.

Code Structure and Algorithms

The numerical implementation consists of three main modules:

1. Metric module: computes $f(r)$ and its derivatives.
2. Geodesic solver: integrates timelike and null geodesics using a fourth- or fifth-order Runge–Kutta scheme with adaptive step control.
3. Ray-tracing module: propagates photon bundles from a distant observer backward in time to construct shadow images.

Energy and angular momentum conservation were monitored throughout the integrations to ensure numerical accuracy.

Robustness Checks

Robustness was verified by varying integration tolerances, grid resolution, and initial conditions. All qualitative features—such as bounded infall trajectories, stable photon spheres, and shadow size—were found to be insensitive to these variations. In the limit $g \to 0$, the code reproduces standard Schwarzschild results, providing an additional consistency check.

Comparison with General Relativity

Conceptual Differences

Classical general relativity allows singular solutions that signal a breakdown of the theory at finite proper time. In contrast, the present framework imposes structural stability as a physical requirement, excluding singular spacetimes from the space of admissible solutions. This represents a shift from accepting singularities as physical endpoints to treating them as indicators of theoretical incompleteness.

Domain of Validity

Both general relativity and the singularity-free models considered here agree in the low-curvature regime, where all experimental tests have been performed. Differences arise only at extremely high curvature, deep inside compact objects, a domain currently inaccessible to direct observation.

Empirical Consistency

All observable predictions examined—light deflection, perihelion shift, black hole shadows, and orbital dynamics—are consistent with existing data. The framework therefore satisfies current empirical constraints while extending the theoretical domain of validity beyond that of classical general relativity.

Taken together, these appendices support the claim that singularity-free gravity provides a mathematically consistent, numerically robust, and empirically viable extension of classical gravitational theory.