arXiv subjects

Ali Övgün

Publications and source records attributed to Ali Övgün.

At least 19 recordsLinked to original sources

arXiv2026

Energy extraction from NED-deformed rotating black holes via the Comisso-Asenjo reconnection process

We study rotating black holes in general relativity coupled to nonlinear electrodynamics (NED), focusing on an axisymmetric solution with deformation parameter $g$. On the spherical seed, weak-field lensing via the Gauss-Bonnet method and the shadow radius yield a spin-insensitive bound by enforcing a conservative $\sim 10\%$ tolerance on the Sgr A* ring size, namely $g/M \lesssim 1.26$. In the eikonal regime we derive analytic quasinormal-mode shifts, even in $g$, and obtain an independent ceiling consistent with the shadow constraint. For the rotating geometry, we provide closed-form ZAMO scalars, chart horizons and the ergoregion, and analyze equatorial geodesics (photon orbits and ISCO). We then formulate the Comisso-Asenjo reconnection channel in the ZAMO frame, identify the negative-energy window, and integrate the extracted power over the allowed radii; from the tolerated fractional departure from the Kerr power we define a spin-dependent extraction bound $g_δ(a\midσ_0,ξ)$. Taken together, the QNM/shadow ceiling and the extraction bound appreciably narrow the admissible region for $g/M$ in the $(a/M,g/M)$ plane, so even within our deliberately simplified, single-layer equatorial setup, the two complementary probes already provide informative constraints on NED deformations, testable with present data and upcoming horizon-scale and ringdown campaigns.

gr-qc↗

arXiv2026

Acceleration radiation, quasinormal modes and quasi-bound states in generic rotating regular black holes

We study horizon-brightened acceleration radiation for a neutral scalar field in separable Kerr-like rotating metrics with nonlinear-electrodynamic mass profiles. The principal branch has $M=0$ and $μ=3$, with finite but direction-dependent central curvature limits. An analytic extremal charge--spin parametrization fixes the black hole domain. At a simple outer horizon, the scalar equation reduces to local conformal quantum mechanics with corotating energy $\epsh=ω-m\OmH$. For an isolated outgoing channel we derive the first finite-gap correction in proper time, including the radial Frobenius coefficient, angular variation, geodesic terms, and switching slope. This replaces a coordinate-dependent trajectory-phase multiplier by a protocol-defined detailed-balance affinity. Direct integration of the outgoing scalar equation and an equatorial infalling trajectory validates the expansion for an explicitly specified finite gate; a solvable logarithmic-mode limit separates switching from geometric contributions. We also analyze forced equatorial circular atoms and show that co-rotating and counter-rotating ultra-relativistic motion yields a mode-dependent orbital response rather than the infalling Hawking affinity. We identify the shrinking near-extremal radial scale and quantify the effect of an unselected second radial branch, without asserting a uniform extremal limit. Random atomic injection gives a selected-mode master equation and a nonnegative population entropy-production identity. A direct first-law integrability test shows why horizon-area bookkeeping remains conditional for this off-shell metric. Independently converged Chebyshev scalar quasinormal frequencies validate the WKB results and probe the exterior geometry beyond the leading horizon data. The calculation distinguishes branch-resolved horizon thermality, finite detector protocol, and exterior scalar dynamics.

gr-qc↗

arXiv2026

Black Hole Shadow and Light Deflection in Generalized Heisenberg-Euler Nonlinear Electrodynamics

We study the optical properties of electrically charged black holes sourced by generalized Heisenberg-Euler nonlinear electrodynamics (NLED) in the second-order formalism. Using the exact parametric form of the static spherically symmetric solution, we analyze the photon sphere, the black hole shadow, and the light deflection angle, and compare the results with those of the Reissner-Nordström and Schwarzschild cases. We find that the photon sphere and shadow radii decrease with increasing charge and increase with black hole mass, while NLED effects become significant mainly near extremality. By confronting the dimensionless shadow radius with the Event Horizon Telescope (EHT) bounds from Sgr A* and M87*, we derive phenomenological upper limits on the charge, with Sgr A* providing the stronger constraint. We further show that, for fixed impact parameter and charge, the deflection angle is systematically larger than in the corresponding Reissner-Nordström spacetime, and that the critical impact parameter is shifted to larger values. Our results show that NLED corrections produce observable modifications in the optical properties of charged black holes, especially close to extremality.

