Geodesic Structure, Thermodynamics and Scalar Perturbations of Mod(A)Max black hole Surrounded by Perfect Fluid Dark Matter

TL;DR

This work analyzes a spherically symmetric Mod(A)Max black hole embedded in perfect fluid dark matter (PFDM), characterized by charge $Q$, ModMax nonlinearity $\gamma$, PFDM strength $\lambda$, and the branch indicator $\eta=\pm1$. It develops and analyzes null and timelike geodesics to extract the photon sphere, shadow, ISCO, and effective potentials, and couples these optical diagnostics to the black hole's thermodynamics, including an extended Smarr relation and topological phase structure. Scalar perturbations are treated via the massless Klein–Gordon equation, and the QNM spectrum is connected to shadow properties through the eikonal QNM–shadow correspondence, with explicit dependence on PFDM and ModMax parameters. The study demonstrates how PFDM and ModMax/Mod(A)Max deformations imprint observable changes on shadows, ringdowns, and thermodynamic stability, offering a unified framework for multi-messenger tests of gravity and dark-sector physics near compact objects.

Abstract

In this work, we investigate the optical properties of a spherically symmetric Mod(A)Max black hole surrounded by perfect fluid dark matter, focusing on key features such as the photon sphere radius, shadow, photon trajectories, and the effective radial force experienced by photons. We also study the dynamics of massive particles around the black hole, deriving the effective potential and, from it, the specific energy and angular momentum of particles moving in circular orbits of fixed radii is discussed. The conditions for marginally stable circular orbits are analyzed, highlighting how the geometric parameters that modify the spacetime curvature influence both the optical and dynamical features. Furthermore, we explore the thermodynamic behavior of the black hole by examining its temperature, Gibbs free energy, and heat capacity, as well as its thermodynamic topology. Finally, scalar field perturbations are considered through the massless Klein-Gordon equation, and the quasinormal modes (QNMs) in the eikonal regime are computed, illustrating how the geometric parameters affect the potential and the QNM spectra.

Figures (23)

• Figure 1: The lapse function $f(r)$ as a function of the radial coordinate $r/M$ for the Mod(A)Max black hole surrounded by perfect fluid dark matter. Panel (a) shows the effect of varying the PFDM parameter $\lambda$ while keeping $Q = 0.5M$ and $\gamma = 0.3$ fixed. Negative values of $\lambda$ increase the horizon radius, while positive values decrease it. Panel (b) illustrates the influence of the electric charge $Q$ on the metric function, demonstrating how larger charges lead to the emergence of inner and outer horizons characteristic of Reissner-Nordström-like solutions. Panel (c) compares the ModMax ($\eta = +1$) and phantom ModMax ($\eta = -1$) cases, revealing that the phantom configuration produces a larger event horizon due to the sign reversal in the electromagnetic contribution. The horizontal dashed line at $f(r) = 0$ marks the location of the event horizons.
• Figure 2: The effective potential $V_{\rm eff}$ for photon motion as a function of the radial coordinate $r/M$ in the Mod(A)Max black hole spacetime surrounded by perfect fluid dark matter. The angular momentum is set to $\mathrm{L} = 5M$. Panel (a) demonstrates the influence of the PFDM parameter $\lambda$: negative values of $\lambda$ increase the peak height and shift it outward, while positive values suppress the potential barrier. Panel (b) shows the effect of the electric charge $Q$, where increasing charge raises the potential barrier, indicating stronger photon deflection. Panel (c) illustrates how the ModMax parameter $\gamma$ modifies the potential; larger values of $\gamma$ reduce the electromagnetic contribution through the $e^{-\gamma}$ factor, lowering the potential peak. The maximum of $V_{\rm eff}$ corresponds to the unstable photon sphere, whose location and height directly influence the black hole shadow size.
• Figure 3: The shadow radius $R_{\rm sh}$ as a function of the model parameters for the Mod(A)Max black hole surrounded by perfect fluid dark matter. Panel (a) shows $R_{\rm sh}$ versus the PFDM parameter $\lambda$, revealing a monotonically increasing relationship: more negative (positive) values of $\lambda$ lead to larger (smaller) shadow radii. Panel (b) displays the shadow radius as a function of the electric charge $Q/M$, demonstrating that the shadow size decreases with increasing charge due to the enhanced electromagnetic repulsion that pushes the photon sphere inward. Panel (c) illustrates the dependence on the ModMax parameter $\gamma$; as $\gamma$ increases, the electromagnetic contribution is suppressed by the factor $e^{-\gamma}$, causing the shadow radius to approach that of a Schwarzschild black hole with PFDM. These results provide testable predictions for comparing with EHT observations of supermassive black hole shadows.
• Figure 4: Photon-sphere radius $r_s$ (in units of $M$) as a function of the deformation parameter $\lambda$ and of the model couplings, for both branches $\eta=\pm1$. Panel (a) shows $r_s/M$ versus $\lambda$ at fixed $(Q,\gamma)$; panel (b) shows $r_s/M$ versus the charge-to-mass ratio $Q/M$ at fixed $(\lambda,\gamma)$; panel (c) shows $r_s/M$ versus $\gamma$ at fixed $(Q,\lambda)$. In all cases, the photon-sphere radius is obtained from the null-circular-orbit condition $2f(r_s)=r_s f'(r_s)$, selecting the physically relevant solution outside the event horizon.
• Figure 5: Complete photon trajectories around the Mod(A)Max black hole surrounded by perfect fluid dark matter, with parameters $Q = 0.5M$, $\gamma = 0.3$, $\lambda = 0.1$, and $\eta = +1$. The black filled circle represents the event horizon at $r_h = 1.61M$, and the green dashed circle marks the photon sphere at $r_s = 2.42M$. Red and orange curves show captured photons with impact parameters $b < b_c$ that spiral into the black hole. The yellow and gold curves represent near-critical orbits ($b \approx b_c = 4.17M$) that execute multiple revolutions around the photon sphere before either falling in or escaping. Blue curves depict scattered photons with $b > b_c$ that are deflected by the gravitational field but escape to infinity. The bright green curve on the photon sphere illustrates the unstable circular orbit where photons can theoretically orbit indefinitely. Light-colored trajectories show additional orbits from different incident angles, demonstrating the azimuthal symmetry of the scattering process.