arXiv subjects

Nikko John Leo S. Lobos

Publications and source records attributed to Nikko John Leo S. Lobos.

18 recordsLinked to original sources

arXiv2026

Photon sphere jet equivalence and the non-uniqueness of black-hole thermodynamics

We examine whether scalar eikonal quasinormal-mode data determine black-hole thermodynamics. For static, spherically symmetric, asymptotically flat, nonextremal geometries, we prove that the formal fixed-overtone expansion of a neutral, minimally coupled, massless scalar through order $L^{-N}$ depends only on the $(2N+2)$ jet of the lapse at the unstable photon sphere. For every finite $N$, real-analytic deformations can preserve this jet and the asymptotic Reissner--Nordström coefficients while changing the surface gravity and Hawking temperature. We also construct a nonanalytic $C^\infty$ deformation that is flat at the photon sphere. This deformation preserves the complete formal eikonal series to every algebraic order but changes the horizon position, temperature, and Einstein area entropy. A nonempty charged subfamily satisfies the weak energy condition as an effective Einstein source. Together, these four results establish the new finite-jet and all-orders thermodynamic obstruction. To separate geometric information from action-dependent information, we incorporate a previously established Reissner--Nordström--MOG comparison. The mapped backgrounds have identical normalized metrics, neutral-scalar boundary-value problems, exact scalar spectra, and Hawking temperatures, yet their Wald entropies differ because their curvature couplings differ. The resulting hierarchy shows that local photon-sphere data do not determine horizon thermodynamics and that even exact action-blind scalar data do not determine gravitational entropy. Agreement of the formal eikonal expansions for the deformed metrics does not imply exact finite-$\ell$ isospectrality. Global differences may appear beyond every algebraic order. Thermodynamic reconstruction from eikonal ringdown is therefore conditional on assumptions about the global metric and the gravitational action.

gr-qc↗

arXiv2026

Scalar decoupling and critical-curve invariance in a reduced MOG black-hole theory

Linear perturbations of a Schwarzschild-MOG black hole separate into scalar and gravito-vector sectors in the reduced strong-field theory at leading order in $μ\bar G M\ll1$. The field redefinition $\widehat h_{μν}=h_{μν}-ψ\bar g_{μν}$ gives a decoupled Klein--Gordon equation for the scalar and the Reissner--Nordström equations for the gravito-vector sector. The scalar is stable under the stated spectral boundary conditions for $m_G^2\ge0$. A scalar-only excitation produces a conformal metric perturbation with null Hamiltonian contribution $H_1^{(s)}=-ψH_0$. For fixed observer worldlines and conformally corresponding orthonormal frames, every linear scalar-to-critical-curve transfer coefficient vanishes, including monopole breathing, provided the conformal factor is smooth, positive, and preserves the capture boundary. The exceptional charged axial dipole instead induces a constrained metric perturbation. A first-order radial integral reconstructs this perturbation in a gauge regular at the future horizon and on the sphere, separating the radiative amplitude from stationary angular momentum without division by frequency. For a unit-incident canonical vector amplitude with zero angular momentum, matched asymptotic solutions give an explicit charge-dependent local metric response that vanishes as $ω^2$ at fixed radius and fixed subextremal background. A separate infinitesimal Kerr--Newman benchmark yields the stationary impact-parameter midpoint shift and recovers $δC_b=-2a_{\rm rot}$ in the neutral limit. These results exclude linear scalar breathing of the specified angular critical curve and distinguish the low-frequency vector-induced metric response from a stationary rotation effect.

gr-qc↗

arXiv2026

Energy, Passivity, and Diffusive Representations in Anisotropic Fractional Maxwell--Cole--Cole Media