gr-qc↗

arXiv2026

Maxwell-J$ü$ttner averaged nonaffine theory of relativistic shear viscosity

We formulate a Maxwell-Jüttner averaged extension of relativistic nonaffine shear-viscosity theory for a classical gas of massive particles. The starting point is the relativistic nonaffine response formula derived from generalized Langevin dynamics, where proper-momentum dissipation produces a Lorentz-factor enhancement of the viscosity. Since a thermal relativistic gas is not characterized by a single Lorentz factor, we replace this scalar enhancement by an equilibrium momentum average over the Maxwell-Jüttner distribution. Within the scalar transverse mean-field closure used in this work, the minimal relativistic correction is the Maxwell-Jüttner mean of the Lorentz factor and becomes a finite-mass enhancement function of $ζ=mc^2/(k_BT)$. In the general closure, the averaged object is the full product of the Lorentz factor, the zero-frequency memory kernel, and the nonaffine affine-force correlator. This construction separates kinematic relativistic enhancement from dynamical dissipation encoded in the bath memory and collective mode spectrum. We derive the nonrelativistic and ultrarelativistic limits, show that a specified effective hard-sphere scaling reproduces the classical square-root law as a consistency test, and identify the assumptions under which a cubic high-temperature law follows. Comparison with relativistic kinetic-theory examples clarifies that the memory kernel is not equivalent to a relaxation time unless a microscopic closure is specified. Retaining the full frequency dependence instead yields a relaxation spectrum. A single dominant pole reduces this spectrum to one shear relaxation time and gives the usual transient shear equation.

cond-mat.stat-mech↗

arXiv2026

Redshift Spectroscopy as a Probe of Regular Black Holes, Black Bounces, and Scalar-Hair Compact Objects

Motivated by [Phys. Rev. D. 107, 064019 (2023)], we develop a unified and model-independent framework for the spectroscopy of photon frequency shifts in generic static, spherically symmetric spacetimes. Working with a line element characterized by three arbitrary radial metric functions, we derive exact expressions for the conserved quantities of massive and massless probes, the conditions for circular timelike geodesics, the local-emission-angle-dependent photon impact parameter, and the corresponding local redshift and blueshift branches measured by distant static observers. The formalism is further extended to include the line-of-sight peculiar motion of the source and the local propagation of photons in a nonmagnetized cold plasma, thereby identifying the gravitational, orbital, and dispersive factors entering the frequency-shift signal under the stated assumptions. We also show that, in vacuum, the same geometric structures governing orbital spectroscopy determine the photon sphere and the shadow impact parameter whenever an external null critical orbit is present. To make the framework suitable for deformed compact-object models, we construct a perturbative expansion around Schwarzschild geometry up to second order in a dimensionless deformation parameter, obtaining explicit corrections to the orbital energy, angular momentum, emitter four-velocity, photon impact parameter, and the vacuum and plasma frequency shifts. We apply the formalism to regular black holes from nonlinear electrodynamics, the Simpson--Visser black-bounce spacetime, and the Fisher--Janis--Newman--Winicour--Wyman geometry. These examples show that the same local spectroscopic language can be used across regular-black-hole, black-bounce, wormhole, and scalar-supported horizonless sectors.

gr-qc↗

arXiv2026

Fixed ADM reconstruction of extended uncertainty principle black holes

We study a phenomenological fixed-ADM prescription for an extended-uncertainty-principle (EUP) correction to the Schwarzschild exterior. Rather than treat mass constancy as a consequence of the EUP, we hold the charge measured at spatial infinity fixed so that we can compare exteriors at the same physical mass. We impose the EUP-shifted horizon radius and Hawking temperature and select the slowest admissible inverse-power representative. Its Einstein tensor defines an effective anisotropic tensor, but we do not infer an independent matter model, quantum state, or perturbation theory from that definition. For a positive EUP parameter, the effective energy density is negative throughout the exterior, while the transverse null and strong energy conditions fail over an extended radial region. The area entropy and the formal entropy obtained from the reduced relation $\mathrm{d}M=T\,\mathrm{d}S$ disagree at leading order; we interpret this mismatch as evidence that the thermodynamic description lacks work terms or an action-based entropy law, not as a definite EUP entropy correction. An explicit one-parameter family with the same mass, horizon, and temperature shows that the orbit, lensing, and test-field eikonal shifts depend on the chosen representative. Our results therefore characterize the minimal member of a phenomenological family rather than universal consequences of the EUP.

