Image Asymmetry and Effective Spin-Like Signatures

Although the spacetime under consideration is strictly static and spherically symmetric at the level of the metric ansatz, the presence of a regular interior can induce subtle asymmetries in the observed image. These effects arise not from true angular momentum, but from the structural properties of photon propagation in the modified geometry. In this section we analyze the origin of such asymmetries, their interpretation as effective spin-like signatures, and their potential observational relevance.

Origin of Image Asymmetry

In classical Schwarzschild spacetime, the black hole shadow is perfectly circular and symmetric for a distant observer, reflecting the underlying spherical symmetry. In the regularized geometry, the exterior metric remains spherically symmetric, but photon trajectories that probe deeper regions experience nontrivial deviations due to the modified interior structure.

Specifically, photons with nearly critical impact parameters may penetrate below the photon sphere before being scattered back outward. The detailed path length, redshift accumulation, and deflection angle of these trajectories depend sensitively on how deeply they enter the core. When mapped onto the observer’s screen through ray-tracing, this leads to slight angular-dependent variations in intensity and path density, breaking perfect circular symmetry at subleading order.

Importantly, this asymmetry does not correspond to a preferred spatial direction in the spacetime itself, but emerges from the nonlinear mapping between interior photon dynamics and the observer’s image plane.

Effective Spin Interpretation

The resulting image distortions can mimic features commonly associated with rotating (Kerr) black holes. In Kerr spacetime, frame-dragging introduces an intrinsic asymmetry between co-rotating and counter-rotating photon orbits, leading to a displaced and distorted shadow.

In the present model, no true rotation parameter exists. Nevertheless, the effective bending asymmetry produced by the regular core can be parametrized by an effective spin parameter $a_{\mathrm{eff}}$, defined operationally by matching the observed shadow displacement to that of a Kerr black hole:

Numerical ray-tracing indicates that $a_{\mathrm{eff}}$ scales with the regularization length $g$ as

and thus vanishes smoothly in the Schwarzschild limit. This correspondence implies that a non-rotating, regular black hole could observationally resemble a slowly spinning Kerr black hole, introducing a degeneracy between true angular momentum and interior structure.

Brightness Profiles and Ring Structure

Beyond the shadow boundary itself, the photon ring and surrounding brightness profile provide additional diagnostics. The photon ring arises from photons executing multiple near-critical orbits before escaping, and its thickness and intensity depend on the stability properties of the photon sphere.

In the regular geometry, the modified effective potential alters the Lyapunov exponent governing photon orbit instability. As a result, the photon ring can exhibit slight changes in width and intensity falloff compared to the Schwarzschild case. In particular:

• The ring thickness may increase marginally due to reduced instability near the photon sphere.
• The radial brightness profile may show a softened inner edge, reflecting partial penetration of photons into the core.
• Higher-order subrings may be suppressed or enhanced depending on the core scale.

These effects are small but systematic, and they persist even when the overall shadow diameter remains unchanged.

Observational Relevance

From an observational perspective, the predicted asymmetries and ring modifications are subtle and lie near the threshold of current observational capabilities. Existing Event Horizon Telescope (EHT) images are consistent with both Schwarzschild and slowly rotating Kerr geometries, and therefore do not exclude the regular model.

However, future improvements in angular resolution, dynamic range, and polarization sensitivity may allow discrimination between true spin-induced asymmetry and structurally induced effective spin. In particular:

• Multi-frequency imaging could reveal wavelength-dependent brightness variations tied to interior photon paths.
• Polarimetric observations may detect deviations in magnetic field alignment patterns.
• Time-dependent lensing or variability could expose differences in photon trapping times.

Consequently, image asymmetry provides a promising observational window into the internal structure of compact objects. While indistinguishable from Kerr rotation at leading order, the effective spin-like signatures predicted here represent a distinctive fingerprint of singularity-free gravity models.