This study develops an energy formulation for linear anisotropic Maxwell--Cole--Cole media with multiple fractional polarization channels, passive conductivity, prescribed currents, and finite-terminal Caputo memory with zero prehistory. A weighted solution class captures the integrable fractional initial layer generated by quiescent polarization. Fractional polarization power then separates into a nonnegative history storage and a nonnegative constitutive dissipation rate. Although the instantaneous rate may diverge at initialization, it remains integrable, while stored history energy and accumulated loss vanish initially. Combined with the Poynting theorem, this decomposition yields a global field-plus-memory identity that establishes electromagnetic passivity, closed-system energy decay, and conditional uniqueness under zero-supply boundary conditions. The framework also recovers the Debye, instantaneous-dielectric, vanishing-susceptibility, and vacuum limits under convergence assumptions. A positive diffusive representation provides a second decomposition whose storage never exceeds the history storage. Their difference is an exact nonnegative weighted variance that vanishes only for a zero polarization history. In agreement with these time-domain results, the finite-time harmonic response approaches the anisotropic multichannel Cole--Cole permittivity, whose Hermitian imaginary part ensures nonnegative cycle-averaged absorption. A reproducible single-channel study shows that fractional order broadens the dielectric-loss band, redistributes wave attenuation, strengthens the integrable initial power layer, and prolongs the step-response tail. An exact electric-step response verifies both energy representations. The formulation connects fractional initial layers, hereditary storage, passivity, diffusive relaxation, dispersion, absorption, and attenuation in a local constitutive model.

math-ph↗

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

Exact Spectrum of a Curvature-Adapted $\mathbb{Z}_{2}^{3}$ Dunkl--Deng--Fan/Eckart System

This study constructs and exactly solves a curvature-adapted radial Dunkl Hamiltonian for the coordinate-reflection group $\mathbb{Z}*{2}^{3}$ on 3D hyperbolic space, determining how curvature and Dunkl multiplicities govern the discrete spectrum and admissible radial states. The warped-product kinetic operator is defined by its Friedrichs extension, which selects the regular boundary condition in singular radial channels. The restriction $α=2κ$ is an exact-solvability condition, not a general relation between molecular range and spatial curvature, and reduces the centrifugal and Deng--Fan terms exactly to a generalized Eckart problem. We derive the finite spectrum, Jacobi-polynomial radial eigenfunctions, normalization integrals, and normalizability condition $η*{n\ell}^{2}<q_κ/2$. Dunkl multiplicities affect the radial spectrum only through $γ=μ_{1}+μ_{2}+μ_{3}$, whereas reflection eigenvalues restrict the angular degrees to $\ell=p+2k$, with total multiplicity $2\ell+1$. On each surviving branch, energy increases with $\ell$ and $γ$, while the number of bound states decreases. Quadratic-form Hellmann--Feynman identities give exact hyperbolic-interaction expectation values, and adjacent-level spacings follow directly from the spectrum. The undeformed and correlated flat limits recover the ordinary curvature-matched Eckart and Dunkl--Kratzer spectra, respectively. Finite-difference calculations for $γ=0$ and $1/2$ confirm the energies, continuum thresholds, and state counts. This tuned curvature-adapted construction is not the generic Euclidean Dunkl--Deng--Fan problem. It isolates the effects of negative curvature and reflection deformation on level ordering, binding thresholds, and bound-state counts, and provides analytic benchmarks for numerical and approximate treatments of singular curvature- and reflection-deformed radial Hamiltonians.

math-ph↗

arXiv2026

Two-branch detector response for Dirac infall into a Schwarzschild--MOG black hole

We study a localized spin-$1/2$ detector falling into a Schwarzschild--MOG black hole. The detector interacts with a neutral massless scalar field through a charge-preserving two-level transition, while its translational wave packet obeys the MOG-charged Dirac equation. Near the outer horizon, the separated radial system reduces to an inverse-square equation with index $Θ_{\rm D}=E_{\rm H}/(2\hbarκ_α)$. The gauge-invariant horizon energy satisfies $E_{\rm H}=m_ψU_{\rm H}>0$ for every future-directed crossing trajectory. We quantize the scalar field in a globally normalized Boulware scattering basis and retain both radial-flux branches of a mode that is outgoing at infinity. This treatment does not identify a local outgoing ansatz with a complete mode. A finite radial gate $χ_p(x)=(x/L_χ)^p e^{-x/L_χ}$ gives closed-form excitation and absorption probability densities that include the ingoing contribution and scattering-phase interference. Within the controlled near-horizon approximation, the full detailed-balance ratio factorizes into a branch-resolved outgoing ratio and a two-branch factor. Only the outgoing ratio approaches $\exp(-2πν/κ_α)$ when the near-horizon, adiabatic, high-gap, and branch-isolation conditions all hold. The leading switching correction is controlled by $(ν/κ_α)/(S_\pm L_χ)$, not by $(S_\pm L_χ)^{-1}$ alone. In a weak-MOG expansion, the local correction separates into surface-gravity, trajectory-prefactor, and finite-gate terms. The full response also contains a contribution from global scattering. This separation shows which terms follow from the local horizon geometry and which depend on the detector protocol and scalar propagation outside the horizon region.