gr-qc↗

arXiv2026

Chromatic Weak Lensing by Charged Black Holes with Two Lorentz-Violating Kalb-Ramond Couplings

We study weak gravitational lensing and steady spherical test--fluid accretion by a static charged black hole in a Lorentz--violating Kalb--Ramond background with two nonminimal curvature couplings, assuming minimally coupled probe radiation. Using the Gauss--Bonnet theorem with the correct boundary term and perturbed ray boundary, we obtain the complete local deflection angle through second post--Minkowskian order. In a homogeneous cold plasma, the mass and charge sectors acquire different frequency dependences, producing distinct chromatic signatures. The two Lorentz--violating couplings are also separated: one controls the conical geometry and mass normalization, while the other first enters through the effective charge. For neutral adiabatic accretion, we derive the conserved fluxes, Bernoulli relation, Hamiltonian flow, and sonic--point conditions. These results disentangle local, global, dispersive, and accretion effects and establish the calibrations required for phenomenological constraints.

gr-qc↗

arXiv2026

Phase-plane formulation of weak gravitational deflection in static spherical spacetimes

This paper develops a phase-plane formulation of gravitational light deflection by static and spherically symmetric black holes, with Schwarzschild spacetime as the principal case. Instead of perturbing the null trajectory and locating the displaced outgoing asymptote, we represent the orbit through an amplitude and an intrinsic phase. The bending angle then follows from the excess physical azimuth accumulated while the intrinsic phase advances between two fixed asymptotic endpoints. For Schwarzschild spacetime, the exact radial first integral reduces the amplitude evolution to a cubic algebraic relation. Its physical branch generates the local phase factor through a single inverse algebraic map. Lagrange inversion then yields an explicit all-order weak-deflection coefficient formula in powers of the invariant ratio $M/b$. Each coefficient separates into an algebraic phase contribution and a universal trigonometric moment, which explains the alternating rational and $π$-dependent structure of the Schwarzschild series. Independent comparison with the exact radial scattering integral and conventional orbit perturbation reproduces the standard weak-bending coefficients. The same phase framework extends to general static spherical geometries, while the Schwarzschild cubic represents an especially simple member of a broader algebraic class. The branch singularity of the phase map also coincides with the critical photon orbit and governs the convergence of the weak-deflection expansion.

gr-qc↗

arXiv2026

Spinning Particle Dynamics and Observational Redshift around an Asymptotically Flat Symmergent Black Hole

We investigate timelike particle dynamics, collision energetics, and photon frequency shifts in the perturbative variable-scalar curvature branch of asymptotically flat symmergent gravity. The low-energy vacuum action contains an $R^{2}$ correction whose coefficient is set by the boson--fermion imbalance of the underlying quantum field theory. At linear order, the exterior geometry is conformal to Schwarzschild spacetime through a radial mode satisfying a linear equation. We retain an independent boundary-condition-dependent amplitude and restrict the analysis to the perturbative domain. The two signs of the symmergent parameter $γ$ yield distinct profiles: $γ>0$ gives a Yukawa-suppressed deformation, whereas $γ<0$ produces an oscillatory inverse-radius deformation. We derive radial equations, effective potentials, circular-orbit and marginal-stability conditions for neutral, electrically charged, and spinning massive particles. Charged particles are treated in the test-field approximation, while spinning particles obey the Mathisson--Papapetrou--Dixon equations with the Tulczyjew condition. We also compute the center-of-mass energy of neutral-particle collisions and the frequency shifts of photons emitted tangentially by circular geodesic sources and detected by a static observer at infinity. The redshift and blueshift factors satisfy $(1+z_{+})(1+z_{-})=1/A(r_e)$, directly linking their product to the lapse function at emission. The $γ>0$ branch yields smooth, short-range deviations from Schwarzschild dynamics, whereas the $γ<0$ branch can generate oscillatory radial bands admitting circular-orbit solutions whose stability must be tested independently. These observables provide complementary probes of the variable-curvature sector, although their quantitative interpretation also depends on the deformation amplitude and, for the oscillatory branch, its phase.

