Observable Optical Signatures, Particle Dynamics and Epicyclic Frequencies of Mod(A)Max Black Holes

TL;DR

This work investigates observable signatures and orbital dynamics around Mod(A)Max black holes, a model combining ModMax nonlinear electrodynamics with phantom anti-Maxwell fields in general relativity. The authors derive the spherically symmetric solution, analyze horizons and thermodynamics, and study null geodesics to determine photon spheres, shadow radii, and light deflection, highlighting how the parameters $M$, $Q$, $\gamma$, and $\eta$ shape these observables. They then examine neutral test-particle dynamics, deriving the effective potential, circular-orbit energies and angular momenta, and computing the ISCO via a cubic equation whose solution depends on the same parameters. Finally, they compute epicyclic frequencies and periastron precession to connect to QPO phenomenology, showing distinct ModMax versus Mod(A)Max predictions. These results provide a framework to constrain ModMax-type theories with black hole shadow measurements (e.g., EHT) and QPO observations in accreting systems, enabling tests of nonlinear electrodynamics in strong gravity.

Abstract

In this work, we investigate the observable optical signatures of the Mod(A)Max black hole spacetime. We analyze key optical features, including the photon sphere, black hole shadow, and photon trajectories, and examine how these observables depend on the underlying geometric parameters, such as the electric charge and the Mod(A)Max coupling parameter. We further study the dynamics of neutral test particles in the vicinity of the black hole by deriving the effective potential within the Hamiltonian formalism. Using this potential, we obtain the specific energy and specific angular momentum for test particles on circular orbits of fixed radius, as well as the innermost stable circular orbit (ISCO), and explore how the geometric parameters influence these quantities and the ISCO radius. Finally, we derive the epicyclic (azimuthal, radial, and vertical) frequencies to analyze quasi-periodic oscillations (QPOs) exploring how the geometric parameters influences these and discuss their physical implications.

Figures (13)

• Figure 1: Behavior of the metric function $f(r)$ for a Mod(A)Max black hole as a function of the radial coordinate $r$. Figure (a) illustrates the ModMax case ($\eta = +1$), while Fig. (b) shows the Mod(A)Max case ($\eta = -1$). Different curves correspond to different values of the ModMax parameter $\gamma$. The horizontal dashed gray line indicates $f(r) = 0$, where the event horizons are located. Here $M = 1$ and $Q = 1$.
• Figure 2: Behavior of the Hawking temperature for Mod(A)Max black hole by varying the electric charge $Q$, while keeping ModMax's parameter fixed $\gamma=0.5$.
• Figure 3: Behavior of the specific heat capacity for ModMax black hole by varying the electric charge $Q$, while keeping the ModMax's parameter $\eta=+1$.
• Figure 4: Effective potential $V_{\rm eff}$ for photons as a function of the radial coordinate $r/M$ in the Mod(A)Max black hole spacetime. Figures (a) and (b) show the variation with electric charge $Q$ for fixed $\gamma = 0.5$, while Figs. (c) and (d) display the variation with the ModMax parameter $\gamma$ for fixed $Q = 0.7$. The left column corresponds to the ModMax case ($\eta = +1$) and the right column to the Mod(A)Max case ($\eta = -1$). In all panels, $M = 1$ and $L = 1$.
• Figure 5: Behavior of the photon sphere radius $r_s$ for a Mod(A)Max black hole as a function of the charge parameter $Q$. Panel (a) illustrates the ModMax case ($\eta = +1$), while panel (b) shows the Mod(A)Max case ($\eta = -1$). Different curves correspond to different values of the ModMax parameter $\gamma$. Here $M = 1$.