gr-qc↗

arXiv2026

Optical-area minimum method for static spherical black hole shadows

We formulate a global optical-area method for shadows of static, spherically symmetric black holes. For the metric $ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+C(r)dΩ^{2}$, spherical sections of the optical geometry have area $\mathcal{A}_{\rm opt}=4πC/A$. A null ray with impact parameter $b$ can cross a spherical section only if $b^{2}\leq C/A$. The capture threshold is fixed by the infimum of $C/A$ on the connected interval between the observer and the black hole horizon. When attained at an interior point, the infimum gives $b_{\rm sh}^{2}=\min(C/A)$, while a static observer outside the controlling minimum, on the inward-sky branch, measures $\sin^{2}α_{\rm sh}=\mathcal{A}_{*}/ \mathcal{A}_{\rm opt}(r_{\rm o})$. The usual photon-sphere equation follows when the minimum occurs at a smooth interior point. Exponential instability additionally requires the minimum to be nondegenerate. The local optical-radius and photon-sphere formulas are established results. Our contribution is to organize them into an observer-to-horizon global selection rule that compares all stationary candidates and relevant endpoint limits. The radial function $B(r)$ does not affect the shadow angle, although it enters the coordinate-time instability rate, whose numerical value also depends on the normalization of the static time coordinate. We derive compact first- and second-order formulas for deformed metrics, demonstrate candidate comparison with a synthetic two-minimum profile, and apply the construction to Reissner--Nordström, Bardeen, charged dilaton, and Kottler black holes. An explicit transformation of the charged-dilaton example from a nonareal to an areal radial coordinate verifies radial-coordinate invariance, while the Kottler example probes a nonasymptotically flat static region.

gr-qc↗

arXiv2026

Breathing Black Hole Shadows in Modified Gravity (MOG)

In this paper, we investigate the dynamic phenomenological signatures of a Schwarzschild-MOG black hole shadow perturbed by passing gravitational waves. By perturbing the Hamilton-Jacobi equation for photon null geodesics, we demonstrate that the unique field content of MOG breaks the observational degeneracy with standard General Relativity. We mathematically prove two distinct, time-dependent signatures. First, the massless MOG scalar field induces a volumetric ``breathing mode'' polarization, causing the total apparent area of the shadow to rhythmically expand and contract. Second, the massive MOG vector field undergoes quantum vacuum dispersion, arriving at the observer with a predictable time delay. This delayed massive wave sources secondary longitudinal metric perturbations that manifest as a sudden, asymmetric translational wobble of the shadow on the celestial screen. These dynamic geometric shifts offer a robust observational template for next-generation interferometry to strictly test the existence of massive force carriers and scalar fields in gravity.

gr-qc↗

arXiv2026

Shadow dependent phenomenology framework for rotating black hole metric

We establish a formal thermodynamic-optical duality that bridges the semiclassical quantum evaporation of black holes with their classical macroscopic geometry. The physical viability of this framework is anchored by a stable multivariate coordinate transformation and a non-vanishing Jacobian determinant, which allows for a diffeomorphic inversion mapping that decouples intrinsic physical quantities such as bare mass from the unobservable spacetime interior. By re-parameterizing black hole properties entirely in terms of the analytical shadow radius ($R_{sh}$), we derive explicit, observable-based expressions for the weak deflection angle, Hawking temperature, and integrated semiclassical luminosity. We demonstrate the framework's predictive utility by applying it to standard Kerr, Kerr-MOG (Scalar-Tensor-Vector Gravity), and rotating Horndeski spacetimes. Our results provide a definitive mathematical solution to parameter degeneracy, revealing that distinct fundamental fields (vector vs. scalar) leave unique observational fingerprints on far-field astrometry and horizon-scale quantum thermodynamics. By confronting these models with Event Horizon Telescope (EHT) M87* data, we show that this formalism successfully breaks mass-parameter degeneracies, offering a robust and computationally efficient operational tool for testing the Kerr hypothesis and probing modified gravity theories with next-generation very-long-baseline interferometry (VLBI).

