Photon Dynamics and Black Hole Shadow

Photon trajectories provide one of the most sensitive probes of strong-field gravity. In particular, the existence of a photon sphere and the resulting black hole shadow encode detailed information about the underlying spacetime geometry. In this section we analyze null geodesics in the regular interior metric, construct the corresponding shadow via ray-tracing, and compare the results with the Schwarzschild prediction.

Null Geodesics in Regular Spacetimes

Photon motion is governed by null geodesics,

where $\lambda$ is an affine parameter. For a static, spherically symmetric metric of the form

the equations of motion reduce to an effective radial equation

with photon effective potential

where $E$ and $L$ are conserved energy and angular momentum.

In the regularized geometry, $f(r)$ remains finite and smooth for all $r \ge 0$. Consequently, the photon effective potential is finite everywhere and admits well-defined extrema. Unlike the Schwarzschild case, where the interior region is classically inaccessible, null geodesics can be extended smoothly through the core without encountering divergences.

Photon Sphere and Shadow Formation

The photon sphere is defined by unstable circular null orbits satisfying

In Schwarzschild spacetime this yields $r_{\text{ph}} = 3GM$. In the regular model, the photon sphere radius is modified to

where $\delta r(g)$ depends on the regularization scale $g$ and vanishes in the limit $g \to 0$.

The black hole shadow observed by a distant observer is determined by the critical impact parameter $b_c = L/E$ associated with this unstable orbit. To leading order,

For $g \ll GM$, the correction to $b_c$ is parametrically suppressed, implying that the angular size of the shadow deviates only minimally from the Schwarzschild value. Thus, current observational constraints from black hole imaging are naturally satisfied.

Ray-Tracing and Interior Trajectories

To visualize the shadow and explore potential interior signatures, we perform numerical ray-tracing by integrating null geodesics backward from a distant observer's screen. Each light ray is classified according to whether it escapes to infinity or probes the deep interior region.

In the regular geometry, a subset of null geodesics penetrates below the photon sphere, reaches the core, and is deflected back outward. This behavior is impossible in Schwarzschild spacetime, where such rays terminate at the singularity. As a result, the regular model admits a notion of interior photon trajectories, which can in principle generate faint secondary images or modifications to the inner brightness profile of the shadow.

However, numerical simulations indicate that these interior contributions are highly suppressed in intensity. The dominant shadow boundary remains controlled by the photon sphere, ensuring near-indistinguishability from a classical black hole in current observations.

Comparison with Schwarzschild Geometry

Comparing the regular spacetime with the Schwarzschild solution reveals both similarities and crucial differences. Externally, the photon sphere, critical impact parameter, and overall shadow diameter coincide with Schwarzschild predictions up to corrections of order $\mathcal{O}(g^3/(GM)^3)$. This guarantees consistency with existing black hole images, such as those from the Event Horizon Telescope.

Internally, however, the conceptual picture changes radically. The Schwarzschild solution predicts geodesic incompleteness and absorption of photons at the singularity, whereas the regular model allows complete null trajectories with finite curvature everywhere. From an observational standpoint, this distinction is subtle but potentially testable in future high-resolution or multi-wavelength imaging, polarization measurements, or time-dependent lensing phenomena.

In summary, photon dynamics in the regularized spacetime reproduce the classical shadow structure while providing a singularity-free and geodesically complete description of light propagation in the deepest strong-field regime.