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

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.

💡 Research Summary

The paper presents a new static, spherically symmetric black‑hole solution that simultaneously incorporates three well‑motivated extensions of General Relativity: (i) ModMax nonlinear electrodynamics characterized by a dimensionless screening parameter γ, (ii) Kalb‑Ramond (KR) gravity with a Lorentz‑violating background encoded in a dimensionless parameter ℓ, and (iii) a cloud of strings described by the Letelier parameter α. The resulting metric function is

f(r)=1−α/(1−ℓ)−2M/r+e^{−γ}(Q_e^2+Q_m^2)/(1−ℓ)^2 r^2,

where M is the black‑hole mass, Q_e and Q_m are electric and magnetic charges, respectively. The constant term (1−α)/(1−ℓ) replaces the usual asymptotic value of unity, producing a global conical deficit that breaks asymptotic flatness. The charge term is exponentially screened by γ (the ModMax effect) and rescaled by (1−ℓ)^{−2} (the KR effect). Setting α=ℓ=γ=0 recovers the standard Reissner‑Nordström solution.

Photon dynamics and shadow – Using the conserved energy E and angular momentum L, the effective potential for null geodesics is V_eff=E^2−L^2 f(r)/r^2. The photon sphere radius r_s satisfies the extremum condition ∂_r