gr-qc↗

arXiv2026

Dynamical Black Hole Thermodynamics in Modified Gravity

We investigate the dynamical and thermodynamic evolution of a Schwarzschild black hole in Modified Gravity (MOG) perturbed by a scalar gravitational wave breathing mode. By evaluating the linearized modified Einstein equations at the near-horizon boundary, we reduce the spatial wave operator to a closed-form temporal ordinary differential equation, thereby explicitly deriving the damped-oscillatory kinematics of the scalar strain. Using a quasi-adiabatic approximation, we show that the effective surface gravity and dynamical temperature are linearly modulated by the perturbation amplitude and velocity. These rapid geometric fluctuations break the semiclassical adiabatic regime, triggering explicitly non-thermal particle creation analogous to the dynamical Casimir effect. Furthermore, we resolve a local thermodynamic paradox concerning apparent horizon area fluctuations. We prove that first-order geometric perturbations $\mathcal{O}(h_b)$ are fully reversible kinematic artifacts, whereas irreversible entropy generation is a strictly second-order $\mathcal{O}(h_b^2)$ effect driven by the Raychaudhuri expansion, thereby preserving the Generalized Second Law. Finally, we apply these mechanisms to the black hole information paradox. We show that treating the MOG deformation parameter as a quantum-scale running coupling, $α(M)$, mathematically decouples the effective gravitational charge from linear mass scaling. This dynamically forces the evaporating black hole toward the extremal limit ($M_G \to Q_G$), smoothly quenching the Hawking temperature to zero and yielding a thermodynamically stable, information-preserving remnant.

gr-qc↗

arXiv2026

Observational signatures of quantum-corrected RN blackhole

We investigate the observational signatures of a quantum-corrected Reissner-Nordström (RN) black hole to constrain Planck-scale modifications to spacetime geometry using current astrophysical data. By analyzing the null geodesic structure, we demonstrate that the quantum correction parameter, $\mathrm{a}$, acts as a repulsive geometric potential that opposes the gravitational compactification induced by the electric charge, $Q$. This competition leads to a parameter degeneracy wherein a highly charged, quantum-corrected black hole can mimic the shadow size of a classical Schwarzschild black hole. To resolve this, we employ the strong-field limit formalism to derive the deflection angle and the observables associated with relativistic Einstein rings. Our analysis reveals that while the electric charge enhances the deflection angle, the quantum correction suppresses it, providing a theoretical mechanism to distinguish the two effects. Confronting these predictions with the latest Event Horizon Telescope (EHT) observations, we derive robust constraints on the dimensionless parameter $Π= \mathrm{a}/Q$. We find that consistency with the shadow angular diameter of Sgr. A$^*$ requires $0 \le Π\lesssim 0.7$, implying that quantum geometric corrections cannot exceed approximately $70\%$ of the black hole charge without violating empirical bounds. These results highlight the potential of strong-field lensing to place precise phenomenological limits on quantum gravity candidates.

gr-qc↗

arXiv2026

Phenomenology of Rotating GEUP Black Holes

We investigate the phenomenological implications of quantum gravity on rotating black holes within the framework of the Generalized Extended Uncertainty Principle (GEUP), which incorporates both minimal length (ultraviolet) and large-scale (infrared) corrections. Lacking a full non-perturbative formulation of quantum gravity, we adopt a metric-based approach. We construct a stationary, axisymmetric ansatz via the Newman-Janis algorithm to model the kinematic features of a rotating black hole subject to Generalized Extended Uncertainty Principle (GEUP) corrections. The thermodynamic analysis reveals that in the infrared-dominated regime, the Hawking temperature scales as $T_H \sim M^{-3}$, leading to a rapid cooling phase that significantly prolongs the lifetime of supermassive black holes. We derive the modified Teukolsky Master Equation for gravitational perturbations and demonstrate that the background geometry preserves the isospectrality between axial and polar modes. In the eikonal limit, the quasinormal mode (QNM) spectrum exhibits orthogonal shifts: the minimal length parameter $β$ induces a spectral blueshift and enhanced damping, while the large-scale parameter $α$ induces a spectral redshift and suppressed damping. Finally, we constrain the theory using observational data from LIGO/Virgo and the Event Horizon Telescope. We establish that the shadow of M87* is approximately $10^6$ times more sensitive to large-scale corrections than Sgr A*, placing stringent bounds on the EUP parameter, while gravitational wave spectroscopy provides complementary constraints on the GUP sector.

