Numerical Methods

All results presented in this work are supported by explicit numerical integration of the spacetime geometry and particle dynamics. The numerical framework is designed to isolate the effects of the regular interior while maintaining full consistency with classical general relativity in the exterior. In this section we describe the construction of the metric, the geodesic integration algorithms, the ray-tracing procedure used to generate shadow images, and the tests performed to ensure numerical robustness.

Metric Construction and Boundary Conditions

We consider a static, spherically symmetric line element of the form

where the metric function $f(r)$ is chosen to interpolate smoothly between a regular interior and an asymptotically Schwarzschild exterior. A representative choice used in the simulations is

with $M$ the ADM mass and $g$ a length scale controlling the size of the regular core.

Boundary conditions are imposed as follows:

• As $r \to \infty$, $f(r) \to 1 - 2M/r$, ensuring asymptotic flatness.
• As $r \to 0$, $f(r) \to 1 - \Lambda_{\rm eff} r^2$, yielding a regular, de Sitter–like core.

This construction guarantees finiteness of curvature invariants and geodesic completeness by design.

Geodesic Integration Algorithms

Particle and photon trajectories are computed by integrating the geodesic equations

where $\lambda$ is an affine parameter. Exploiting spherical symmetry, motion is restricted to the equatorial plane without loss of generality.

The equations are reduced to a system of first-order ordinary differential equations using conserved energy $E$ and angular momentum $L$. Numerical integration is performed using a fourth- or fifth-order adaptive Runge–Kutta method with dynamic step-size control. This ensures accurate handling of both weak-field trajectories at large radius and rapidly varying dynamics near the photon sphere and interior region.

Energy and angular momentum conservation are monitored throughout the integration as internal consistency checks.

Ray-Tracing and Shadow Imaging

To generate black hole shadow images, a backward ray-tracing approach is employed. Light rays are initialized at a distant observer screen with coordinates $(\alpha,\beta)$ corresponding to impact parameters. Each ray is propagated backward in time by integrating the null geodesic equations until one of the following conditions is met:

• The ray escapes to large radius, corresponding to a bright pixel.
• The ray enters a trapping region near the core or undergoes multiple orbits, corresponding to the shadow or photon ring.

The shadow boundary is identified as the set of critical initial conditions separating escaping from trapped trajectories. Intensity maps are constructed by counting trajectory density and accumulated affine time, yielding a first-order approximation to brightness and ring structure.

The same procedure is repeated for the Schwarzschild metric to enable direct comparison under identical numerical conditions.

Convergence and Stability Tests

To ensure reliability of the numerical results, several convergence and stability tests are performed:

• Step-size refinement tests confirm convergence of trajectories and shadow boundaries.
• Grid-resolution studies verify stability of the shadow shape and photon ring profile.
• Limiting-case checks confirm recovery of Schwarzschild results as $g \to 0$.
• Symmetry tests verify invariance of results under rotations of the observer screen.

All reported features—including shadow size, ring thickness, and effective asymmetry—are robust under variations of numerical parameters. Residual numerical uncertainties are significantly smaller than the physical effects discussed in the preceding sections.

The numerical framework thus provides a controlled and reproducible platform for probing the physical consequences of singularity-free interior geometries.