gr-qc↗

arXiv2026

Density-preserving core-Einasto black holes with Event Horizon Telescope bounds

A Newtonian halo density does not uniquely determine a relativistic spacetime, and the resulting completion ambiguity can dominate predicted horizon-scale signals. We make this dependence explicit for the feedback-cored Einasto profile calibrated on FIRE-2 simulations. In Schwarzschild gauge, the Einstein equations give an asymptotically flat, density-preserving geometry supported by an effective anisotropic fluid; no microscopic description of collisionless dark matter is assumed. By contrast, a commonly used rotation-curve completion replaces the finite seed core by an inverse-square source cusp. For the density-preserving solution we derive the enclosed mass in closed form, obtain a sharp one-horizon criterion and the extremal boundaries of a three-horizon phase, and establish the fixed-halo first law. More generally, the leading fractional shift of a spherical black-hole shadow is the environmental mass enclosed within the vacuum photon sphere divided by the central black-hole mass. Applying this result to published Event Horizon Telescope shadow-deviation summaries gives illustrative one-sided 95\% credible limits of $4.8\times10^{23}\,\Msun\,{\rm pc}^{-3}$ for $\sgr$ and $4.1\times10^{17}\,\Msun\,{\rm pc}^{-3}$ for $\mseven$, far above realistic smooth-halo densities. A Milky-Way calibration predicts a fractional shadow shift of $1.7\times10^{-25}$, whereas the rotation-curve completion gives a shift 19 orders of magnitude larger for the same galactic inputs. A covariant polarized thin-disk calculation shows the same weak-core suppression. Thus current horizon-scale images do not constrain a smooth kiloparsec-scale core-Einasto halo; an observable environmental signal would instead require a compact inner component or a physically different relativistic source.

gr-qc↗

arXiv2026

An obstruction and residual-completion theory for Newman--Janis deformations

We formulate an obstruction and residual-completion theory for Newman-Janis-type deformations of black hole seeds, where Newman-Janis-type includes both the original complex-coordinate Newman-Janis algorithm and modified prescriptions such as non-complexification variants. The Newman-Janis algorithm is treated as an off-shell map from a static seed to a stationary-axisymmetric trial geometry, rather than as a solution-generating theorem. Its failure is encoded in a Newman-Janis obstruction tensor, defined as the residual obtained after substituting the trial configuration into the intended field equations. We decompose this residual into dynamical, geometrical/coordinate-admissibility, and source-preservation channels, separating field-equation failure from circularity, Boyer-Lindquist integrability, stress-tensor type, equation-of-state preservation, and matter-model realizability. When the obstruction is nonzero, residual completion asks whether a minimal correction of the metric and matter fields can cancel it. At leading nonzero order in the rotation parameter, this becomes a linear solvability problem: the obstruction must lie in the image of a gauge-fixed completion operator subject to boundary and source-sector constraints, while the cokernel condition gives a no-go criterion in the chosen ansatz class. We illustrate the framework with a Schwarzschild example in vacuum GR. Using an ONJA-type complexification different from the Kerr-generating one, we obtain a non-Kerr rotating trial metric and compute its obstruction. The example gives $n_\star=2$ and $\ell_\star=0$, with an additional quadrupolar component at the same order, and shows how the leading residual is removed by the minimal even-parity correction restoring the Kerr complexification. The framework replaces the search for the correct complexification rule with computable criteria for success, completion, or obstruction.

gr-qc↗

arXiv2026

Quasinormal modes of Kerr-like black bounce spacetime

We investigate the quasinormal mode (QNM) spectrum of a Kerr-like black-bounce spacetime under massive scalar-field perturbations. Starting from the Kerr-like deformation of the Simpson--Visser black-bounce geometry, we derive the corresponding radial and angular equations and obtain the effective potential governing scalar perturbations. The QNM frequencies are computed by means of the Pöschl--Teller potential approximation and the semi-analytic WKB method (up to sixth order), and we demonstrate reasonable agreement between these two approaches. We then analyze in detail how the QNM spectrum depends on the spin parameter $a$, the bounce parameter $p$ that interpolates between black-hole and wormhole geometries, and the scalar-field mass $μ$. Our results show that increasing the spin parameter $a$ raises the oscillation frequency, while increasing the bounce parameter $p$ lowers it, and in both cases the damping rate decreases. Moreover, the mass of the scalar field has a non-negligible impact on the ringdown spectrum. These features suggest that rotating black-bounce geometries may leave distinct imprints in the ringdown phase of gravitational-wave signals, and motivate future studies of echoes and parameter estimation in the context of present and upcoming detectors.