gr-qc↗

arXiv2026

Constraining Lorentz Violation in Kalb-Ramond Gravity via Thermodynamics and Gravitational Wave Analysis

We investigate the observational signatures of a static, spherically symmetric black hole embedded in a spontaneous Kalb-Ramond (KR) background. By normalizing the solution to the physically observable mass $M_{\text{phys}}$, we demonstrate that the thermodynamics of the KR black hole are consistent with General Relativity, with no deviations in the entropy-area law. However, the Lorentz-violating parameter $l$ induces distinct geometric effects: it suppresses the optical shadow radius by a factor of $\sqrt{1-l}$ and hardens the quasinormal mode frequency by the inverse factor. Utilizing Event Horizon Telescope (EHT) data for Sagittarius A*, and assuming the mass prior derived from stellar dynamics, we place a constraint of $l \lesssim 0.19$. While the product of the shadow radius and ringdown frequency remains degenerate with General Relativity, the specific suppression of the shadow size offers a viable pathway to constrain Planck-scale physics with current and future horizon-scale imaging.

gr-qc↗

arXiv2025

Analyzing Deflection Angles and Photon Sphere Dynamics of Magnetically Charged Black Holes in Nonlinear Electrodynamic

In this paper, we investigate the gravitational lensing properties of magnetically charged black holes within the framework of nonlinear electrodynamics. We derive the deflection angle and examine the influence of the nonlinear electrodynamics parameter $ξ$ on light bending. We initially employ a geometric approach based on the Gauss-Bonnet theorem to analyze the gravitational deflection of null and timelike particles. This method encapsulates the global characteristics of the lensing effect in an elegant manner. In the subsequent part of the work, we explore the impact of nonlinear electromagnetic corrections on the black hole shadow. Using numerical techniques, we study the behavior of the photon sphere and demonstrate that a reduction in the photon sphere radius leads to a correspondingly smaller shadow. We compare these results with those for the Schwarzschild and Reissner-Nordström black holes, highlighting the distinctive features introduced by nonlinear electrodynamics. Furthermore, we examine the strong deflection limit for light trajectories near these black holes, focusing on the roles of both the magnetic charge $Q$ and the nonlinear parameter $ξ$. Our analysis reveals that the combined effects of $Q$ and $ξ$ enhance the strong deflection angle, resulting in a more pronounced lensing effect than that predicted by the classical Reissner-Nordström solution. These findings suggest that the nonlinear interactions may provide a potential observational signature for identifying NED black holes.

gr-qc↗

arXiv2025

Testing black holes with cosmological constant in Einstein-bumblebee gravity through the black hole shadow using EHT data and deflection angle

This study probes spacetime solutions within Einstein-Bumblebee gravity, a modified gravitational framework incorporating spontaneous Lorentz symmetry violation through a vector field mechanism. By introducing a cosmological constant into this model, the research scrutinizes thermodynamic properties of black holes in both anti-de Sitter (AdS) and de Sitter (dS) geometries. The investigation demonstrates how Lorentz-violating parameters alter foundational thermodynamic principles, including revisions to the first law of black hole mechanics and shifts in critical phenomena during phase transitions. Notably, the bumblebee coupling parameter emerges as a critical factor governing horizon structure and thermal emission characteristics, with pronounced deviations from general relativity (GR) predictions observed as this parameter increases. The analysis extends to observational signatures by calculating shadow profiles of these modified black holes. Shadow morphology exhibits dual dependence on the cosmological constant and the bumblebee parameter, presenting measurable discrepancies from classical relativity that could be constrained through Event Horizon Telescope (EHT) observational data. Furthermore, using geometric formalisms, the study quantifies light deflection phenomena in weak and strong gravitational regimes. Results reveal that both the cosmological constant and Lorentz-violating parameter induce detectable modifications to lensing angles compared to Schwarzschild or Kerr benchmarks. These deviations, while subtle, underscore the necessity for next-generation astronomical instruments capable of resolving fine-scale spacetime curvature effects.

