Light Deflection due to Spinoptic Effects in Parametrized and Spherically Symmetric Hairy Black Holes

Abstract: In the standard geometric optics approximation, null rays propagating in a spherically symmetric black hole background follow planar geodesics. This picture changes, however, when the helicity-dependent effects of light are incorporated into the dynamics. Specifically, the interaction between the helicity of light and the spacetime curvature induces a significant angular deflection out of the geodesic plane. In this paper, we employ the spinoptics formalism to study light deflection due to the helicity-curvature interaction in two spherically symmetric geometries: the Rezzolla--Zhidenko (RZ) parametrized metric, and a hairy regular black hole solution obtained via gravitational decoupling. Our results reveal clear imprints of both the RZ parametrization coefficients and the hairy black hole parameter on the deflection angle. Furthermore, we assess the viability of using the RZ parametrization to mimic the regular hairy black hole, discussing the validity and limitations of such an approximation.

• The paper demonstrates that spinoptics corrections lead to helicity-dependent deviations from conventional null geodesics, refining photon trajectory models.
• It applies both Rezzolla-Zhidenko parametrization and gravitational decoupling to derive metrics, highlighting the limitations of first-order approximations for large hair parameter B.
• Numerical integrations reveal pronounced gravitational birefringence near the photon sphere, offering new insights for black hole shadow imaging and tests of beyond-GR effects.

Light Deflection via Spinoptic Effects in Parametrized and Hairy Black Holes

Background and Motivation

The propagation of electromagnetic radiation in curved spacetime is central to both theoretical investigations and observational astrophysics, including the interpretation of black hole shadow images. Conventional geometric optics treats photon trajectories as null geodesics, insensitive to photon helicity. However, it omits the gravitational spin-Hall effect, wherein light rays acquire helicity-dependent deviations from the geodesic plane due to helicity-curvature coupling. The spinoptics formalism corrects this by including finite-frequency effects, yielding modifications of order $O(1/\omega)$ to ray dynamics. Prior works established spinoptics results for Schwarzschild and Kerr spacetimes; this paper extends the analysis to spherically symmetric black holes described by both parametrized metrics (Rezzolla-Zhidenko, RZ) and regular, hairy solutions constructed via gravitational decoupling (GD).

Regular Hairy Black Hole Construction

The paper reproduces the derivation of a regular hairy black hole metric using the GD method, where the energy-momentum tensor is decomposed into a seed component and an additional source. By deforming both the radial and temporal components of the Schwarzschild metric and imposing the weak energy condition, a non-singular hairy metric emerges, parameterized by $B$. This parameter moderates both the event horizon location and the central curvature, with the spacetime regularity improved as $B$ increases. The Kretschmann scalar is shown to decrease with rising $B$, indicative of diminished curvature and regularization at the black hole core. Critical values of $B$ determine whether the solution possesses two horizons, one horizon (extremal configuration), or none.

Rezzolla-Zhidenko Parametrization and Its Fidelity

The RZ parametrization employs continued fraction expansions in compact radial coordinates to efficiently describe deviations from Schwarzschild, achieving high accuracy with minimal expansion coefficients. The paper rigorously matches the hairy GD solution to the RZ form by expanding the metric function and relating RZ parameters ($\epsilon, a_1$) to the hair parameter $B$. Numerically, shadow radius computations reveal that approximation accuracy deteriorates with larger $B$, with relative errors reaching ~6.8% near the critical hair parameter. The first-order RZ parametrization is shown to be insufficient for high $B$ regimes, requiring higher-order corrections to maintain fidelity.

Spinoptics Formalism in Spherically Symmetric Spacetimes

Adopting the effective action approach (Frolov), the spinoptics equations are derived for high-frequency electromagnetic waves in general spherically symmetric metrics. The key result is that null rays interacting with spacetime curvature via helicity are no longer confined to planar geodesics. The formalism is adapted to metrics with $g_{rr}g_{tt} \neq 1$, requiring the construction of generalized parallel-propagated null tetrads. The Hamilton-Jacobi equations governing photon propagation acquire helicity-dependent corrections, leading to explicit expressions for the off-planar deviation angle.

Numerical Results: Light Deflection and Birefringence

Numerical integration of the spinoptics equations for both RZ-parametrized and hairy GD black holes demonstrates strong dependence of the off-equatorial deflection angle on black hole parameters ($B$0, $B$1, $B$2, $B$3). For RZ metrics, increasing $B$4 correlates with larger horizon radii and stronger gravitational fields, enhancing spin-curvature induced deflection. Similarly, in the hairy GD metric, increasing $B$5 systematically attenuates the spinoptics effect, especially close to the horizon; distant rays remain largely unaffected. The results robustly reproduce Schwarzschild values in the limit of vanishing deviation parameters.

Spinoptics-induced birefringence is confirmed: photons with opposite helicities experience equal and opposite deflection, leading to trajectory separation. This gravitational birefringence effect is pronounced for photons approaching the photon sphere, with implications for the polarization structure of black hole shadows.

Comparison: Parametrized vs. Hairy Metrics

By comparing spinoptics-induced deflection angles in the RZ and hairy GD metrics, the paper quantitatively establishes the limitations of RZ parametrization in replicating the hairy solution's spinoptics signatures, particularly as $B$6 nears its critical value. Relative errors of up to 500% are observed for large $B$7, clearly demarcating the boundaries of the RZ approach's applicability without higher-order terms. This discrepancy has direct consequences for interpreting electromagnetic signals and imaging black hole shadows where beyond-GR effects are substantial.

Practical and Theoretical Implications

The findings have several immediate implications:

• Photon Trajectory Modeling: Spinoptics must be included in high-precision models of photon trajectories for strong-field regimes, especially near compact objects with non-standard hair.
• Black Hole Shadow Imaging: The presence of spinoptics-induced deviations could modify expected shadow shapes, making polarization-resolved imaging sensitive to underlying spacetime structure and photon helicity effects.
• Testing Beyond-GR Theories: Parametrized metrics, while efficient, may not adequately capture all observable consequences of modified gravity (e.g., hairy metrics), especially for spin-optical effects; observational programs employing shadow radius or lensing data must calibrate these limitations.
• Birefringence as a Diagnostic: Gravitational birefringence could become a discriminating feature in electromagnetic and gravitational wave observations, especially with high-frequency and polarization-sensitive instruments.

Future developments should focus on refining parametrizations to better accommodate strong deviations and systematically incorporating spinoptics corrections in numerical simulations of astrophysical observables. The theoretical framework can be extended to rotating black holes, multi-hairy configurations, and gravitational wave spinoptics.

Conclusion

This work rigorously quantifies the impact of helicity-curvature interactions on photon trajectories in spherically symmetric black holes, using both parametrized RZ metrics and regular hairy GD solutions (2605.20524). The analysis demonstrates the necessity of the spinoptics formalism for accurate light deflection predictions and delineates the limits of first-order parametrizations in reproducing the physics of regular hairy black holes. The highlighted dependence of the deflection angle on metric parameters provides both a practical tool for interpreting future observations and a theoretical impetus to refine metric modeling and expand spinoptic applications.