gr-qc↗

arXiv2026

Shadow of the generalized Vaidya black hole

In this paper, we investigate the shadow of the generalized Vaidya black hole described by the Husain solution. Assuming a barotropic equation of state for the additional matter sector, we first identify the self-similar class admitting a homothetic Killing vector and transform the geometry to a conformally static form. This allows us to derive the effective potential for null geodesics in the conformally static representative, determine the corresponding homothetic photon surface, and then transform the result back to the original generalized Vaidya coordinates. We show that, within the weak-energy-condition branches, the parameter $α$ controlling the equation of state determines whether the shadow is enlarged or reduced relative to the Vaidya case: the branch $0\leqα<1/2$ increases the photon-sphere and shadow radii, whereas the branch $1/2<α\leq 1$ decreases them. We then turn to the genuinely time-dependent Husain metric and derive quasistatic slow-evolution equations for the instantaneous photon surface and critical impact parameter. In particular, we show that the growth or contraction of the shadow is governed by the local effective influx in the photon region rather than by the signs of $\dot M$ and $\dot N$ separately. For the charge-like branch, this yields a transparent criterion for when accretion enlarges the shadow and when rapid charge growth instead causes it to shrink.

gr-qc↗

arXiv2026

Spinning particle dynamics, epicyclic frequencies, and transient QPO signatures in Schwarzschild spacetime

We study the motion of spinning test particles in Schwarzschild spacetime within the Mathisson--Papapetrou--Dixon pole--dipole approximation, imposing the Tulczyjew--Dixon spin supplementary condition. Restricting to equatorial orbits with the particle spin aligned with the orbital angular momentum, and retaining terms through linear order in the specific spin $s$, we derive the spin-corrected radial potential, circular-orbit conditions, bound periodic trajectories, epicyclic frequencies, and Lyapunov exponents of unstable circular orbits. The spin--curvature coupling shifts the circular-orbit energy and angular momentum and moves the innermost stable circular orbit to $r_{\rm ISCO}=6M-2\sqrt{2/3}\,s+\mathcal{O}(s^2)$ in the sign convention adopted here. We construct bound periodic orbits using the Levin--Perez-Giz zoom--whirl taxonomy and show how the particle spin deforms the corresponding energy--angular-momentum map. We then obtain the coordinate-time azimuthal and radial epicyclic frequencies and use them as kinematical inputs for relativistic-precession and resonance prescriptions for quasi-periodic oscillations. Finally, we relate the Lyapunov exponent of unstable circular orbits to the local separatrix structure governing near-homoclinic zoom--whirl motion. The resulting formulation provides a compact analytic connection between linear-in-spin MPD dynamics, periodic-orbit taxonomy, epicyclic-frequency shifts, and transient strong-field phenomenology in a nonrotating black-hole background. Also, we study the gravitational waveforms from the periodic orbits of a massive spinning particle around a black hole, presenting those associated with extreme mass-ratio inspirals involving a stellar-mass compact spinning object orbiting a supermassive black hole.

gr-qc↗

arXiv2026

Lorentz-violating modifications to particle dynamics, thermodynamics and vacuum energy in bumblebee gravity

