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arXiv · 2507.16512

Semi-Analytic Trajectory Analysis of Light in Generic Static Spacetimes

Ali Övgün·Reggie C. Pantig
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Abstract

We study a unified semi-analytical framework to study null geodesics and weak-field light deflection in generic static, spherically symmetric spacetimes of the form \(ds^2 = -\alpha(r,\delta)\,dt^2 + \gamma(r,\delta)\,dr^2 + \beta(r,\delta)\,d\Omega_2^2,\) where $\alpha$, $\beta$, and $\gamma$ encode model-dependent deviations from Schwarzschild gravity inspired from [Phys.Rev.D 112 (2025) 12, 124072]. Starting from the exact first-order orbit equation, we derive a compact master equation for the impact-parameter-dependent trajectory $u(\varphi)\equiv 1/r(\varphi)$ and obtain a model-independent expression for the bending angle $\alpha(b)$ in terms of generic metric functions and their derivatives. This master equation is then solved semi-analytically by three complementary techniques: (i) the homotopy perturbation method (HPM), (ii) the variational iteration method (VIM), and (iii) a calibrated impulse (single-kick) approximation expressed directly in terms of the effective gravitational potential. As nontrivial test beds we consider a scalar-hairy Reissner-Nordstr\"om-like black hole where the scalar hair enters as $Q_s$ in an effective charge parameter. Then we derive closed-form expressions for the deflection angle, identify the leading scalar-hair, and compare the accuracy and convergence properties of HPM, VIM, and the impulse method against the standard Schwarzschild limits. Our results show that the generic formulation in $(\alpha,\beta,\gamma)$ can efficiently accommodate a broad class of modified gravity black hole solutions. We further supplement the analytic treatment with a compact numerical results against the exact null-geodesic integral near the photon sphere in order to delineate the practical range of validity of the three approximation schemes.

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Ali Övgün, Reggie C. Pantig. 2025-07-22. Semi-Analytic Trajectory Analysis of Light in Generic Static Spacetimes. https://doi.org/10.1016/j.nuclphysb.2026.117557

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