Branch structure and nonextensive thermodynamics of Kalb-Ramond-ModMax black holes: observational signatures

TL;DR

This work analyzes a charged black hole in Einstein gravity with Kalb-Ramond Lorentz-symmetry breaking and ModMax nonlinear electrodynamics, uncovering a physically meaningful branch structure governed by $\zeta=\pm1$ that yields ordinary RN-like and phantom single-horizon spacetimes. The authors derive the exact KR-ModMax solution, study Hawking temperature, and frame the thermodynamics in Tsallis non-extensive form, including a Joule–Thomson expansion that reveals branch-dependent heating/cooling behavior. They map optical signatures using a Gauss–Bonnet approach to weak lensing, analyze photon propagation in both homogeneous and inhomogeneous plasmas, and examine tidal forces, uncovering tidal inversion only in the ordinary branch. The results show distinct, observationally testable differences between branches across thermodynamics, lensing, plasma effects, and tides, highlighting how KR and ModMax deformations jointly shape strong-field phenomenology and offering pathways to constrain Lorentz-violating and nonlinear electrodynamics in gravity.

Abstract

We investigate a static, spherically symmetric black hole arising in Einstein gravity coupled to a Kalb-Ramond field and ModMax nonlinear electrodynamics, both of which are independently well motivated extensions of standard electrovacuum gravity. The solution depends, beyond mass and charge, on a Lorentz-violating parameter, a ModMax deformation parameter, and a discrete branch selector $ζ=\pm1$. We show that the ordinary branch admits extremal and non-extremal configurations, while the phantom branch generically supports a single-horizon geometry. Black-hole thermodynamics is analyzed within the Tsallis non-extensive framework, revealing branch-dependent stability and Joule-Thomson behavior. Weak gravitational lensing, photon propagation in plasma, and tidal forces are then studied, revealing clear optical and strong-field signatures that distinguish the two branches. In particular, the ordinary branch exhibits finite-radius tidal inversion, absent in the phantom sector. Our results demonstrate that the combined Kalb-Ramond and ModMax effects lead to a rich and observationally distinguishable black-hole phenomenology.

Figures (12)

• Figure 1: Metric function $f(r)$ for KR-ModMax black holes in the ordinary ($\zeta=+1$) and phantom ($\zeta=-1$) branches. The horizontal dashed line $f(r)=0$ marks the locations of event horizons. Panel (a) illustrates the charge-driven transition from non-extremal to extremal and naked configurations in the ordinary branch. Panel (b) demonstrates the absence of extremal or naked solutions in the phantom sector. Panels (c) and (d) display the sensitivity of the geometry to the Kalb-Ramond parameter $\ell$ and the ModMax parameter $\gamma$, respectively.
• Figure 2: Hawking temperature $T_H$ as a function of the horizon radius $r_h$ for KR-ModMax black holes with fixed parameters $M=1.0$, $Q=0.5$, and $\ell=0.5$. The color scale denotes the ModMax parameter $\gamma \in [0,3]$. The ordinary branch exhibits a $\gamma$-dependent approach to extremality, whereas the phantom branch maintains strictly positive temperature throughout the entire parameter space.
• Figure 3: Tsallis internal energy $E_T$ as a function of the horizon radius $r_h$ for KR-ModMax black holes with fixed parameters $M=1.0$, $Q=0.5$, $\ell=0.5$, and $\gamma=2.0$. The color scale denotes the non-extensivity parameter $\delta \in [1,2]$. Across the entire parameter space, the phantom branch maintains higher internal energy, establishing a clear thermodynamic hierarchy between the two ModMax sectors.
• Figure 4: Tsallis Helmholtz free energy $F_T$ as a function of the horizon radius $r_h$ for KR-ModMax black holes with the same parameters as Fig. \ref{['fig:e_zeta_comparison']}. While both branches show monotonic decay with increasing $r_h$, the phantom branch maintains a larger free-energy magnitude throughout, reinforcing the thermodynamic hierarchy inferred from the internal energy and anticipating the branch-dependent stability behavior discussed in subsequent sections.
• Figure 5: Tsallis thermodynamic pressure $P_T$ as a function of the horizon radius $r_h$ for KR-ModMax black holes. The negative pressure indicates that, in this asymptotically flat setup, the effective cosmological contribution acts as a tension rather than a true pressure. Although both branches converge at large $r_h$ where the mass term dominates, their distinct radial behavior at small $r_h$ plays a crucial role in shaping the Joule-Thomson coefficient and the cooling-heating structure discussed in Sec. \ref{['isec4']}.