Dyonic ModMax Black Holes in Kalb-Ramond gravity with a Cloud of Strings as Source

Abstract

We investigate the geodesic structure, shadow, thermodynamics, and Hawking radiation from a dyonic ModMax black hole in Kalb-Ramond gravity with a cloud of strings. The combined presence of ModMax nonlinear electrodynamics, the Lorentz-violating Kalb-Ramond background, and the string cloud breaks asymptotic flatness and introduces a global conical deficit that modifies all observables through a single geometric prefactor. We derive analytic expressions for the photon sphere, critical impact parameter, and shadow radius, and show that the shadow size depends on both the non-flat asymptotics and the ModMax screening of the dyonic charge. For massive test particles, we determine the innermost stable circular orbit and the accretion efficiency as functions of all model parameters. We also establish the first law of black hole thermodynamics and the generalized Smarr relation for this solution, identify a Hawking-Page-type phase transition in the specific heat, and compute the spectral energy emission rate, which we show is directly governed by the shadow radius in the geometric-optics limit. Our results demonstrate that the interplay of these three ingredients produces a phenomenology observationally distinguishable from standard Reissner-Nordström black holes.

Figures (12)

• Figure 1: Lapse function $f(r)$ as a function of the radial coordinate $r/M$, for $M=1$. Each panel varies one parameter over 12 equally spaced values (color-coded from dark to light via a continuous colormap) while the remaining parameters are fixed at the reference set $\alpha=0.15$, $\ell=0.10$, $Q_e=0.30$, $Q_m=0.20$, $\gamma=0.10$. (a)$\alpha \in [0.00,\,0.55]$ with $\ell=0.10$, $Q_e=0.30$, $Q_m=0.20$, $\gamma=0.10$. (b)$\ell \in [0.00,\,0.55]$ with $\alpha=0.15$, $Q_e=0.30$, $Q_m=0.20$, $\gamma=0.10$. (c)$Q_e=Q_m \in [0.00,\,0.55]$ with $\alpha=0.15$, $\ell=0.10$, $\gamma=0.10$. (d)$\gamma \in [0.00,\,2.00]$ with $\alpha=0.15$, $\ell=0.10$, $Q_e=0.30$, $Q_m=0.20$. The dashed horizontal line marks $f(r)=0$.
• Figure 2: Photon effective potential $V_{\rm eff}$ as a function of $r/M$ for $M=1$ and $E=1$. Each panel shows 12 curves varying one parameter continuously (colormap from dark to light). (a)$L \in [2.0,\,7.0]\,M$ with $\alpha=0.15$, $\ell=0.10$, $Q_e=0.30$, $Q_m=0.20$, $\gamma=0.10$. (b)$\alpha \in [0.00,\,0.50]$ with $L=4\,M$, $\ell=0.10$, $Q_e=0.30$, $Q_m=0.20$, $\gamma=0.10$. (c)$Q_e=Q_m \in [0.05,\,0.55]$ with $L=4\,M$, $\alpha=0.15$, $\ell=0.10$, $\gamma=0.10$. (d)$\gamma \in [0.00,\,2.00]$ with $L=4\,M$, $\alpha=0.15$, $\ell=0.10$, $Q_e=0.30$, $Q_m=0.20$. The dashed line marks $V_{\rm eff}=0$.
• Figure 3: Photon sphere radius $r_s/M$ as a function of each free parameter, for $M=1$. The remaining parameters are fixed at $\alpha=0.15$, $\ell=0.10$, $Q_e=0.30$, $Q_m=0.20$, $\gamma=0.10$ in each panel. (a)$r_s$ vs. $\alpha \in [0.00,\,0.50]$. (b)$r_s$ vs. $\ell \in [0.00,\,0.50]$. (c)$r_s$ vs. $Q_e=Q_m \in [0.01,\,0.55]$. (d)$r_s$ vs. $\gamma \in [0.00,\,2.50]$.
• Figure 4: Critical impact parameter $\beta_c/M$ as a function of each free parameter, for $M=1$. The remaining parameters are fixed at $\alpha=0.15$, $\ell=0.10$, $Q_e=0.30$, $Q_m=0.20$, $\gamma=0.10$ in each panel. (a)$\beta_c$ vs. $\alpha \in [0.00,\,0.50]$. (b)$\beta_c$ vs. $\ell \in [0.00,\,0.50]$. (c)$\beta_c$ vs. $Q_e=Q_m \in [0.01,\,0.55]$. (d)$\beta_c$ vs. $\gamma \in [0.00,\,2.50]$.
• Figure 5: Black hole shadow radius $R_{\rm sh}/M$ for $M=1$. (a)$R_{\rm sh}$ as a function of the observer distance $r_O\in[10,\,200]\,M$ for 12 values of $Q_e=Q_m\in[0.05,\,0.55]$ (colormap), with $\alpha=0.15$, $\ell=0.10$, $\gamma=0.10$. (b)$R_{\rm sh}$ for a distant observer vs. $\ell\in[0.00,\,0.50]$, with $\alpha=0.15$, $Q_e=0.30$, $Q_m=0.20$, $\gamma=0.10$. (c)$R_{\rm sh}$ vs. $Q_e=Q_m\in[0.01,\,0.55]$, with $\alpha=0.15$, $\ell=0.10$, $\gamma=0.10$. (d)$R_{\rm sh}$ vs. $\gamma\in[0.00,\,2.50]$, with $\alpha=0.15$, $\ell=0.10$, $Q_e=0.30$, $Q_m=0.20$.