gr-qc↗

arXiv2025

Analyzing the effect of higher dimensions on the black hole silhouette, deflection angles, and PINN approximated quasinormal modes

This study investigates the effects of higher dimensions on the observable properties of Schwarzschild-Tangherlini black holes, focusing on the photonsphere, shadow radius, deflection angles, and quasinormal modes (QNMs). By extending classical methods with Physics-Informed Neural Networks (PINNs), the research examines how increasing dimensionality alters these properties, causing shadow size reduction, weaker deflection angles, and shifts in QNM frequencies. The findings suggest that as black holes increase in dimensionality, their gravitational influence diminishes, particularly affecting light deflection and the stability of photon orbits. Through both weak and strong deflection analyses, this study indicates the need for ultrasensitive technology to detect these higher-dimensional signatures. Remarks on the observational data constraints currently favor four-dimensional spacetime; however, the exploration of additional dimensions remains vital in advancing models of quantum gravity. This work provides a theoretical framework for understanding black hole behavior in higher dimensions, potentially informing future astrophysical observations.

gr-qc↗

arXiv2025

Spacetime-curvature induced uncertainty principle: linking the large-structure global effects to the local black hole physics

This paper links the advanced formulation of the Generalized Uncertainty Principle, termed the Asymptotic Generalized Extended Uncertainty Principle (AGEUP), to the corpuscular framework to derive the AGEUP-inspired black hole metric. The former incorporates spacetime curvature effects to explore black hole dynamics under quantum gravitational corrections, while the latter is a view that black holes are Bose-Einstein condensates of weakly interacting gravitons. In a particular case, the phenomenological union between the AGEUP with cosmological constant $Λ$ to the corpuscular framework enabled a black hole metric that has a scaled mass, which depends on $Λ$ and the Planck length $l_{\rm Pl}$. Interesting implications occur, such as the maximum limit for mass $M$ where $Λ$ ceases to influence the black hole. Another is the derived value of the modulation factor of the EUP term, $α$, if the large-scale fundamental length is defined solely as the cosmological horizon. Additional analyses were done through the shadow and deflection angle phenomena, deriving constraints on the quantum gravity modulation parameter $β$. Constraints from the Event Horizon Telescope (EHT) and Very Long Baseline Interferometry (VLBI) are discussed as avenues for verifying AGEUP-related deviations in black hole shadow radius and deflection angles, offering potential observational evidence of quantum gravitational effects at astrophysical scales. The findings suggest that AGEUP could be instrumental in providing hints on the quantum gravity nature of black holes, particularly in high-energy astrophysical contexts. By linking local black hole physics with large-scale curvature effects, AGEUP paves the way for further research at the intersection of quantum gravity and cosmology, with implications for observational astrophysics and the fundamental structure of spacetime.

gr-qc↗

arXiv2023

Generalized Extended Uncertainty Principle Black Holes: Shadow and lensing in the macro- and microscopic realms

Motivated by the recent work about the Extended Uncertainty Principle (EUP) black holes \cite{Mureika:2018gxl}, we present in this study its extension called the Generalized Extended Uncertainty Principle (GEUP) black holes. In particular, we investigated the GEUP effects on astrophysical and micro-black holes. First, we derive the expression for the shadow radius to investigate its behavior as perceived by a static observer located near and far from the black hole. Constraints to the large fundamental length scale $L*$ up to $2σ$ level were also found using the EHT data: for Sgr. A*, $L* = 5.716\text{x}10^{10}$ m, while for M87*, $L* = 3.264\text{x}10^{13}$ m. Under the GEUP effect, the value of the shadow radius behaves the same way as the Schwarzschild case due to a static observer, and the effect only emerges if the mass of the black hole $M$ is around the order of magnitude of $L_*$ (or $l_\text{Pl}$). In addition, the GEUP effect increases the shadow radius for astrophysical black holes, but the reverse happens for micro-black holes. We also explored GEUP effects to the weak and strong deflection angles as an alternative analysis. For both realms, a time-like particle gives a higher value for the weak deflection angle. Similar to the shadow, the deviation is seen when the values of $L_*$ and $M$ are close. The strong deflection angle gives more sensitivity to GEUP deviation at smaller masses in the astrophysical scenario. However, the weak deflection angle is a better probe in the micro world.

gr-qc↗