We investigate how spontaneous Lorentz symmetry breaking in bumblebee gravity modifies particle dynamics, thermodynamics, and vacuum energy around a static black hole background. Starting from the optical-mechanical correspondence, we derive a modified dispersion relation that encodes the influence of the Lorentz-violating parameter $λ$ on the propagation of massive and massless modes. We analyze the resulting optical properties, including the effective refractive index, group velocity, and energy-dependent time delay, and show how the non-asymptotically flat geometry reshapes signal propagation. From the same dispersion relation, we construct the interparticle potential for massive and massless excitations and evaluate the electron scattering cross section within the Born approximation, identifying characteristic Lorentz-violating corrections. We then develop a statistical-ensemble description based on the deformed energy-momentum relation and obtain analytic expressions for the thermodynamic observables of a massless bosonic gas. The pressure, mean energy, entropy, and heat capacity are examined in three representative regimes -- extremely close to the horizon, near the photon sphere, and in the asymptotic region -- where Lorentz violation systematically increase the magnitude of these quantities and leads to finite asymptotic plateaus. Finally, we analyze the vacuum state in the curved background and compute the regularized Casimir energy at zero and finite temperature.

gr-qc↗

arXiv2026

Near extremal RN-AdS control of holographic Josephson transport

We formulate a holographic weak-link construction in which Josephson transport is controlled by the charge sector of a Reissner--Nordstrom-AdS black brane. The model is an Einstein-Maxwell-charged-scalar theory in asymptotically AdS$_4$, with a spatially inhomogeneous boundary chemical potential that creates two superconducting banks separated by a normal or weakly superconducting barrier. The Josephson phase difference is defined from gauge-invariant boundary observable of the charged condensate, rather than from the black hole charge itself, allowing a controlled extension of the standard holographic SNS junction to charged AdS backgrounds. We identify the current-phase relation, critical current, coherence length, and small-phase stiffness as the main observables. In the SNS regime, the critical current and midpoint condensate probe the same proximity scale, while higher harmonics in the current-phase relation diagnose enhanced transparency or departure from the opaque weak-link limit. The new mechanism is the near-extremal RN-AdS throat: as extremality is approached, the emergent AdS$_2\times\mathbb{R}^2$ region can radially modify the Josephson coupling before it reaches the ultraviolet boundary. After removing the ordinary spatial suppression due to the junction width, the residual critical current and phase stiffness are expected to exhibit scaling governed by the infrared dimension of the charged scalar in AdS$_2$. This separates ordinary proximity suppression, smooth finite-density corrections, and genuinely near-extremal throat control, providing a framework for phase-sensitive transport in charged holographic matter.

hep-th↗

arXiv2026

Josephson's effect in the Schwarzschild background

We develop a fully covariant, analytic framework for Josephson phenomena in static curved spacetimes and specialize it to the Schwarzschild exterior. The formulation rests on two invariant elements: the gauge-invariant condensate momentum that governs phase dynamics and the conserved current whose hypersurface flux encodes transport for an observer at infinity. Using the timelike Killing field to relate proper and asymptotic quantities, we derive a redshifted AC Josephson law in which the asymptotic phase-evolution rate is proportional to the difference of redshifted voltage drops, i.e. to $V_i^\infty \equiv α_i V_i^{\rm proper}$; equivalently, it depends on $α_i V_i^{\rm proper}$ for local control. Under RF drive specified at infinity, the Shapiro-step loci are invariant (expressed in asymptotic voltages) while propagation phases set any apparent lobe translation. For DC transport, a short-junction solution on a static slice yields the proper current-phase relation; mapping to asymptotic observables gives a single-power redshift scaling of critical currents, $I_{c,\infty}\propto αI_c^{\rm proper}$, whereas power scales as $P_\infty\propto α^2 P_{\rm proper}$. In a "vertical" dc-SQUID with junctions at different radii, gravity does not shift the DC interference pattern at linear order; it produces a small envelope deformation and an amplitude rescaling. Gravity does not alter the local Josephson microphysics; it reshapes the clocks and energy accounting that define measurements at infinity. The resulting predictions are gauge- and coordinate-invariant, operationally stated in terms of an experimenter who can control (proper vs. asymptotic bias), and remain analytic from the weak-field regime to the near-horizon limit.

hep-th↗

arXiv2026

Semi-Analytic Trajectory Analysis of Light in Generic Static Spacetimes

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 $ds^2 = -α(r,δ)\,dt^2 + γ(r,δ)\,dr^2 + β(r,δ)\,dΩ_2^2,$ where $α$, $β$, and $γ$ 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 $u(φ)\equiv 1/r(φ)$ and obtain a model-independent expression for the bending angle $α(b)$ 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-Nordström-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 $(α,β,γ)$ 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.

gr-qc↗